7 Placeholder Substructures III: A
Bit-String-Driven “Recipe Theory” for
Infinite-Dimensional Zero-Divisor Spaces
Robert P. C. de Marrais ∗
Thothic Technology Partners, P.O.Box 3083, Plymouth MA 02361
October 29, 2018
Abstract
Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-
dimensional hypercomplex numbers (N a power of 2, and at least 4) can
represent singularities and, as N → ∞, fractals – and thereby, scale-free net-
works. Any integer > 8 and not a power of 2 generates a meta-fractal or
Sky when it is interpreted as the strut constant (S) of an ensemble of octahe-
dral vertex figures called Box-Kites (the fundamental ZD building blocks).
Remarkably simple bit-manipulation rules or recipes provide tools for trans-
forming one fractal genus into others within the context of Wolfram’s Class
4 complexity.
1 The Argument So Far
In Parts I[1] and II[2], the basic facts concerning zero-divisors (ZDs) as they arise
in the hypercomplex context were presented and proved. “Basic,” in the context of
this monograph, means seven things. First, they emerged as a side-effect of apply-
ing CDP a minimum of 4 times to the Real Number Line, doubling dimension to
the Complex Plane, Quaternion 4-Space, Octonion 8-Space, and 16-D Sedenions.
With each such doubling, new properties were found: as the price of sacrificing
∗Email address: rdemarrais@alum.mit.edu
http://arxiv.org/abs/0704.0112v3
counting order, the Imaginaries made a general theory of equations and solution-
spaces possible; the non-commutative nature of Quaternions mapped onto the re-
alities of the manner in which forces deploy in the real world, and led to vector
calculus; the non-associative nature of Octonions, meanwhile, has only come into
its own with the need for necessarily unobservable quantities (because of confor-
mal field-theoretical constraints)in String Theory. In the Sedenions, however, the
most basic assumptions of all – well-defined notions of field and algebraic norm
(and, therefore, measurement) – break down, as the phenomena correlated with
their absence, zero-divisors, appear onstage (never to leave it for all higher CDP
dimension-doublings).
Second thing: ZDs require at least two differently-indexed imaginary units
to be defined, the index being an integer larger than 0 (the CDP index of the
Real Unit) and less than 2N for a given CDP-generated collection of 2N-ions. In
“pure CDP,” the enormous number of alternative labeling schemes possible in
any given 2N-ion level are drastically reduced by assuming that units with such
indices interact by XOR-ing: the index of the product of any two is the XOR
of their indices. Signing is more tricky; but, when CDP is reduced to a 2-rule
construction kit, it becomes easy: for index u < G, G the Generator of the 2N-ions
(i.e., the power of 2 immediately larger than the highest index of the predecessor
2N−1-ions), Rule 1 says iu · iG = +i(u+G). Rule 2 says take an associative triplet
(a,b,c), assumed written in CPO (short for “cyclically positive order”: to wit,
a · b = +c, b · c = +a, and c · a = +b). Consider, for instance, any (u,G,G+ u)
index set. Then three more such associative triplets (henceforth, trips) can be
generated by adding G to two of the three, then switching their resultants’ places
in the CPO scheme. Hence, starting with the Quaternions’ (1,2,3) (which we’ll
call a Rule 0 trip, as it’s inherited from a prior level of CDP induction), Rule
1 gives us the trips (1,4,5), (2,4,6), and (3,4,7), while Rule 2 yields up the
other 4 trips defining the Octonions: (1,7,6), (2,5,7), and (3,6,5). Any ZD
in a given level of 2N-ions will then have units with one index < G, written in
lowercase, and the other index > G, written in uppercase. Such pairs, alternately
called “dyads” or “Assessors,” saturate the diagonal lines of their planes, which
diagonals never mutually zero-divide each other (or make DMZs, for ”divisors (or
dyads) making zero”), but only make DMZs with other such diagonals, in other
such Assessors. (This is, of course, the opposite situation from the projection
operators of quantum mechanics, which are diagonals in the planes formed by
Reals and dimensions spanned by Pauli spin operators contained within the 4-
space created by the Cartesian product of two standard imaginaries.)
Third thing: Such ZDs are not the only possible in CDP spaces; but they define
the “primitive” variety from which ZD spaces saturating more than 1-D regions
can be articulated. A not quite complete catalog of these can be found in our
first monograph on the theme [3]; a critical kind which was overlooked there,
involving the Reals (and hence, providing the backdrop from which to see the
projection-operator kind as a degenerate type), were first discussed more recently
[4]. (Ironically, these latter are the easiest sorts of composites to derive of any:
place the two diagonals of a DMZ pairing with differing internal signing on axes
of the same plane, and consider the diagonals they make with each other!) All the
primitive ZDs in the Sedenions can be collected on the vertices of one of 7 copies
of an Octahedron in the Box-Kite representation, each of whose 12 edges indicates
a two-way “DMZ pathway,” evenly divided between 2 varieties. For any vertex V,
and k any real scalar, indicate the diagonals this way: (V,/) = k · (iv+ iV ), while
(V, \)= k · (iv− iV ). 6 edges on a Box-Kite will always have negative edge-sign
(with unmarked ET cell entries: see the “sixth thing”). For vertices M and N,
exactly two DMZs run along the edge joining them, written thus:
(M,/) ·(N, \) = (M, \) · (N,/) = 0
The other 6 all have positive edge-sign, the diagonals of their two DMZs hav-
ing same slope (and marked – with leading dashes – ET cell entries):
(Z,/) ·(V, /)= (Z, \) ·(V,\)= 0
Fourth thing: The edges always cluster similarly, with two opposite faces
among the 8 triangles on the Box-Kite being spanned by 3 negative edges (con-
ventionally painted red in color renderings), with all other edges being positive
(painted blue). One of the red triangles has its vertices’ 3 low-index units forming
a trip; writing their vertex labels conventionally as A, B, C, we find there are in
fact always 4 such trips cycling among them: (a,b,c), the L-trip; and the three
U-trips obtained by replacing all but one of the lowercase labels in the L-trip with
uppercase: (a,B,C); (A,b,C); (A,B,c). Such a 4-trip structure is called a Sail, and
a Box-Kite has 4 of them: the Zigzag, with all negative edges, and the 3 Trefoils,
each containing two positive edges extending from one of the Zigzag vertices to
the two vertices opposite its Sailing partners. These opposite vertices are always
joined by one of the 3 negative edges comprising the Vent which is the Zigzag’s
opposite face. Again by convention, the vertices opposite A, B, C are written F,
E, D in that order; hence, the Trefoil Sails are written (A,D,E); (F,D,B), and
(F,C,E), ordered so that their lowercase renderings are equivalent to their CPO
L-trips. The graphical convention is to show the Sails as filled in, while the other 4
faces, like the Vent, are left empty: they show “where the wind blows” that keeps
the Box-Kite aloft. A real-world Box-Kite, meanwhile, would be held together
by 3 dowels (of wood or plastic, say) spanning the joins between the only vertices
left unconnected in our Octahedral rendering: the Struts linking the strut-opposite
vertices (A, F); (B, E); (C, D).
Fifth thing: In the Sedenions, the 7 isomorphic Box-Kites are differentiated by
which Octonion index is missing from the vertices, and this index is designated
by the letter S, for “signature,” “suppressed index,” or strut constant. This last
designation derives from the invariant relationship obtaining in a given Box-Kite
between S and the indices in the Vent and Zigzag termini (V and Z respectively)
of any of the 3 Struts, which we call the “First Vizier” or VZ1. This is one of
3 rules, involving the three Sedenion indices always missing from a Box-Kite’s
vertices: G, S, and their simple sum X (which is also their XOR product, since G
is always to the left of the left-most bit in S). The Second Vizier tells us that the
L-index of either terminus with the U-index of the other always form a trip with
G, and it true as written for all 2N-ions. The Third shows the relationship between
the L- and U- indices of a given Assessor, which always form a trip with X. Like
the First, it is true as written only in the Sedenions, but as an unsigned statement
about indices only, it is true universally. (For that reason, references to VZ1 and
VZ3 hereinout will be assumed to refer to the unsigned versions.) First derived in
the last section of Part I, reprised in the intro of Part II, we write them out now for
the third and final time in this monograph:
VZ1: v · z =V ·Z = S
VZ2: Z · v =V · z = G
VZ3: V · v = z ·Z = X.
Rules 1 and 2, the Three Viziers, plus the standard Octonion labeling scheme
derived from the simplest finite projective group, usually written as PSL(2,7), pro-
vide the basis of our toolkit. This last becomes powerful due to its capacity for
recursive re-use at all levels of CDP generation, not just the Octonions. The sim-
plest way to see this comes from placing the unique Rule 0 trip provided by the
Quaternions on the circle joining the 3 sides’ midpoints, with the Octonion Gen-
erator’s index, 4, being placed in the center. Then the 3 lines leading from the
Rule 0 trip’s (1, 2, 3) midpoints to their opposite angles – placed conventionally
in clockwise order in the midpoints of the left, right, and bottom sides of a triangle
whose apex is at 12 o’clock – are CPO trips forming the Struts, while the 3 sides
themselves are the Rule 2 trips. These 3 form the L-index sets of the Trefoil Sails,
while the Rule 0 trip provides the same service for the Zigzag. By a process analo-
gized to tugging on a slipcover (Part I) and pushing things into the central zone
of hot oil while wok-cooking (Part II), all 7 possible values of S in the Sedenions,
not just the 4, can be moved into the center while keeping orientations along all
7 lines of the Triangle unchanged. Part II’s critical Roundabout Theorem tells us,
moreover, that all 2N-ion ZDs, for all N > 3, are contained in Box-Kites as their
minimal ensemble size. Hence, by placing the appropriate G, S, or X in the center
of a PSL(2,7) triangle, with a suitable Rule 0 trip’s indices populating the circle,
any and all candidate primitive ZDs can be discovered and situated.
Sixth thing: The word “candidate” in the above is critical; its exploration was
the focus of Part II. For, starting with N = 5 and hence G = 16 (which is to say, in
the 32-D Pathions), whole Box-Kites can be suppressed (meaning, all 12 edges,
and not just the Struts, no longer serve as DMZ pathways). But for all N, the full
set of candidate Box-Kites are viable when S≤ 8 or equal to some higher power of
2. For all other S values, though, the phenomenon of carrybit overflow intervenes
– leading, ultimately, to the “meta-fractal” behavior claimed in our abstract. To
see this, we need another mode of representation, less tied to 3-D visualizing, than
the Box-Kite can provide. The answer is a matrix-like method of tabulating the
products of candidate ZDs with each other, called Emanation Tables or ETs. The
L-indices only of all candidate ZDs are all we need indicate (the U-indices being
forced once G is specified); these will saturate the list of allowed indices < G,
save for the value of S whose choice, along with that of G, fixes an ET. Hence,
the unique ET for given G and S will fill a square spreadsheet whose edge has
length 2N−1 −2. Moreover, a cell entry (r,c) is only filled when row and column
labels R and C form a DMZ, which can never be the case along an ET’s long
diagonals: for the diagonal starting in the upper left corner, R xor R = 0, and
the two diagonals within the same Assessor, can never zero-divide each other; for
the righthand diagonal, the convention for ordering the labels (ascending counting
order from the left and top, with any such label’s strut-opposite index immediately
being entered in the mirror-opposite positions on the right and bottom) makes R
and C strut-opposites, hence also unable to form DMZs.
For the Sedenions, we get a 6 x 6 table, 12 of whose cells (those on long
diagonals) are empty: the 24 filled cells, then, correspond to the two-way traffic
of “edge-currents” one imagines flowing between vertices on a Box-Kite’s 12
edges. A computational corollary to the Roundabout Theorem, dubbed the Trip-
Count Two-Step, is of seminal importance. It connects this most basic theorem of
ETs to the most basic fact of associative triplets, indicated in the opening pages
of Part I, namely: for any N, the number TripN of associative triplets is found,
by simple combinatorics, to be (2N −1)(2N −2)/3! – 35 for the Sedenions, 155
for the Pathions, and so on. But, by Trip-Count Two-Step, we also know that the
maximum number of Box-Kites that can fill a 2N-ion ET = TripN−2. For S a power
of 2, beginning in the Pathions (for S= 25−2 = 8), the Number Hub Theorem says
the upper left quadrant of the ET is an unsigned multiplication table of the 2N−2-
ions in question, with the 0’s of the long diagonal (indicated Real negative units)
replaced by blanks – a result effectively synonymous with the Trip-Count Two-
Step.
Seventh thing: We found, as Part II’s argument wound down, that the 2 classes
of ETs found in the Pathions – the “normal” for S ≤ 8, filled with indices for all
7 possible Box-Kites, and the “sparse” so-called Sand Mandalas, showing only
3 Box-Kites when 8 < S < 16, were just the beginning of the story. A simple
formula involving just the bit-string of s and g, where the lowercase indicates the
values of S and G modulo G/2, gave the prototype of our first recipe: all and
only cells with labels R or C, or content P ( = R xor C ), are filled in the ET. The
4 “missing Box-Kites” were those whose L-index trip would have been that of a
Sail in the 2N−1 realm with S = s and G = g. The sequence of 7 ETs, viewed in
S-increasing succession, had an obvious visual logic leading to their being dubbed
a flip-book. These 7 were obviously indistinguishable from many vantages, hence
formed a spectrographic band. There were 3 distinct such bands, though, each
typified by a Box-Kite count common to all band-members, demonstrable in the
ETs for the 64-D Chingons. Each band contained S values bracketed by multiples
of 8 (either less than or equal to the higher, depending upon whether the latter
was or wasn’t a power of 2). These were claimed to underwrite behaviors in all
higher 2N-ion ETs, according to 3 rough patterns in need of algorithmic refining
in this Part III. Corresponding to the first unfilled band, with ETs always missing
4N−4 of their candidate Box-Kites for N > 4, we spoke of recursivity, meaning
the ETs for constant S and increasing N would all obey the same recipe, properly
abstracted from that just cited above, empirically found among the Pathions for
S > 8. The second and third behaviors, dubbed, for S ascending, (s,g)-modularity
and hide/fill involution respectively, make their first showings in the Chingons, in
the bands where 16 < S ≤ 24, and then where 24 < S < 32. In all such cases,
we are concerned with seeing the “period-doubling” inherent in CDP and Chaotic
attractors both become manifest in a repeated doubling of ET edge-size, leading
to the fixed-S, N increasing analog of the fixed-N,S increasing flip-books first ob-
served in the Pathions, which we call balloon-rides. Specifying and proving their
workings, and combining all 3 of the above-designated behaviors into the “funda-
mental theorem of zero-division algebra,” will be our goals in this final Part III.
Anyone who has read this far is encouraged to bring up the graphical complement
to this monograph, the 78-slide Powerpoint show presented at NKS 2006 [5], in
another window. (Slides will be referenced by number in what follows.)
2 8 < S < 16,N → ∞ : Recursive Balloon Rides in
the Whorfian Sky
We know that any ET for the 2N-ions is a square whose edge is 2N−1 − 2 cells.
How, then, can any simply recursive rule govern exporting the structure of one
such box to analogous boxes for progressively higher N? The answer: include
the label lines – not just the column and row headers running across the top and
left margins, but their strut-opposite values, placed along the bottom and right
margins, which are mirror-reversed copies of the label-lines (LLs) proper to which
they are parallel. This increases the edge-size of the ET box to 2N−1.
Theorem 11. For any fixed S > 8 and not a power of 2, the row and column indices
comprising the Label Lines (LLs) run along the left and top borders of the 2N-
ion ET ”spreadsheet” for that S. Treat them as included in the spreadsheet, as
labels, by adding a row and column to the given square of cells, of edge 2N−1−2,
which comprises the ET proper. Then add another row and column to include
the strut-opposite values of these labels’ indices in “mirror LLs,” running along
the opposite edges of a now 2N−1-edge-length box, whose four corner cells, like
the long diagonals they extend, are empty. When, for such a fixed S, the ET for
the 2N+1-ions is produced, the values of the 4 sets of LL indices, bounding the
contained 2N-ion ET, correspond, as cell values, to actual DMZ P-values in the
bigger ET, residing in the rows and columns labeled by the contained ET’s G and
X (the containing ET’s g and g+S). Moreover, all cells contained in the box they
bound in the containing ET have P-values (else blanks) exactly corresponding to –
and including edge-sign markings of – the positionally identical cells in the 2N-ion
ET: those, that is, for which the LLs act as labels.
Proof. For all strut constants of interest, S < g(= G/2); hence, all labels up
to and including that immediately adjoining its own strut constant (that is, the
first half of them) will have indices monotonically increasing, up to and at least
including the midline bound, from 1 to g − 1. When N is incremented by 1,
the row and column midlines separating adjoining strut-opposites will be cut and
pulled apart, making room for the labels for the 2N+1-ion ET for same S, which
middle range of label indices will also monotonically increase, this time from
the current 2N-ion generation’s g (and prior generation’s G), up to and at least
including its own midline bound, which will be g plus the number of cells in
the LL inherited from the prior generation, or g/2− 1. The LLs are therefore
contained in the rows and columns headed by g and its strut opposite, g+S. To
say that the immediately prior CDP generation’s ET labels are converted to the
current generation’s P-values in the just-specified rows and columns is equivalent
to asserting the truth of the following calculation:
(g+u)+(sg) · (G+g+uopp)
g + (G+g+S)
−(vz) · (G+uopp) +(vz) · (sg) ·u
+u − (sg) · (G+uopp)
0 only if vz = (−sg)
Here, we use two binary variables, the inner-sign-setting sg, and the Vent-or-
Zigzag test, based on the First Vizier. Using the two in tandem lets us handle the
normal and “Type II” box-kites in the same proof. Recall (and see Appendix B of
Part II for a quick refresher) that while the “Type I” is the only type we find in the
Sedenions, we find that a second variety emerges in the Pathions, indistinguishable
from Type I in most contexts of interest to us here: the orientation of 2 of the
3 struts will be reversed (which is why VZ1 and VZ3 are only true generally
when unsigned). For a Type I, since S < g, we know by Rule 1 that we have the
trip (S,g,g+S); hence, g – for all 2N-ions beyond the Pathions, where the Sand
Mandalas’ g = 8 is the L-index of the Zigzag B Assessor – must be a Vent (and
its strut-opposite, g+S, a Zigzag). For a Type II, however, this is necessarily so
only for 1 of the 3 struts – which means, per the equation above, that sg must be
reversed to obtain the same result. Said another way, we are free to assume either
signing of vz means +1, so the “only if” qualifying the zero result is informative.
It is u and its relationship to g+ u that is of interest here, and this formulation
makes it easier to see that the products hold for arbitrary LL indices u or their
strut-opposites. But for this, the term-by-term computations should seem routine:
the left bottom is the Rule 1 outcome of (u,g,g+ u): obviously, any u index
must be less than g. To its right, we use the trip (uopp,g,g+ uopp) → (G+ g+
uopp,g,G+uopp), whose CPO order is opposite that of the multiplication. For the
top left, we use (u,S,uopp) as limned above, then augment by g, then G, leaving
uopp unaffected in the first augmenting, and g + u in the second. Finally, the
top right (ignoring sg and vz momentarily) is obtained this way: (u,S,uopp) →
(u,g+uopp,g+S)→ (u,G+g+ s,G+g+uopp); ergo, +u.
Note that we cannot eke out any information about edge-sign marks from this
setup: since labels, as such, have no marks, we have nothing to go on – unlike all
other cells which our recursive operations will work on. Indeed, the exact algo-
rithmic determination of edge-sign marks for labels is not so trivial: as one iterates
through higher N values, some segments of LL indexing will display reversals of
marks found in the ascending or descending left midline column, while other seg-
ments will show them unchanged – with key values at the beginnings and ends
of such octaves (multiples of 8, and sums of such multiples with S mod 8) some-
times being reversed or kept the same irrespective of the behavior of the terms
they bound. Fortunately, such behaviors are of no real concern here – but they are,
nevertheless, worth pointing out, given the easy predictability of other edge-sign
marks in our recursion operations.
Now for the ET box within the labels: if all values (including edge-sign marks)
remain unchanged as we move from the 2N-ion ET to that for the 2N+1-ions, then
one of 3 situations must obtain: the inner-box cells have labels u,v which belong to
some Zigzag L-trip (u,v,w); or, on the contrary, they correspond to Vent L-indices
– the first two terms in the CPO triplet (wopp,vopp,u), for instance; else, finally,
one term is a Vent, the other a Zigzag (so that inner-signs of their multiplied dyads
are both positive): we will write them, in CPO order, vopp and u, with third trip
member wopp. Clearly, we want all the products in the containing ET to indicate
DMZs only if the inner ET’s cells do similarly. This is easily arranged: for the
containing ET’s cells have indices identical to those of the contained ET’s, save
for the appending of g to both (and ditto for the U-indices).
Case 1: If (u,v,w) form a Zigzag L-index set, then so do (g+ v,g+u,w), so
markings remain unchanged; and if the (u,v) cell entry is blank in the contained,
so will be that for (g+ u,g+ v) in its container. In other words, the following
holds:
(g+ v)+(sg) · (G+g+ vopp)
(g+u) + (G+g+uopp)
−(G+wopp) − (sg) ·w
−w − (sg) · (G+wopp)
0 only if sg = (−1)
(g+u) · (g+ v) = P : (u,v,w)→ (g+ v,g+u,w); hence, (−w).
(g+ u) · (sg) · (G+ g+ vopp) = P : (u,wopp,vopp) → (g+ vopp,wopp,g+ u)
→ (G+wopp,G+g+ vopp,g+u); hence, (sg) · (−(G+wopp)).
(G+g+uopp) · (g+ v) = P : (uopp,wopp,v)→ (g+ v,wopp,g+uopp) → (g+
v,G+g+uopp,G+wopp); hence, (−(G+wopp)).
(G+g+uopp) ·(G+g+vopp) = P : Rule 2 twice to the same two terms yields
the same result as the terms in the raw, hence (−w).
Clearly, cycling through (u,v,w) to consider (g+ v) · (g+ w) will give the
exactly analogous result, forcing two (hence three) negative inner-signs in the
candidate Sail; hence, if we have DMZs at all, we have a Zigzag Sail.
Case 2: The product of two Vents must have negative edge-sign, and there’s
no cycling through same-inner-signed products as with the Zigzag, so we’ll just
write our setup as a one-off, with upper inner-sign explicitly negative, and claim
its outcome true.
(g+ vopp)− (G+g+ v)
(g+wopp) + (G+g+w)
+(G+uopp) +u
−u − (G+uopp)
(g+wopp) · (g+ vopp) = P : (wopp,vopp,u) → (g+ vopp,g+wopp,u); hence,
(−u).
(g+wopp) · (G+g+v) = P : (wopp,v,uopp)→ (g+v,g+wopp,uopp)→ (G+
uopp,g+wopp,G+ g+ v); but inner sign of upper dyad is negative, so (−(G+
uopp)).
(G+g+w) · (g+vopp) = P : (vopp,uopp,w)→ (g+w,uopp,g+vopp)→ (G+
uopp,G+g+w,g+ vopp); hence, (+(G+uopp)).
(G+g+w) · (G+g+ v) = P : Rule 2 twice to the same two terms yields the
same result as the terms in the raw; but inner sign of upper dyad is negative, so
(+u).
Case 3: The product of Vent and Zigzag displays same inner sign in both
dyads; hence the following arithmetic holds:
(g+u)+(G+g+uopp)
(g+ vopp) + (G+g+ v)
−(G+w) +wopp
−wopp +(G+w)
The calculations are sufficiently similar to the two prior cases as to make their
writing out tedious. It is clear that, in each of our three cases, content and marking
of each cell in the contained ET and the overlapping portion of the container ET
are identical. �
To highlight the rather magical label/content involution that occurs when N is
in- or de- cremented, graphical realizations of such nested patterns, as in Slides
60-61, paint LLs (and labels proper) a sky-blue color. The bottom-most ET being
overlaid in the central box has g = the maximum high-bit in S, and is dubbed the
inner skybox. The degree of nesting is strictly measured by counting the number
of bits B that a given skybox’s g is to the left of this strut-constant high-bit. If we
partition the inner skybox into quadrants defined by the midlines, and count the
number Q of quadrant-sized boxes along one or the other long diagonal, it is obvi-
ous that the inner skybox itself has B = 0 and Q = 1; the nested skyboxes contain-
ing it have Q = 2B. If recursion of skybox nesting be continued indefinitely – to
the fractal limit, which terminology we will clarify shortly – the indices contained
in filled cells of any skybox can be interpreted in B distinct ways, B → ∞, as rep-
resentations of distinct ZDs with differing G and, therefore, differing U-indices.
By obvious analogy to the theory of Riemann surfaces in complex analysis, each
such skybox is a separate “sheet”; as with even such simple functions as the log-
arithmic, the number of such sheets is infinite. We could then think of the infinite
sequence of skyboxes as so many cross-sections, at constant distances, of a flash-
light beam whose intensity (one over the ET’s cell count) follows Kepler’s inverse
square law. Alternatively, we could ignore the sheeting and see things another
Where we called fixed-N, S varying sequences of ETs flip-books, we refer to
fixed-S, N varying sequences as balloon rides: the image is suggested by David
Niven’s role as Phineas Fogg in the movie made of Jules Vernes’ Around the World
in 80 Days: to ascend higher, David would drop a sandbag over the side of his hot-
air balloon’s basket; if coming down, he would pull a cord that released some of
the balloon’s steam. Each such navigational tactic is easy to envision as a bit-
shift, pushing G further to the left to cross LLs into a higher skybox, else moving
Figure 1: ETs for S=15, N=5,6,7 (nested skyboxes in blue)· · · and “fractal limit.”
it rightward to descend. Using S = 15 as the basis of a 3-stage balloon-ride, we
see how increasing N from 5 to 6 to 7 approaches the white-space complement
of one of the simplest (and least efficient) plane-filling fractals, the Cesàro double
sweep [6, p. 65].
The graphics were programmatically generated prior to the proving of the the-
orems we’re elaborating: their empirical evidence was what informed (indeed,
demanded) the theoretical apparatus. And we are not quite finished with the cur-
rent task the apparatus requires of us. We need two more theorems to finish the
discussion of skybox recursion. For both, suppose some skybox with B = k, k any
non-negative integer, is nested in one with B = k + 1. Divide the former along
midlines to frame its four quadrants, then block out the latter skybox into a 4×4
grid of same-sized window panes, partitioned by the one-cell-thick borders of
its own midlines into quadrants, each of which is further subdivided by the out-
side edges of the 4 one-cell-thick label lines and their extensions to the window’s
frame. These extended LLs are themselves NSLs, and have R,C values of g and
g+S; for S = 15, they also adjoin NSLs along their outer edges whose R,C val-
ues are multiples of 8 plus S mod 8. These pane-framing pairs of NSLs we will
henceforth refer to (as a windowmaker would) as muntins. It is easy to calculate
that while the inner skybox has but one muntin each among its rows and columns,
each further nesting has 2B+1 − 1. But we are getting ahead of ourselves, as we
still have two proofs to finish. Let’s begin with Four Corners, or
Theorem 12. The 4 panes in the corners of the 16-paned B = k + 1 window are
identical in contents and marks to the analogously placed quadrants of the B = k
skybox.
Proof. Invoke the Zero-Padding Lemma with regard to the U-indices, as the labels
of the boxes in the corners of the B = k+1 ET are identical to those of the same-
sized quadrants in the B = k ET, all labels ≥ the latter’s g only occurring in the
newly inserted region. �
Remarks. For N = 6, all filled Four Corners cells indicate edges belonging to
3 Box-Kites, whose edges they in fact exhaust. These 3, not surprisingly, are
the zero-padded versions of the identically L-indexed trio which span the entirety
of the N = 5 ET. By calculations we’ll see shortly, however, the inner skybox,
when considered as part of the N = 6 ET, has filled cells belonging to all the
other 16 Box-Kites, even though the contents of these cells are identical to those
in the N = 5 ET. As B increases, then, the “sheets” covering this same central
region must draw upon progressively more extensive networks of interconnected
Box-Kites. As we approach the fractal limit – and “the Sky is the limit” – these
networks hence become scale-free. (Corollarily, for N = 7, the Four Corners’ cells
exhaust all the edges of the N = 6 ET’s 19 Box-Kites, and so on.)
Unlike a standard fractal, however, such a Sky merits the prefix “meta”: for
each empty ET cell corresponds to a point in the usual fractal variety; and each
pair of filled ET cells, having (r,c) of one = (c,r) of the other), correspond to
diagonal-pairs in Assessor planes, orthogonal to all other such diagonal-pairs be-
longing to the other cells. Each empty ET cell, in other words, not only corre-
sponds to a point in the usual plane-confined fractal, but belongs to the comple-
ment of the filled cells’ infinite number of dimensions framing the Sky’s meta-
fractal.
We’ve one last thing to prove here. The French Windows Theorem shows us
the way the cell contents of the pairs of panes contained between the B = k+ 1
skybox’s corners are generated from those of the analogous pairings of quadrants
in the B = k skybox, by adding g to L-indices.
Theorem 13. For each half-square array of cells created by one or the other midline
(the French windows), each cell in the half-square parallel to that adjoining the
midline (one of the two shutters), but itself adjacent to the label-line delimiting
the former’s bounds, has content equal to g plus that of the cell on the same line
orthogonal to the midline, and at the same distance from it, as it is from the label-
line. All the empty long-diagonal cells then map to g (and are marked), or g+S
(and are unmarked). Filled cells in extensions of the label-lines bounding each
shutter are calculated similarly, but with reversed markings; all other cells in a
shutter have the same marks as their French-window counterparts.
Preamble. Note that there can be (as we shall see when we speak of hide/fill
involution) cells left empty for rule-based reasons other than P = R⊻C = 0 | S.
The shutter-based counterparts of such French-window cells, unlike those of long-
diagonal cells, remain empty.
Proof. The top and left (bottom and right) shutters are equivalent: one merely
switches row for column labels. Top/left and bottom/right shutter-sets are likewise
equivalent by the symmetry of strut-opposites. We hence make the case for the
left shutter only. But for the novelties posed by the initially blank cells and the
label lines (with the only real subtleties involving markings), the proof proceeds
in a manner very similar to Theorem 11: split into 3 cases, based on whether (1)
the L-index trip implied by the R,C,P values is a Zigzag; (2) u,v are both Vents;
or, (3) the edge signified by the cell content is the emanation of same-inner-signed
dyads (that is, one is a Vent, the other a Zigzag).
Case 1: Assume (u,v,w) a Zigzag L-trip in the French window’s contained
skybox; the general product in its shutter is
v − (G+ vopp)
(g+u) + (G+g+uopp)
−(G+g+wopp) +(g+w)
−(g+w) +(G+g+wopp)
(g+u) · v = P : (u,v,w)→ (g+w,v,g+u); hence, (−(g+w)).
(g+ u) · (G+ vopp) = P : (u,wopp,vopp) → (g+wopp,g+ u,vopp) → (G+
vopp,g+u,G+g+wopp); dyads’ opposite inner signs make (G+g+wopp) pos-
itive.
(G+g+uopp) · v = P : (uopp,wopp,v) → (g+wopp,g+uopp,v) → (G+g+
uopp,G+g+wopp,v); hence, (−(G+g+wopp)).
(G+g+uopp) · (G+ vopp) = P : (vopp,uopp,w) → (vopp,g+w,g+uopp) →
(G+g+uopp,g+w,G+vopp); dyads’ opposite inner signs make (g+w) positive.
Case 2: The product of two Vents must have negative edge-sign, hence nega-
tive inner sign in top dyad to lower dyad’s positive. The shutter product thus looks
like this:
(uopp)− (G+u)
(g+ vopp) + (G+g+ v)
+(G+g+wopp) +(g+w)
−(g+w) − (G+g+wopp)
(g+vopp) ·uopp = P : (vopp,uopp,w)→ (g+w,uopp,g+vopp); hence, (−(g+
(g+ vopp) · (G+ u) = P : (vopp,u,wopp) → (g+wopp,u,g+ vopp) → (G+
u,G+ g+wopp,g+ vopp); but dyads’ inner signs are opposite, so (−(G+ g+
wopp)).
(G+g+v) ·uopp = P : (uopp,wopp,v)→ (uopp,g+v,g+wopp)→ (uopp,G+
g+wopp,G+g+ v); hence, (+(G+g+wopp)).
(G+g+v) ·(G+u)= P : (u,v,w)→ (u,g+w,g+v)→ (G+g+v,g+w,G+
u); but dyads’ inner signs are opposite, so (+(g+w)).
Case 3: The product of Vent and Zigzag displays same inner sign in both
dyads; hence the following arithmetic holds:
(uopp)+(G+u)
(g+ v) + (G+g+ vopp)
+(G+g+w) +(g+wopp)
−(g+wopp) − (G+g+w)
As with the last case in Theorem 11, we omit the term-by-term calculations for
this last case, as they should seem “much of a muchness” by this point. What is
clear in all three cases is that index values of shutter cells have same markings
as their French-window counterparts, at least for all cells which have markings in
the contained skybox; but, in all cases, indices are augmented by g.
The assignment of marks to the shutter-cells linked to blank cells in French
windows is straightforward for Type I box-kites: since any containing skybox
must have g > S, and since g+ s has g as its strut opposite, then the First Vizier
tells us that any g must be a Vent. But then the R,C indices of the cell containing
g must belong to a Trefoil in such a box-kite; hence, one is a Vent, the other a
Zigzag, and g must be marked. Only if the R,C,P entry in the ET is necessar-
ily confined to a Type II box-kites will this not necessarily be so. But Part II’s
Appendix B made clear that Type II’s are generated by excluding g from their
L-indices: recall that, in the Pathions, for all S ¡ 8, all and only Type II box-kites
are created by placing one of the Sedenion Zigzag L-trips on the “Rule 0” circle
of the PSL(2,7) triangle with 8 in the middle (and hence excluded). This is a box-
kite in its own right (one of the 7 “Atlas” box-kites with S = 8); its 3 sides are
“Rule 2” triplets, and generate Type II box-kites when made into zigzag L-index
sets. Conversely, all Pathion box-kites containing an ’8’ in an L-index (dubbed
”strongboxes” in Appendix B) are Type I. Whether something peculiar might oc-
cur for large N (where there might be multiple powers of 2 playing roles in the
same box-kite) is a matter of marginal interest to present concerns, and will be left
as an open question for the present. We merely note that, by a similar argument,
and with the same restrictions assumed, g+S must be a Zigzag L-index, and R,C
either both be likewise (hence, g+S is unmarked); or, both are Vents in a Trefoil
(so g+S must be unmarked here too).
The last detail – reversal of label-line markings in their g-augmented shutter-
cell extensions – is demonstrated as follows, with the same caveat concerning
Type II box-kites assumed to apply. Such cells house DMZs (just swap u for g+u
in Theorem 11’s first setup – they form a Rule 1 trip – and compute). The LL
extension on top has row-label g; that along the bottom, the strut-opposite g+S.
Given trip (u,v,w), the shutter-cell index for R,C = (g,u) corresponds to French-
window index for R,C = (g,g+u). But (u,g,g+u) is a Trefoil, since g is a Vent.
So if u is one too, g+u isn’t; hence marks are reversed as claimed. �
3 Maximal High-Bit Singletons: (s,g)-Modularity for
16 < S ≤ 24
The Whorfian Sky, having but one high bit in its strut constant, is the simplest
possible meta-fractal – the first of an infinite number of such infinite-dimensional
zero-divisor-spanned spaces. We can consider the general case of such single-
ton high-bit recursiveness in two different, complementary ways. First, we can
supplement the just-concluded series of theorems and proofs with a calculational
interlude, where we consider the iterative embeddings of the Pathion Sand Man-
dalas in the infinite cascade of boxes-within-boxes that a Sky oversees. Then, we
can generalize what we saw in the Pathions to consider the phenomenology of
strut constants with singleton high-bits, which we take to be any bits representing
a power of 2 ≥ 3 if S contains low bits (is not a multiple of 8), else a power of 2
strictly greater than 3 otherwise. Per our earlier notation, g = G/2 is the highest
such singleton bit possible. We can think of its exponential increments – equiva-
lent to left-shifts in bit-string terms – as the side-effects of conjoint zero-padding
of N and S. This will be our second topic in this section.
Maintaining our use of S = 15 as exemplary, we have already seen that NSLs
come in quartets: a row and column are each headed by S mod g (henceforth, s)
and g, hence 7 and 8 in the Sand Mandalas. But each recursive embedding of the
current skybox in the next creates further quartets. Division down the midlines
to insert the indices new to the next CDP generation induces the Sand Mandala’s
adjoining strut-opposite sets of s and g lines (the pane-framing muntins) to be
displaced to the borders of the four corners and shutters, with the new skybox’s
g and g+ s now adjoining the old s and g to form new muntins, on the right and
left respectively, while g+g/2 (the old G+g) and its strut opposite form a third
muntin along the new midlines. Continuing this recursive nesting of skyboxes
generates 1, 3, 7, · · ·, 2B+1 −1 row-and-column muntin pairs involving multiples
of 8 and their supplementings by s, where (recalling earlier notation) B = 0 for the
inner skybox, and increments by 1 with each further nesting. Put another way, we
then have a muntin number µ = (2N−4 −1), or 4µ NSL’s in all.
The ET for given N has (2N−1 −2) cells in each row and column. But NSLs
divvy them up into boxes, so that each line is crossed by 2µ others, with the 0,
2 or 4 cells in their overlap also belonging to diagonals. The number of cells in
the overlap-free segments of the lines, or ω , is then just 4µ · (2N−1 − 2− 2µ) =
24µ(µ +1): an integer number of Box-Kites. For our S = 15 case, the minimized
line shuffling makes this obvious: all boxes are 6 x 6, with 2-cell-thick boundaries
(the muntins separating the panes), with µ boundaries, and (µ + 1) overlap-free
cells per each row or column, per each quartet of lines.
The contribution from diagonals, or δ , is a little more difficult, but straight-
forward in our case of interest: 4 sets of 1,2,3, · · · ,µ boxes are spanned by mov-
ing along one empty long diagonal before encountering the other, with each box
contributing 6, and each overlap zone between adjacent boxes adding 2. Hence,
δ = 24 · (2N−3 − 1)(2N−3 − 2)/6 – a formula familiar from associative-triplet
counting: it also contributes an integer number of Box-Kites. The one-liner we
want, then, is this:
BKN, 8<S<16 = ω +δ = (2N−4)(2N−4 −1) + (2N−3 −1)(2N−3 −2)/6
For N = 4,5,6,7,8,9,10, this formula gives 0,3,19,91,395, 1643,6699. Add
4N−4 to each – the immediate side-effect of the offing of all four Rule 0 candidate
trips of the Sedenion Box-Kite exploded into the Sand Mandala that begins the
recursion – and one gets “déjà vu all over again”: 1, 7, 35, 155, 651, 2667, 10795
– the full set of Box-Kites for S ≤ 8.
It would be nice if such numbers showed up in unsuspected places, having
nothing to do with ZDs. Such a candidate context does, in fact, present itself, in Ed
Pegg’s regular MAA column on “Math Games” focusing on “Tournament Dice.”
[7] He asks us, “What dice make a non-transitive four player game, so that if three
dice are chosen, a fourth die in the set beats all three? How many dice are needed
for a five player non-transitive game, or more?” The low solution of 3 explicitly
involves PSL(2,7); the next solution of 19 entails calculations that look a lot like
those involved in computing row and column headers in ETs. No solutions to the
dice-selecting game beyond 19 are known. The above formulae, though, suggest
the next should be 91. Here, ZDs have no apparent role save as dummies, like the
infinity of complex dimensions in a Fourier-series convergence problem, tossed
out the window once the solution is in hand. Can a number-theory fractal, with
intrinsically structured cell content (something other, non-meta, fractals lack) be
of service in this case – and, if not in this particular problem, in others like it?
Now let’s consider the more general situation, where the singleton high-bit
can be progressively left-shifted. Reverting to the use of the simplest case as
exemplary, use S = g+1 = 9 in the Pathions, then do tandem left-shifts to pro-
duce this sequence: N = 6, S = g+1 = 17; N = 7, S = g+1 = 33; · · · ; N = K,
S = g+1 = 2K−2 + 1. A simple rule governs these ratchetings: in all cases, the
number of filled cells = 6 · (2N−1 − 4), since there are two sets of parallel sides
which are filled but for long-diagonal intersections, and two sets of g and 1 entries
distributed one per row along orthogonals to the empty long diagonals. Hence,
for the series just given, we have cell counts of 72, 168, · · · , 6 · (2N−1 − 4) for
BKN, S = 3, 7, · · · , 2
N−3 − 1, for g < S < g + 8 = G in the Pathions, and all
g < S ≤ g+8 in the Chingons, 27-ions, and general 2N-ions, in that order.
Algorithmically, the situation is just as easy to see: the splitting of dyads,
sending U- and L- indices to strut-opposite Assessors, while incorporating the S
and G of the current CDP generation as strut-opposites in the next, continues. For
S = 17 in the Chingons, there are now 2N−3 −1 = 7, not 3, Box-Kites sharing the
new g = 16 (at B) and S mod g = 1 (at E) in our running example. The U- indices
of the Sand Mandala Assessors for S = g+1 = 9 are now L-indices, and so on:
every integer < G and 6= S gets to be an L-index of one of the 30(= 2N−1 − 2)
Assessors, as 16 and S mod g = 1 appear in each of the 7 Box-Kites, with each
other eligible integer appearing once only in one of the 7 ·4= 28 available L-index
slots.
As an aside, in all 7 cases, writing the smallest Zigzag L-index at a mandates
all the Trefoil trips be “precessed” – a phenomenon also observed in the S = 8
Pathion case, as tabulated on p. 14 of [8]. For Zigzag L-index set (2,16,18),
for instance, (a,d,e) = (2,3,1) instead of (1,2,3); ( f ,c,e) = (19,18,1) not
(1,19,18); and ( f ,d,b) = (19,3,16). But otherwise, there are no surprises: for
N = 7, there are (27−3 − 1) = 15 Box-Kites, with all 62(= 2N−1 − 2) available
cells in the rows and columns linked to labels g and S mod g being filled, and so
Note that this formulation obtains for any and all S > 8 where the maximum
high-bit (that is, g) is included in its bitstring: for, with g at B and S mod g at
E, whichever R,C label is not one of these suffices to completely determine the
remaining Assessor L-indices, so that no other bits in S play a role in determining
any of them. Meanwhile, cell contents P containing either g or S mod g, but
created by XORing of row and column labels equal to neither, are arrayed in off-
diagonal pairs, forming disjoint sets parallel or perpendicular to the two empty
ones. If we write S mod g with a lower-case s, then we could call the rule in
play here (s,g)-modularity. Using the vertical pipe for logical or, and recalling the
special handling required by the 8-bit when S is a multiple of 8 (which we signify
with the asterisk suffixed to “mod”), we can shorthand its workings this way:
Theorem 14. For a 2N-ion inner skybox whose strut constant S has a singleton
high-bit which is maximal (that is, equal to g = G/2 = 2N−2), the recipe for its
filled cells can be condensed thus:
R | C| P = g | S mod∗ g
Under recursion, the recipe needs to be modified so as to include not just the
inner-skybox g and S mod∗ g (henceforth, simply lowercase s), but all integer
multiples k of g less than the G of the outermost skybox, plus their strut opposites
k ·g+ s.
Proof. The theorem merely boils down the computational arguments of prior para-
graphs in this section, then applies the last section’s recursive procedures to them.
The first claim of the proof is identical to what we’ve already seen for Sand Man-
dalas, with zero-padding injected into the argument. The second claim merely
assumes the area quadrupling based on midline splitting, with the side-effects al-
ready discussed. No formal proof, then, is called for beyond these points. �
Remarks. Using the computations from two paragraphs prior to the theorem’s
statement, we can readily calculate the box-kite count for any skybox, no matter
how deeply nested: recall the formula 6 · (2N−1 − 4) for BKN, S = 2
N−3 − 1. It
then becomes a straightforward matter to calculate, as well, the limiting ratio of
this count to the maximal full count possible for the ET as N → ∞, with each cell
approaching a point in a standard 2-D fractal. Hence, for any S with a singleton
high-bit in evidence, there exists a Sky containing all recursive redoublings of its
inner skybox, and computations like those just considered can further be used to
specify fractal dimensions and the like. (Such computations, however, will not
concern us.) Finally, recall that, by spectrographic equivalence, all such compu-
tations will lead to the same results for each S value in the same spectral band or
octave.
4 Hide/Fill Involution: Further-Right High-Bits with
24 < S < 32.
Recall that, in the Sand Mandala flip-book, each increment of S moved the two
sets of orthogonal parallel lines one cell closer toward their opposite numbers:
while S = 9 had two filled-in rows and columns forming a square missing its cor-
ners, the progression culminating in S = 15 showed a cross-hairs configuration:
the parallel lines of cells now abutted each other in 2-ply horizontal and vertical
arrays. The same basic progression is on display in the Chingons, starting with
S = 17. But now the number of strut-opposite cell pairs in each row and column
is 15, not 7, so the cross-hairs pattern can’t arise until S = 31. Yet it never arises
in quite the manner expected, as something quite singular transpires just after flip-
ping past the ET in the middle, for S = 24. Here, rows and columns labeled 8 and
16 constrain a square of empty cells in the center · · · quickly followed by an ET
which seems to continue the expected trajectory – except that almost all the non-
long-diagonal cells left empty in its predecessor ETs are now inexplicably filled.
More, there is a method to the “almost all” as well: for we now see not 2, but 4
rows and columns, all being blanked out while those labeled with g and S mod g
are being filled in.
This is an inevitable side effect of a second high-bit in S: we call this phe-
nomenon, first appearing in the Chingons, hide/fill involution. There are 4, not 2,
line-pairs, because S and G, modulo a lower power of 2 (because devolving upon
a prior CDP generation’s g), offer twice the possibilities: for S = 25, S mod 16 is
now 9, but S mod 8 can result in either 1 or 17 as well – with correlated multiples
of 8 (8 proper, and 24) defining the other two pairings. All cells with R |C | P
equal to one of these 4 values, but for the handful already set to “on” by the first
high-bit, will now be set to “off,” while all other non-long-diagonal cells set to
“off” in the Pathion Sand Mandalas are suddenly “on.” What results for each
Chingon ET with 24 < S < 32 is an ensemble comprised of 23 Box-Kites. (For
the flip-book, see Slides 40 – 54.) Why does this happen? The logic is as straight-
forward as the effect can seem mysterious, and is akin, for good reason, to the
involutory effect on trip orientation induced by Rule 2 addings of G to 2 of the
trip’s 3 indices.
In order to grasp it, we need only to consider another pair of abstract calcula-
tion setups, of the sort we’ve seen already many times. The first is the core of the
Two-Bit Theorem, which we state and prove as follows:
Theorem 15. 2N-ion dyads making DMZs before augmenting S with a new high-bit
no longer do so after the fact.
Proof. Suppose the high-bit in the bitstring representation of S is 2K, K < (N −
1). Suppose further that, for some L-index trip (u,v,w), the Assessors U and V
are DMZ’s, with their dyads having same inner signs. (This last assumption is
strictly to ease calculations, and not substantive: we could, as earlier, use one
or more binary variables of the sg type to cover all cases explicitly, including
Type I vs. Type II box-kites. To keep things simple, we assume Type I in what
follows.) We then have (u+ u ·X)(v+ v · X) = (u+U)(v+V ) = 0. But now
suppose, without changing N, we add a bit somewhere further to the left to S, so
that S < (2K = L) < G. The augmented strut constant now equals SL = S+L.
One of our L-indices, say v, belongs to a Vent Assessor thanks to the assumed
inner signing; hence, by Rule 2 and the Third Vizier, (V,v,X)→ (X +L,v,V +L).
Its DMZ partner u, meanwhile, must thereby be a Zigzag L-index, which means
(u,U,X)→ (u,X +L,U +L). We claim the truth of the following arithmetic:
v + (V +L)
u + (U +L)
+(W + L) +w
+ w − (W +L)
NOT ZERO (+w’s don’t cancel)
The left bottom product is given. The product to its right is derived as follows:
since u is a Zigzag L-index, the Trefoil U-trip (u,V,W) has the same orientation as
(u,v,w), so that Rule 2 → (u,W +L,V +L), implying the negative result shown.
The left product on the top line, though, has terms derived from a Trefoil U-trip
lacking a Zigzag L-index, so that only after Rule 2 reversal are the letters arrayed
in Zigzag L-trip order: (U +L,v,W +L). Ergo, +(W +L). Similarly for the top
right: Rule 2 reversal “straightens out” the Trefoil U-trip, to give (U+L,V +L,w);
therefore, (+w) results. If we explicitly covered further cases by using an sg
variable, we would be faced with a Theorem 2 situation: one or the other product
pair cancels, but not both. �
Remark. The prototype for the phenomenon this theorem covers is the “explo-
sion” of a Sedenion box-kite into a trio of interconnected ones in a Pathion sand
mandala, with the S of the latter = the X of the former. As part of this process, 4 of
the expected 7 are “hidden” box-kites (HBKs), with no DMZs along their edges.
These have zigzag L-trips which are precisely the L-trips of the 4 Sedenion Sails.
Here, an empirical observation which will spur more formal investigations in a
sequel study: for the 3 HBKs based on trefoil L-trips, exactly 1 strut has reversed
orientation (a different one in each of them), with the orientation of the triangular
side whose midpoint it ends in also being reversed. For the HBK based on the
zigzag L-trip, all 3 struts are reversed, so that the flow along the sides is exactly
the reverse of that shown in the “Rule 0” circle. (Hence, all possible flow patterns
along struts are covered, with only those entailing 0 or 2 reversals corresponding
to functional box-kites: our Type I and Type II designations.) It is not hard to show
that this zigzag-based HBK has another surprising property: the 8 units defined
by its own zigzag’s Assessors plus X and the real unit form a ZD-free copy of the
Octonions. This is also true when the analogous Type II situation is explored, al-
beit for a slightly different reason: in the former case, all 3 Catamaran “twistings”
take the zigzag edges to other HBKs; in the latter, though, the pair of Assessors in
some other Type II box-kite reached by “twisting” – (a,B) and (A,b), say, if the
edge be that joining Assessors A and B, with strut-constant copp = d – are strut
opposites, and hence also bereft of ZDs. The general picture seems to mirror this
concrete case, and will be studied in “Voyage by Catamaran” with this expecta-
tion: the bit-twiddling logic that generates meta-fractal “Skies” also underwrites
a means for jumping between ZD-free Octonion clones in an infinite number of
HBKs housed in a Sky. Given recent interest in pure “E8” models giving a privi-
leged place to the basis of zero-divisor theory, namely “G2” projections (viz., A.
Garrett Lisi’s “An Exceptionally Simple Theory of Everything”); a parallel vogue
for many-worlds approaches; and, the well-known correspondence between 8-D
closest-packing patterns, the loop of the 240 unit Octonions which Coxeter dis-
covered, and E8 algebras – given all this, tracking the logic of the links across
such Octonionic “brambles” might prove of great interest to many researchers.
Now, we still haven’t explained the flipside of this off-switch effect, to which
prior CDP generation Box-Kites – appropriately zero-padded to become Box-
Kites in the current generation until the new high-bit is added to the strut-constant
– are subjected. How is it that previously empty cells not associated with the sec-
ond high-bit’s blanked-out R, C, P values are now full? The answer is simple, and
is framed in the Hat-Trick Theorem this way.
Theorem 16. Cells in an ET which represent DMZ edges of some 2N-ion Box-
Kites for some fixed S, and which are offed in turn upon augmenting of S by a
new leftmost bit, are turned on once more if S is augmented by yet another new
leftmost bit.
Proof. We begin an induction based upon the simplest case (which the Chingons
are the first 2N-ions to provide): consider Box-Kites with S ≤ 8. If a high-bit
be appended to S, then the associated Box-Kites are offed. However, if another
high-bit be affixed, these dormant Box-Kites are re-awakened – the second half of
hide/fill involution. We simply assume an L-index set (u,v,w) underwriting a Sail
in the ET for the pre-augmented S, with Assessors (u,U) and (v,V ). Then, we
introduce a more leftified bit 2Q = M, where pre-augmented S < L < M < G, then
compute the term-by term products of (u+(U +L+M)) and (v+ sg · (V +L+
M)), using the usual methods. And as these methods tell us that two applications
of Rule 2 have the same effect as none in such a setup, we have no more to prove.
Corollary. The induction just invoked makes it clear that strut constants equal to
multiples of 8 not powers of 2 are included in the same spectral band as all other
integers larger than the prior multiple. The promissory note issued in the second
paragraph of Part II’s concluding section, on 64-D Spectrography, can now be
deemed redeemed.
In the Chingons, high-bits L and M are necessarily adjacent in the bitstring for
S < G = 32; but in the general 2N-ion case, N large, zero-padding guarantees that
things will work in just the same manner, with only one difference: the recursive
creation of “harmonics” of relatively small-g (s,g)-modular R,C,P values will
propagate to further levels, thereby effecting overall Box-Kite counts.
In general terms, we have echoes of the formula given for (s,g)-modular cal-
culations, but with this signal difference: there will be one such rule for each
high-bit 2H in S, where residues of S modulo 2H will generate their own near-
solid lines of rows and columns, be they hidden or filled. Likewise for multiples
of 2H <G which are not covered by prior rules, and multiples of 2H supplemented
by the bit-specific residue (regardless of whether 2H itself is available for treat-
ment by this bit-specific rule). In the simplest, no-zero-padding instances, all even
multiples are excluded, as they will have occurred already in prior rules for higher
bits, and fills or hides, once fixed by a higher bit’s rule, cannot be overridden.
Cases with some zero-padding are not so simple. Consider this two-bit in-
stance, S = 73,N = 8: the fill-bit is 64, the hide-bit is just 8, so that only 9 and 64
generate NSLs of filled values; all other multiples of 8, and their supplementing
by 1 (including 65) are NSLs of hidden values. Now look at a variation on this
example, with the single high-bit of zero-padding removed – i. e., S = 41,N = 8.
Here, the fill-bit is 32, and its multiples 64 and 96, as well as their supplements by
S modulo 32 = 9, or 9 and 73 and 105, label NSLs of filled values; but all other
multiples of 8, plus all multiples of 8 supplemented by 1 not equal to 9 or 73 or
105, label NSLs of hidden values. Cases with multiple fill and hide bits, with or
without additional zero-padding, are obviously even more complicated to handle
explicitly on a case-by-case basis, but the logic framing the rules remain simple;
hence, even such messy cases are programmatically easy to handle.
Hide/fill involution means, then, that the first, third, and any further odd-
numbered high-bits (counting from the left) will generate “fill” rules, whereas
all the even-numbered high-bits generate “hide” rules – with all cells not touched
by a rule being either hidden (if the total number of high-bits B is odd) or filled (B
is even).
Two further examples should make the workings of this protocol more clear.
First, the Chingon test case of S = 25: for (R | C | P = 9 | 16), all the ET cells are
filled; however, for (R | C | P = 1 | 8 | 17 | 24), ET cells not already filled by the
first rule (and, as visual inspection of Slide 48 indicates, there are only 8 cells in
the entire 840-cell ET already filled by the prior rule which the current rule would
like to operate on) are hidden from view. Because the 16- and 8- bits are the only
high-bits, the count of same is even, meaning all remaining ET cells not covered
by these 2 rules are filled.
We get 23 for Box-Kite count as follows. First, the 16-bit rule gives us 7 Box-
Kites, per earlier arguments; the 8-bit rule, which gives 3 filled Box-Kites in the
Pathions, recursively propagates to cover 19 hidden Box-Kites in the Chingons,
according to the formula produced last section. But hide/fill involution says that,
of the 35 maximum possible Box-Kites in a Chingon ET, 35− 19 = 16 are now
made visible. As none of these have the Pathion G = 16 as an L-index, and all the
7 Box-Kites from the 16-bit rule do, we therefore have a grand total of 7+16= 23
Box-Kites in the S = 25 ET, as claimed (and as cell-counting on the cited Slide
will corroborate).
The concluding Slides 76–78 present a trio of color-coded “histological slices”
of the hiding and filling sequence (beginning with the blanking of the long diago-
nals) for the simplest 3-high-bit case, N = 7,S = 57. Here, the first fill rule works
on 25 and 32; the first hide rule, on 9, 16, 41, and 48; the second fill rule, on 1, 8,
17, 24, 33, 40, 49, and 56; and the rest of the cells, since the count of high-bits is
odd, are left blank.
We do not give an explicit algorithmic method here, however, for computing
the number of Box-Kites contained in this 3,720-cell ET. Such recursiveness is
best handled programmatically, rather than by cranking out an explicit (hence,
long and tedious) formula, meant for working out by a time-consuming hand cal-
culation. What we can do, instead, is conclude with a brief finale, embodying
all our results in the simple “recipe theory” promised originally, and offer some
reflections on future directions.
5 Fundamental Theorem of Zero-Divisor Algebra
All of the prior arguments constitute steps sufficient to demonstrate the Funda-
mental Theorem of Zero-Divisor Algebra. Like the role played by its Gaussian
predecessor in the legitimizing of another “new kind of [complex] number the-
ory,” its simultaneous simplicity and generality open out on extensive new vistas
at once alien and inviting. The Theorem proper can be subdivided into a Proposi-
tion concerning all integers, and a “Recipe Theory” pragmatics for preparing and
“cooking” the meta-fractal entities whose existence the proposition asserts, but
cannot tell us how to construct.
Proposition: Any integer K > 8 not a power of 2 can uniquely be associated with
a Strut Constant S of ZD ensembles, whose inner skybox resides in the 2N-ions
with 2N−2 < K < 2N−1. The bitstring representation of S completely determines
an infinite-dimensional analog of a standard plane-confined fractal, with each of
the latter’s points associated with an empty cell in the infinite Emanation Table,
with all non-empty cells comprised wholly of mutually orthogonal primitive zero-
divisors, one line of same per cell.
Preparation: Prepare each suitable S by producing its bitstring representation,
then determining the number of high-bits it contains: if S is a multiple of 8, right-
shift 4 times; otherwise, right-shift 3 times. Then count the number B of 1’s
in the shortened bitstring that results. For this set {B} of B elements, construct
two same-sized arrays, whose indices range from 1 to B: the array {i} which
indexes the left-to-right counting order of the elements of {B}; and, the array
{P} which indexes the powers of 2 of the same element in the same left-to-right
order. (Example: if K = 613, the inner skybox is contained in the 211-ions; as
the number is not a multiple of 8, the bistring representation 1001100101 is right-
shifted thrice to yield the substring of high-bits 1001100; B = 3, and for 1 ≤ i ≤
3, P1 = 9, P2 = 6;P3 = 5.)
Cookbook Instructions:
[0] For a given strut-constant S, compute the high-bit count B and bitstring
arrays {i} and {P}, per preparation instructions.
[1] Create a square spreadsheet-cell array, of edge-length 2I , where I ≥G/2= g
of the inner skybox for S, with the Sky as the limit when I → ∞.
[2] Fill in the labels along all four edges, with those running along the right
(bottom) borders identical to those running along the left (top), except in
reversed left-right (top-bottom) order. Refer to those along the top as col-
umn numbers C, and those along the left edge, as row numbers R, setting
candidate contents of any cell (r,c) to R⊻C = P.
[3] Paint all cells along the long diagonals of the spreadsheet just constructed
a color indicating BLANK, so that all cells with R =C (running down from
upper left corner) else R⊻C = S (running down from upper right) have their
P-values hidden.
[4] For 1 ≤ i ≤ B, consider for painting only those cells in the spreadsheet
created in [1] with R | C | P = m · 2γ | m · 2γ + σ , where γ = Pi,σ =
S mod∗ 2γ , and m is any integer ≥ 0 (with m= 0 only producing a legitimate
candidate for the right-hand’s second option, as an XOR of 0 indicates a
long-diagonal cell).
[5] If a candidate cell has already been painted by a prior application of these
instructions to a prior value of i, leave it as is. Otherwise, paint it with R⊻C
if i = odd, else paint it BLANK.
[6] Loop to [4] after incrementing i. If i < B, proceed until this step, then
reloop, reincrement, and retest for i = B. When this last condition is met,
proceed to the next step.
[7] If B is odd, paint all cells not already painted, BLANK; for B even, paint
them with R⊻C.
In these pseudocode instructions, no attention is given to edge-mark gener-
ation, performance optimization, or other embellishments. Recursive expansion
beyond the chosen limits of the 2N-ion starting point is also not addressed. (Just
keep all painted cells as is, then redouble until the expanded size desired is at-
tained; compute appropriate insertions to the label lines, then paint all new cells
according to the same recipe.) What should be clear, though, is any optimization
cannot fail to be qualitatively more efficient than the code in the appendix to [9],
which computes on a cell-by-cell basis. For S > 8, N > 4, we’ve reached the
onramp to the Metafractal Superhighway: new kinds of efficiency, synergy, con-
nectedness, and so on, would seem to more than compensate for the increase in
dimension.
It is well-known that Chaotic attractors are built up from fractals; hence, our
results make it quite thinkable to consider Chaos Theory from the vantage of pure
Number · · · and hence the switch from one mode of Chaos to another as a bitstring-
driven – or, put differently, a cellular automaton-type – process, of Wolfram’s
Class 4 complexity. Such switching is of the utmost importance in coming to
terms with the most complex finite systems known: human brains. The late Fran-
cisco Varela, both a leading visionary in neurological research and its computer
modeling, and a long-time follower of Madhyamika Buddhism who’d collabo-
rated with the Dalai Lama in his “Tibetan Buddhists talk with brain scientists”
dialogues [10], pointed to just the sorts of problems being addressed here as the
next frontier. In a review essay he co-authored in 2001 just before his death [11,
p. 237], we read these concluding thoughts on the theme of what lies “Beyond
Synchrony” in the brain’s workings:
The transient nature of coherence is central to the entire idea of large-
scale synchrony, as it underscores the fact that the system does not be-
have dynamically as having stable attractors [e.g., Chaos], but rather
metastable patterns – a succession of self-limiting recurrent patterns.
In the brain, there is no “settling down” but an ongoing change marked
only by transient coordination among populations, as the attractor it-
self changes owing to activity-dependent changes and modulations of
synaptic connections.
Varela and Jean Petitot (whose work was the focus of the intermezzo conclud-
ing Part I, in which semiotically inspired context the Three Viziers were intro-
duced) were long-time collaborators, as evidenced in the last volume on Naturaliz-
ing Phenomenology [12] which they co-edited. It is only natural then to re-inscribe
the theme of mathematizing semiotics into the current context: Petitot offers sepa-
rate studies, at the “atomic” level where Greimas’ “Semiotic Square” resides; and
at the large-scale and architectural, where one must place Lévi-Strauss’s “Canon-
ical Law of Myths.” But the pressing problem is finding a smooth approach that
lets one slide the same modeling methodology from the one scale to the other:
a fractal-based “scale-free network” approach, in other words. What makes this
distinct from the problem we just saw Varela consider is the focus on the structure,
rather than dynamics, of transient coherence – a focus, then, in the last analysis, on
a characterization of database architecture that can at once accommodate meta-
chaotic transiency and structural linguists’ cascades of “double articulations.”
Starting at least with C. S. Peirce over a century ago, and receiving more
recent elaboration in the hands of J. M. Dunn and the research into the “Semantic
Web” devolving from his work, data structures which include metadata at the
same level as the data proper have led to a focus on “triadic logic,” as perhaps best
exemplified in the recent work of Edward L. Robertson. [13] His exploration of
a natural triadic-to-triadic query language deriving from Datalog, which he calls
Trilog, is not (unlike our Skies) intrinsically recursive. But his analysis depends
upon recursive arguments built atop it, and his key constructs are strongly resonant
with our own (explicitly recursive) ones. We focus on just a few to make the point,
with the aim of provoking interest in fusing approaches, rather than in proving any
particular results.
The still-standard technology of relational databases based on SQL statements
(most broadly marketed under the Oracle label) was itself derived from Peirce’s
triadic thinking: the creator of the relational formalism, Edgar F. “Ted” Codd, was
a PhD student of Peirce editor and scholar Arthur W. Burks. Codd’s triadic “re-
lations,” as Robertson notes (and as Peirce first recognized, he tells us, in 1885),
are “the minimal, and thus most uniform” representations “where metadata, that
is data about data, is treated uniformly with regular data.” In Codd’s hands (and in
those of his market-oriented imitators in the SQL arena), metadata was “relegated
to an essentially syntactic role” [13, p. 1] – a role quite appropriate to the appli-
cations and technological limitations of the 1970’s, but inadequate for the huge
and/or highly dynamic schemata that are increasingly proving critical in bioinfor-
matics, satellite data interpretation, Google server-farm harvesting, and so on. As
Robertson sums up the situation motivating his own work,
Heterogeneous situations, where diverse schemata represent semanti-
cally similar data, illustrate the problems which arise when one per-
son’s semantics is another’s syntax – the physical “data dependence”
that relational technology was designed to avoid has been replaced by
a structural data dependence. Hence we see the need to [use] a simple,
uniform relational representation where the data/metadata distinction
is not frozen in syntax. [13, pp. 1-2]
As in relational database theory and practice, the forming and exploiting of
inner and outer joins between variously keyed tables of data is seminal to Robert-
son’s approach as well as Codd’s. And while the RDF formalism of the Semantic
Web (the representational mechanism for describing structures as well as contents
of web artifacts on the World Wide Web) is likewise explicitly triadic, there has,
to date, been no formal mechanism put in place for manipulating information in
RDF format. Hence, “there is no natural way to restrict output of these mecha-
nisms to triples, except by fiat” [13, p. 4], much less any sophisticated rule-based
apparatus like Codd’s “normal forms” for querying and tabulating such data. It is
no surprise, then, that Robertson’s “fundamental operation on triadic relations is
a particular three-way join which takes explicit advantage of the triadic structure
of its operands.” This triadic join, meanwhile, “results in another triadic relation,
thus providing the closure required of an algebra.” [13, p. 6]
Parsing Robertson’s compact symbolic expressions into something close to
standard English, the trijoin of three triadic relations R, S, T is defined as some
(a,b,c) selected from the universe of possibilities (x,y,z), such that (a,x,z) ∈ R,
(x,b,y) ∈ S, and (z,y,c) ∈ T . This relation, he argues, is the most fundamental
of all the operators he defines. When supplemented with a few constant relations
(analogs of Tarski’s “infinite constants” embodied in the four binary relations of
universality of all pairs, identity of all equal pairs, diversity of all unequal pairs,
and the empty set), it can express all the standard monotonic operators (thereby
excluding, among his primitives, only the relative complement).
How does this compare with our ZD setup, and the workings of Skies? For one
thing, Infinite constants, of a type akin to Tarski’s, are embodied in the fact that
any full meta-fractal requires the use of an infinite G, which sits atop an endless
cascade of singleton leftmost bits, determining for any given S an indefinite tower
of ZDs. One of the core operators massaging Robertson’s triads is the flip, which
fixes one component of a relation while interchanging the other two · · · but our
Rule 2 is just the recursive analog of this, allowing one to move up and down
towers of values with great flexibilty (allowing, as well, on and off switching
effecting whole ensembles). The integer triads upon which our entire apparatus
depends are a gift of nature, not dictated “by fiat,” and give us a natural basis for
generating and tracking unique IDs with which to “tag” and “unpack” data (with
“storage” provided free of charge by the empty spaces of our meta-fractals: the
“atoms” of Semiotic Squares have four long-diagonal slots each, one per each of
the “controls” Petitot’s Catastrophe Theory reading calls for, and so on.)
Finally, consider two dual constructions that are the core of our own triadic
number theory: if the (a,b,c) of last paragraph, for instance, be taken as a Zigzag’s
L-index set, then the other trio of triples correlates quite exactly with the Zigzag
U-trips. And this 3-to-1 relation, recall, exactly parallels that between the 3 Tre-
foil, and 1 Zigzag, Sails defining a Box-Kite, with this very parallel forming the
support for the recursion that ultimately lifts us up into a Sky. We can indeed
make this comparison to Robinson’s formalism exceedingly explicit: if his X, Y,
Z be considered the angular nodes of PSL(2,7) situated at the 12 o’clock apex and
the right and left corners respectively, then his (a,b,c) correspond exactly to our
own Rule 0 trip’s same-lettered indices!
Here, we would point out that these two threads of reflection – on underwriting
Chaos with cellular-automaton-tied Number Theory, and designing new kinds of
database architectures – are hardly unrelated. It should be recalled that two years
prior to his revolutionary 1970 paper on relational databases [14], Codd published
a pioneering book on cellular automata [15]. It is also worth noting that one
of the earliest technologies to be spawned by fractals arose in the arena of data
compression of images, as epitomized in the work of Michael Barnsley and his
Iterative Systems company. The immediate focus of the author’s own commercial
efforts is on fusing meta-fractal mathematics with the context-sensitive adaptive-
parsing “Meta-S” technology of business associate Quinn Tyler Jackson. [16] And
as that focus, tautologically, is not mathematical per se, we pass it by and leave it,
like so many other themes just touched on here, for later work.
References
[1] Robert P. C. de Marrais, “Placeholder Substructures I: The Road From NKS
to Scale-Free Networks is Paved with Zero Divisors,” Complex Systems, 17
(2007), 125-142; arXiv:math.RA/0703745
[2] Robert P. C. de Marrais, “Placeholder Substructures II: Meta-Fractals, Made
of Box-Kites, Fill Infinite-Dimensional Skies,” arXiv:0704.0026 [math.RA]
[3] Robert P. C. de Marrais, “The 42 Assessors and the Box-Kites They Fly,”
arXiv:math.GM/0011260
[4] Robert P. C. de Marrais, “The Marriage of Nothing and All: Zero-Divisor
Box-Kites in a ‘TOE’ Sky,” in Proceedings of the 26th International Col-
loquium on Group Theoretical Methods in Physics, The Graduate Center
of the City University of New York, June 26-30, 2006, forthcoming from
Springer–Verlag.
[5] Robert P. C. de Marrais, “Placeholder Substructures: The Road from NKS
to Small-World, Scale-Free Networks Is Paved with Zero-Divisors,” http://
wolframscience.com/conference/2006/ presentations/materials/demarrais.ppt
(Note: the author’s surname is listed under “M,” not “D.”)
[6] Benoit Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and
Company, San Francisco, 1983)
[7] Ed Pegg, Jr., “Tournament Dice,” Math Games column for July 11, 2005, on
the MAA website at http://www.maa.org/editorial/ mathgames/mathgames
_07_11_05.html
[8] Robert P. C. de Marrais, “The ‘Something From Nothing’ Insertion Point,”
http://www.wolframscience.com/conference/2004/ presentations/
materials/rdemarrais.pdf
[9] Robert P. C. de Marrais, “Presto! Digitization,” arXiv:math.RA/0603281
[10] Francisco Varela, editor, Sleeping, Dreaming, and Dying: An Exploration
of Consciousness with the Dalai Lama (Wisdom Publications: Boston, 1997).
[11] F. J. Varela, J.-P. Lachauz, E. Rodrigues and J. Martinerie, “The brainweb:
phase synchronization and large-scale integration,” Nature Reviews Neuro-
science, 2 (2001), pp. 229-239.
http://arxiv.org/abs/math/0703745
http://arxiv.org/abs/0704.0026
http://arxiv.org/abs/math/0011260
http://www.maa.org/editorial/
http://www.wolframscience.com/conference/2004/
http://arxiv.org/abs/math/0603281
[12] Jean Petitot, Francisco J. Varela, Bernard Pachoud and Jean-Michel Roy,
Naturalizing Phenomenology: Issues in Contemporary Phenomenology and
Cognitive Science (Stanford University Press: Stanford, 1999)
[13] Edward L. Robertson, “An Algebra for Triadic Relations,” Technical Re-
port No. 606, Computer Science Department, Indiana University, Bloom-
ington IN 47404-4101, January 2005; online at http://www.cs.indiana.edu/
pub/techreports/TR606.pdf
[14] E. F. Codd, The Relational Model for Database Management: Version 2
(Addison-Wesley: Reading MA, 1990) is the great visionary’s most recent
and comprehensive statement.
[15] E. F. Codd, Cellular Automata (Academic Press: New York, 1968)
[16] Quinn Tyler Jackson, Adapting to Babel – Adaptivity and Context-Sensiti-
vity in Parsing: From anbncn to RNA (Ibis Publishing: P.O. Box3083, Ply-
mouth MA 02361, 2006; for purchasing information, contact Thothic Tech-
nology Partners, LLC, at their website, www.thothic.com).
http://www.cs.indiana.edu/
	The Argument So Far
	8 <S < 16, N  : Recursive Balloon Rides in the Whorfian Sky
	Maximal High-Bit Singletons: (s,g)-Modularity for 16 < S 24
	Hide/Fill Involution: Further-Right High-Bits with 24 < S < 32.
	Fundamental Theorem of Zero-Divisor Algebra

  Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from
N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent
singularities and, as N approaches infinite, fractals -- and thereby,scale-free
networks. Any integer greater than 8 and not a power of 2 generates a
meta-fractal or "Sky" when it is interpreted as the "strut constant" (S) of an
ensemble of octahedral vertex figures called "Box-Kites" (the fundamental
building blocks of ZDs). Remarkably simple bit-manipulation rules or "recipes"
provide tools for transforming one fractal genus into others within the context
of Wolfram's Class 4 complexity.

7 Placeholder Substructures III: A
Bit-String-Driven “Recipe Theory” for
Infinite-Dimensional Zero-Divisor Spaces
Robert P. C. de Marrais ∗
Thothic Technology Partners, P.O.Box 3083, Plymouth MA 02361
October 29, 2018
Abstract
Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-
dimensional hypercomplex numbers (N a power of 2, and at least 4) can
represent singularities and, as N → ∞, fractals – and thereby, scale-free net-
works. Any integer > 8 and not a power of 2 generates a meta-fractal or
Sky when it is interpreted as the strut constant (S) of an ensemble of octahe-
dral vertex figures called Box-Kites (the fundamental ZD building blocks).
Remarkably simple bit-manipulation rules or recipes provide tools for trans-
forming one fractal genus into others within the context of Wolfram’s Class
4 complexity.
1 The Argument So Far
In Parts I[1] and II[2], the basic facts concerning zero-divisors (ZDs) as they arise
in the hypercomplex context were presented and proved. “Basic,” in the context of
this monograph, means seven things. First, they emerged as a side-effect of apply-
ing CDP a minimum of 4 times to the Real Number Line, doubling dimension to
the Complex Plane, Quaternion 4-Space, Octonion 8-Space, and 16-D Sedenions.
With each such doubling, new properties were found: as the price of sacrificing
∗Email address: rdemarrais@alum.mit.edu
http://arxiv.org/abs/0704.0112v3
counting order, the Imaginaries made a general theory of equations and solution-
spaces possible; the non-commutative nature of Quaternions mapped onto the re-
alities of the manner in which forces deploy in the real world, and led to vector
calculus; the non-associative nature of Octonions, meanwhile, has only come into
its own with the need for necessarily unobservable quantities (because of confor-
mal field-theoretical constraints)in String Theory. In the Sedenions, however, the
most basic assumptions of all – well-defined notions of field and algebraic norm
(and, therefore, measurement) – break down, as the phenomena correlated with
their absence, zero-divisors, appear onstage (never to leave it for all higher CDP
dimension-doublings).
Second thing: ZDs require at least two differently-indexed imaginary units
to be defined, the index being an integer larger than 0 (the CDP index of the
Real Unit) and less than 2N for a given CDP-generated collection of 2N-ions. In
“pure CDP,” the enormous number of alternative labeling schemes possible in
any given 2N-ion level are drastically reduced by assuming that units with such
indices interact by XOR-ing: the index of the product of any two is the XOR
of their indices. Signing is more tricky; but, when CDP is reduced to a 2-rule
construction kit, it becomes easy: for index u < G, G the Generator of the 2N-ions
(i.e., the power of 2 immediately larger than the highest index of the predecessor
2N−1-ions), Rule 1 says iu · iG = +i(u+G). Rule 2 says take an associative triplet
(a,b,c), assumed written in CPO (short for “cyclically positive order”: to wit,
a · b = +c, b · c = +a, and c · a = +b). Consider, for instance, any (u,G,G+ u)
index set. Then three more such associative triplets (henceforth, trips) can be
generated by adding G to two of the three, then switching their resultants’ places
in the CPO scheme. Hence, starting with the Quaternions’ (1,2,3) (which we’ll
call a Rule 0 trip, as it’s inherited from a prior level of CDP induction), Rule
1 gives us the trips (1,4,5), (2,4,6), and (3,4,7), while Rule 2 yields up the
other 4 trips defining the Octonions: (1,7,6), (2,5,7), and (3,6,5). Any ZD
in a given level of 2N-ions will then have units with one index < G, written in
lowercase, and the other index > G, written in uppercase. Such pairs, alternately
called “dyads” or “Assessors,” saturate the diagonal lines of their planes, which
diagonals never mutually zero-divide each other (or make DMZs, for ”divisors (or
dyads) making zero”), but only make DMZs with other such diagonals, in other
such Assessors. (This is, of course, the opposite situation from the projection
operators of quantum mechanics, which are diagonals in the planes formed by
Reals and dimensions spanned by Pauli spin operators contained within the 4-
space created by the Cartesian product of two standard imaginaries.)
Third thing: Such ZDs are not the only possible in CDP spaces; but they define
the “primitive” variety from which ZD spaces saturating more than 1-D regions
can be articulated. A not quite complete catalog of these can be found in our
first monograph on the theme [3]; a critical kind which was overlooked there,
involving the Reals (and hence, providing the backdrop from which to see the
projection-operator kind as a degenerate type), were first discussed more recently
[4]. (Ironically, these latter are the easiest sorts of composites to derive of any:
place the two diagonals of a DMZ pairing with differing internal signing on axes
of the same plane, and consider the diagonals they make with each other!) All the
primitive ZDs in the Sedenions can be collected on the vertices of one of 7 copies
of an Octahedron in the Box-Kite representation, each of whose 12 edges indicates
a two-way “DMZ pathway,” evenly divided between 2 varieties. For any vertex V,
and k any real scalar, indicate the diagonals this way: (V,/) = k · (iv+ iV ), while
(V, \)= k · (iv− iV ). 6 edges on a Box-Kite will always have negative edge-sign
(with unmarked ET cell entries: see the “sixth thing”). For vertices M and N,
exactly two DMZs run along the edge joining them, written thus:
(M,/) ·(N, \) = (M, \) · (N,/) = 0
The other 6 all have positive edge-sign, the diagonals of their two DMZs hav-
ing same slope (and marked – with leading dashes – ET cell entries):
(Z,/) ·(V, /)= (Z, \) ·(V,\)= 0
Fourth thing: The edges always cluster similarly, with two opposite faces
among the 8 triangles on the Box-Kite being spanned by 3 negative edges (con-
ventionally painted red in color renderings), with all other edges being positive
(painted blue). One of the red triangles has its vertices’ 3 low-index units forming
a trip; writing their vertex labels conventionally as A, B, C, we find there are in
fact always 4 such trips cycling among them: (a,b,c), the L-trip; and the three
U-trips obtained by replacing all but one of the lowercase labels in the L-trip with
uppercase: (a,B,C); (A,b,C); (A,B,c). Such a 4-trip structure is called a Sail, and
a Box-Kite has 4 of them: the Zigzag, with all negative edges, and the 3 Trefoils,
each containing two positive edges extending from one of the Zigzag vertices to
the two vertices opposite its Sailing partners. These opposite vertices are always
joined by one of the 3 negative edges comprising the Vent which is the Zigzag’s
opposite face. Again by convention, the vertices opposite A, B, C are written F,
E, D in that order; hence, the Trefoil Sails are written (A,D,E); (F,D,B), and
(F,C,E), ordered so that their lowercase renderings are equivalent to their CPO
L-trips. The graphical convention is to show the Sails as filled in, while the other 4
faces, like the Vent, are left empty: they show “where the wind blows” that keeps
the Box-Kite aloft. A real-world Box-Kite, meanwhile, would be held together
by 3 dowels (of wood or plastic, say) spanning the joins between the only vertices
left unconnected in our Octahedral rendering: the Struts linking the strut-opposite
vertices (A, F); (B, E); (C, D).
Fifth thing: In the Sedenions, the 7 isomorphic Box-Kites are differentiated by
which Octonion index is missing from the vertices, and this index is designated
by the letter S, for “signature,” “suppressed index,” or strut constant. This last
designation derives from the invariant relationship obtaining in a given Box-Kite
between S and the indices in the Vent and Zigzag termini (V and Z respectively)
of any of the 3 Struts, which we call the “First Vizier” or VZ1. This is one of
3 rules, involving the three Sedenion indices always missing from a Box-Kite’s
vertices: G, S, and their simple sum X (which is also their XOR product, since G
is always to the left of the left-most bit in S). The Second Vizier tells us that the
L-index of either terminus with the U-index of the other always form a trip with
G, and it true as written for all 2N-ions. The Third shows the relationship between
the L- and U- indices of a given Assessor, which always form a trip with X. Like
the First, it is true as written only in the Sedenions, but as an unsigned statement
about indices only, it is true universally. (For that reason, references to VZ1 and
VZ3 hereinout will be assumed to refer to the unsigned versions.) First derived in
the last section of Part I, reprised in the intro of Part II, we write them out now for
the third and final time in this monograph:
VZ1: v · z =V ·Z = S
VZ2: Z · v =V · z = G
VZ3: V · v = z ·Z = X.
Rules 1 and 2, the Three Viziers, plus the standard Octonion labeling scheme
derived from the simplest finite projective group, usually written as PSL(2,7), pro-
vide the basis of our toolkit. This last becomes powerful due to its capacity for
recursive re-use at all levels of CDP generation, not just the Octonions. The sim-
plest way to see this comes from placing the unique Rule 0 trip provided by the
Quaternions on the circle joining the 3 sides’ midpoints, with the Octonion Gen-
erator’s index, 4, being placed in the center. Then the 3 lines leading from the
Rule 0 trip’s (1, 2, 3) midpoints to their opposite angles – placed conventionally
in clockwise order in the midpoints of the left, right, and bottom sides of a triangle
whose apex is at 12 o’clock – are CPO trips forming the Struts, while the 3 sides
themselves are the Rule 2 trips. These 3 form the L-index sets of the Trefoil Sails,
while the Rule 0 trip provides the same service for the Zigzag. By a process analo-
gized to tugging on a slipcover (Part I) and pushing things into the central zone
of hot oil while wok-cooking (Part II), all 7 possible values of S in the Sedenions,
not just the 4, can be moved into the center while keeping orientations along all
7 lines of the Triangle unchanged. Part II’s critical Roundabout Theorem tells us,
moreover, that all 2N-ion ZDs, for all N > 3, are contained in Box-Kites as their
minimal ensemble size. Hence, by placing the appropriate G, S, or X in the center
of a PSL(2,7) triangle, with a suitable Rule 0 trip’s indices populating the circle,
any and all candidate primitive ZDs can be discovered and situated.
Sixth thing: The word “candidate” in the above is critical; its exploration was
the focus of Part II. For, starting with N = 5 and hence G = 16 (which is to say, in
the 32-D Pathions), whole Box-Kites can be suppressed (meaning, all 12 edges,
and not just the Struts, no longer serve as DMZ pathways). But for all N, the full
set of candidate Box-Kites are viable when S≤ 8 or equal to some higher power of
2. For all other S values, though, the phenomenon of carrybit overflow intervenes
– leading, ultimately, to the “meta-fractal” behavior claimed in our abstract. To
see this, we need another mode of representation, less tied to 3-D visualizing, than
the Box-Kite can provide. The answer is a matrix-like method of tabulating the
products of candidate ZDs with each other, called Emanation Tables or ETs. The
L-indices only of all candidate ZDs are all we need indicate (the U-indices being
forced once G is specified); these will saturate the list of allowed indices < G,
save for the value of S whose choice, along with that of G, fixes an ET. Hence,
the unique ET for given G and S will fill a square spreadsheet whose edge has
length 2N−1 −2. Moreover, a cell entry (r,c) is only filled when row and column
labels R and C form a DMZ, which can never be the case along an ET’s long
diagonals: for the diagonal starting in the upper left corner, R xor R = 0, and
the two diagonals within the same Assessor, can never zero-divide each other; for
the righthand diagonal, the convention for ordering the labels (ascending counting
order from the left and top, with any such label’s strut-opposite index immediately
being entered in the mirror-opposite positions on the right and bottom) makes R
and C strut-opposites, hence also unable to form DMZs.
For the Sedenions, we get a 6 x 6 table, 12 of whose cells (those on long
diagonals) are empty: the 24 filled cells, then, correspond to the two-way traffic
of “edge-currents” one imagines flowing between vertices on a Box-Kite’s 12
edges. A computational corollary to the Roundabout Theorem, dubbed the Trip-
Count Two-Step, is of seminal importance. It connects this most basic theorem of
ETs to the most basic fact of associative triplets, indicated in the opening pages
of Part I, namely: for any N, the number TripN of associative triplets is found,
by simple combinatorics, to be (2N −1)(2N −2)/3! – 35 for the Sedenions, 155
for the Pathions, and so on. But, by Trip-Count Two-Step, we also know that the
maximum number of Box-Kites that can fill a 2N-ion ET = TripN−2. For S a power
of 2, beginning in the Pathions (for S= 25−2 = 8), the Number Hub Theorem says
the upper left quadrant of the ET is an unsigned multiplication table of the 2N−2-
ions in question, with the 0’s of the long diagonal (indicated Real negative units)
replaced by blanks – a result effectively synonymous with the Trip-Count Two-
Step.
Seventh thing: We found, as Part II’s argument wound down, that the 2 classes
of ETs found in the Pathions – the “normal” for S ≤ 8, filled with indices for all
7 possible Box-Kites, and the “sparse” so-called Sand Mandalas, showing only
3 Box-Kites when 8 < S < 16, were just the beginning of the story. A simple
formula involving just the bit-string of s and g, where the lowercase indicates the
values of S and G modulo G/2, gave the prototype of our first recipe: all and
only cells with labels R or C, or content P ( = R xor C ), are filled in the ET. The
4 “missing Box-Kites” were those whose L-index trip would have been that of a
Sail in the 2N−1 realm with S = s and G = g. The sequence of 7 ETs, viewed in
S-increasing succession, had an obvious visual logic leading to their being dubbed
a flip-book. These 7 were obviously indistinguishable from many vantages, hence
formed a spectrographic band. There were 3 distinct such bands, though, each
typified by a Box-Kite count common to all band-members, demonstrable in the
ETs for the 64-D Chingons. Each band contained S values bracketed by multiples
of 8 (either less than or equal to the higher, depending upon whether the latter
was or wasn’t a power of 2). These were claimed to underwrite behaviors in all
higher 2N-ion ETs, according to 3 rough patterns in need of algorithmic refining
in this Part III. Corresponding to the first unfilled band, with ETs always missing
4N−4 of their candidate Box-Kites for N > 4, we spoke of recursivity, meaning
the ETs for constant S and increasing N would all obey the same recipe, properly
abstracted from that just cited above, empirically found among the Pathions for
S > 8. The second and third behaviors, dubbed, for S ascending, (s,g)-modularity
and hide/fill involution respectively, make their first showings in the Chingons, in
the bands where 16 < S ≤ 24, and then where 24 < S < 32. In all such cases,
we are concerned with seeing the “period-doubling” inherent in CDP and Chaotic
attractors both become manifest in a repeated doubling of ET edge-size, leading
to the fixed-S, N increasing analog of the fixed-N,S increasing flip-books first ob-
served in the Pathions, which we call balloon-rides. Specifying and proving their
workings, and combining all 3 of the above-designated behaviors into the “funda-
mental theorem of zero-division algebra,” will be our goals in this final Part III.
Anyone who has read this far is encouraged to bring up the graphical complement
to this monograph, the 78-slide Powerpoint show presented at NKS 2006 [5], in
another window. (Slides will be referenced by number in what follows.)
2 8 < S < 16,N → ∞ : Recursive Balloon Rides in
the Whorfian Sky
We know that any ET for the 2N-ions is a square whose edge is 2N−1 − 2 cells.
How, then, can any simply recursive rule govern exporting the structure of one
such box to analogous boxes for progressively higher N? The answer: include
the label lines – not just the column and row headers running across the top and
left margins, but their strut-opposite values, placed along the bottom and right
margins, which are mirror-reversed copies of the label-lines (LLs) proper to which
they are parallel. This increases the edge-size of the ET box to 2N−1.
Theorem 11. For any fixed S > 8 and not a power of 2, the row and column indices
comprising the Label Lines (LLs) run along the left and top borders of the 2N-
ion ET ”spreadsheet” for that S. Treat them as included in the spreadsheet, as
labels, by adding a row and column to the given square of cells, of edge 2N−1−2,
which comprises the ET proper. Then add another row and column to include
the strut-opposite values of these labels’ indices in “mirror LLs,” running along
the opposite edges of a now 2N−1-edge-length box, whose four corner cells, like
the long diagonals they extend, are empty. When, for such a fixed S, the ET for
the 2N+1-ions is produced, the values of the 4 sets of LL indices, bounding the
contained 2N-ion ET, correspond, as cell values, to actual DMZ P-values in the
bigger ET, residing in the rows and columns labeled by the contained ET’s G and
X (the containing ET’s g and g+S). Moreover, all cells contained in the box they
bound in the containing ET have P-values (else blanks) exactly corresponding to –
and including edge-sign markings of – the positionally identical cells in the 2N-ion
ET: those, that is, for which the LLs act as labels.
Proof. For all strut constants of interest, S < g(= G/2); hence, all labels up
to and including that immediately adjoining its own strut constant (that is, the
first half of them) will have indices monotonically increasing, up to and at least
including the midline bound, from 1 to g − 1. When N is incremented by 1,
the row and column midlines separating adjoining strut-opposites will be cut and
pulled apart, making room for the labels for the 2N+1-ion ET for same S, which
middle range of label indices will also monotonically increase, this time from
the current 2N-ion generation’s g (and prior generation’s G), up to and at least
including its own midline bound, which will be g plus the number of cells in
the LL inherited from the prior generation, or g/2− 1. The LLs are therefore
contained in the rows and columns headed by g and its strut opposite, g+S. To
say that the immediately prior CDP generation’s ET labels are converted to the
current generation’s P-values in the just-specified rows and columns is equivalent
to asserting the truth of the following calculation:
(g+u)+(sg) · (G+g+uopp)
g + (G+g+S)
−(vz) · (G+uopp) +(vz) · (sg) ·u
+u − (sg) · (G+uopp)
0 only if vz = (−sg)
Here, we use two binary variables, the inner-sign-setting sg, and the Vent-or-
Zigzag test, based on the First Vizier. Using the two in tandem lets us handle the
normal and “Type II” box-kites in the same proof. Recall (and see Appendix B of
Part II for a quick refresher) that while the “Type I” is the only type we find in the
Sedenions, we find that a second variety emerges in the Pathions, indistinguishable
from Type I in most contexts of interest to us here: the orientation of 2 of the
3 struts will be reversed (which is why VZ1 and VZ3 are only true generally
when unsigned). For a Type I, since S < g, we know by Rule 1 that we have the
trip (S,g,g+S); hence, g – for all 2N-ions beyond the Pathions, where the Sand
Mandalas’ g = 8 is the L-index of the Zigzag B Assessor – must be a Vent (and
its strut-opposite, g+S, a Zigzag). For a Type II, however, this is necessarily so
only for 1 of the 3 struts – which means, per the equation above, that sg must be
reversed to obtain the same result. Said another way, we are free to assume either
signing of vz means +1, so the “only if” qualifying the zero result is informative.
It is u and its relationship to g+ u that is of interest here, and this formulation
makes it easier to see that the products hold for arbitrary LL indices u or their
strut-opposites. But for this, the term-by-term computations should seem routine:
the left bottom is the Rule 1 outcome of (u,g,g+ u): obviously, any u index
must be less than g. To its right, we use the trip (uopp,g,g+ uopp) → (G+ g+
uopp,g,G+uopp), whose CPO order is opposite that of the multiplication. For the
top left, we use (u,S,uopp) as limned above, then augment by g, then G, leaving
uopp unaffected in the first augmenting, and g + u in the second. Finally, the
top right (ignoring sg and vz momentarily) is obtained this way: (u,S,uopp) →
(u,g+uopp,g+S)→ (u,G+g+ s,G+g+uopp); ergo, +u.
Note that we cannot eke out any information about edge-sign marks from this
setup: since labels, as such, have no marks, we have nothing to go on – unlike all
other cells which our recursive operations will work on. Indeed, the exact algo-
rithmic determination of edge-sign marks for labels is not so trivial: as one iterates
through higher N values, some segments of LL indexing will display reversals of
marks found in the ascending or descending left midline column, while other seg-
ments will show them unchanged – with key values at the beginnings and ends
of such octaves (multiples of 8, and sums of such multiples with S mod 8) some-
times being reversed or kept the same irrespective of the behavior of the terms
they bound. Fortunately, such behaviors are of no real concern here – but they are,
nevertheless, worth pointing out, given the easy predictability of other edge-sign
marks in our recursion operations.
Now for the ET box within the labels: if all values (including edge-sign marks)
remain unchanged as we move from the 2N-ion ET to that for the 2N+1-ions, then
one of 3 situations must obtain: the inner-box cells have labels u,v which belong to
some Zigzag L-trip (u,v,w); or, on the contrary, they correspond to Vent L-indices
– the first two terms in the CPO triplet (wopp,vopp,u), for instance; else, finally,
one term is a Vent, the other a Zigzag (so that inner-signs of their multiplied dyads
are both positive): we will write them, in CPO order, vopp and u, with third trip
member wopp. Clearly, we want all the products in the containing ET to indicate
DMZs only if the inner ET’s cells do similarly. This is easily arranged: for the
containing ET’s cells have indices identical to those of the contained ET’s, save
for the appending of g to both (and ditto for the U-indices).
Case 1: If (u,v,w) form a Zigzag L-index set, then so do (g+ v,g+u,w), so
markings remain unchanged; and if the (u,v) cell entry is blank in the contained,
so will be that for (g+ u,g+ v) in its container. In other words, the following
holds:
(g+ v)+(sg) · (G+g+ vopp)
(g+u) + (G+g+uopp)
−(G+wopp) − (sg) ·w
−w − (sg) · (G+wopp)
0 only if sg = (−1)
(g+u) · (g+ v) = P : (u,v,w)→ (g+ v,g+u,w); hence, (−w).
(g+ u) · (sg) · (G+ g+ vopp) = P : (u,wopp,vopp) → (g+ vopp,wopp,g+ u)
→ (G+wopp,G+g+ vopp,g+u); hence, (sg) · (−(G+wopp)).
(G+g+uopp) · (g+ v) = P : (uopp,wopp,v)→ (g+ v,wopp,g+uopp) → (g+
v,G+g+uopp,G+wopp); hence, (−(G+wopp)).
(G+g+uopp) ·(G+g+vopp) = P : Rule 2 twice to the same two terms yields
the same result as the terms in the raw, hence (−w).
Clearly, cycling through (u,v,w) to consider (g+ v) · (g+ w) will give the
exactly analogous result, forcing two (hence three) negative inner-signs in the
candidate Sail; hence, if we have DMZs at all, we have a Zigzag Sail.
Case 2: The product of two Vents must have negative edge-sign, and there’s
no cycling through same-inner-signed products as with the Zigzag, so we’ll just
write our setup as a one-off, with upper inner-sign explicitly negative, and claim
its outcome true.
(g+ vopp)− (G+g+ v)
(g+wopp) + (G+g+w)
+(G+uopp) +u
−u − (G+uopp)
(g+wopp) · (g+ vopp) = P : (wopp,vopp,u) → (g+ vopp,g+wopp,u); hence,
(−u).
(g+wopp) · (G+g+v) = P : (wopp,v,uopp)→ (g+v,g+wopp,uopp)→ (G+
uopp,g+wopp,G+ g+ v); but inner sign of upper dyad is negative, so (−(G+
uopp)).
(G+g+w) · (g+vopp) = P : (vopp,uopp,w)→ (g+w,uopp,g+vopp)→ (G+
uopp,G+g+w,g+ vopp); hence, (+(G+uopp)).
(G+g+w) · (G+g+ v) = P : Rule 2 twice to the same two terms yields the
same result as the terms in the raw; but inner sign of upper dyad is negative, so
(+u).
Case 3: The product of Vent and Zigzag displays same inner sign in both
dyads; hence the following arithmetic holds:
(g+u)+(G+g+uopp)
(g+ vopp) + (G+g+ v)
−(G+w) +wopp
−wopp +(G+w)
The calculations are sufficiently similar to the two prior cases as to make their
writing out tedious. It is clear that, in each of our three cases, content and marking
of each cell in the contained ET and the overlapping portion of the container ET
are identical. �
To highlight the rather magical label/content involution that occurs when N is
in- or de- cremented, graphical realizations of such nested patterns, as in Slides
60-61, paint LLs (and labels proper) a sky-blue color. The bottom-most ET being
overlaid in the central box has g = the maximum high-bit in S, and is dubbed the
inner skybox. The degree of nesting is strictly measured by counting the number
of bits B that a given skybox’s g is to the left of this strut-constant high-bit. If we
partition the inner skybox into quadrants defined by the midlines, and count the
number Q of quadrant-sized boxes along one or the other long diagonal, it is obvi-
ous that the inner skybox itself has B = 0 and Q = 1; the nested skyboxes contain-
ing it have Q = 2B. If recursion of skybox nesting be continued indefinitely – to
the fractal limit, which terminology we will clarify shortly – the indices contained
in filled cells of any skybox can be interpreted in B distinct ways, B → ∞, as rep-
resentations of distinct ZDs with differing G and, therefore, differing U-indices.
By obvious analogy to the theory of Riemann surfaces in complex analysis, each
such skybox is a separate “sheet”; as with even such simple functions as the log-
arithmic, the number of such sheets is infinite. We could then think of the infinite
sequence of skyboxes as so many cross-sections, at constant distances, of a flash-
light beam whose intensity (one over the ET’s cell count) follows Kepler’s inverse
square law. Alternatively, we could ignore the sheeting and see things another
Where we called fixed-N, S varying sequences of ETs flip-books, we refer to
fixed-S, N varying sequences as balloon rides: the image is suggested by David
Niven’s role as Phineas Fogg in the movie made of Jules Vernes’ Around the World
in 80 Days: to ascend higher, David would drop a sandbag over the side of his hot-
air balloon’s basket; if coming down, he would pull a cord that released some of
the balloon’s steam. Each such navigational tactic is easy to envision as a bit-
shift, pushing G further to the left to cross LLs into a higher skybox, else moving
Figure 1: ETs for S=15, N=5,6,7 (nested skyboxes in blue)· · · and “fractal limit.”
it rightward to descend. Using S = 15 as the basis of a 3-stage balloon-ride, we
see how increasing N from 5 to 6 to 7 approaches the white-space complement
of one of the simplest (and least efficient) plane-filling fractals, the Cesàro double
sweep [6, p. 65].
The graphics were programmatically generated prior to the proving of the the-
orems we’re elaborating: their empirical evidence was what informed (indeed,
demanded) the theoretical apparatus. And we are not quite finished with the cur-
rent task the apparatus requires of us. We need two more theorems to finish the
discussion of skybox recursion. For both, suppose some skybox with B = k, k any
non-negative integer, is nested in one with B = k + 1. Divide the former along
midlines to frame its four quadrants, then block out the latter skybox into a 4×4
grid of same-sized window panes, partitioned by the one-cell-thick borders of
its own midlines into quadrants, each of which is further subdivided by the out-
side edges of the 4 one-cell-thick label lines and their extensions to the window’s
frame. These extended LLs are themselves NSLs, and have R,C values of g and
g+S; for S = 15, they also adjoin NSLs along their outer edges whose R,C val-
ues are multiples of 8 plus S mod 8. These pane-framing pairs of NSLs we will
henceforth refer to (as a windowmaker would) as muntins. It is easy to calculate
that while the inner skybox has but one muntin each among its rows and columns,
each further nesting has 2B+1 − 1. But we are getting ahead of ourselves, as we
still have two proofs to finish. Let’s begin with Four Corners, or
Theorem 12. The 4 panes in the corners of the 16-paned B = k + 1 window are
identical in contents and marks to the analogously placed quadrants of the B = k
skybox.
Proof. Invoke the Zero-Padding Lemma with regard to the U-indices, as the labels
of the boxes in the corners of the B = k+1 ET are identical to those of the same-
sized quadrants in the B = k ET, all labels ≥ the latter’s g only occurring in the
newly inserted region. �
Remarks. For N = 6, all filled Four Corners cells indicate edges belonging to
3 Box-Kites, whose edges they in fact exhaust. These 3, not surprisingly, are
the zero-padded versions of the identically L-indexed trio which span the entirety
of the N = 5 ET. By calculations we’ll see shortly, however, the inner skybox,
when considered as part of the N = 6 ET, has filled cells belonging to all the
other 16 Box-Kites, even though the contents of these cells are identical to those
in the N = 5 ET. As B increases, then, the “sheets” covering this same central
region must draw upon progressively more extensive networks of interconnected
Box-Kites. As we approach the fractal limit – and “the Sky is the limit” – these
networks hence become scale-free. (Corollarily, for N = 7, the Four Corners’ cells
exhaust all the edges of the N = 6 ET’s 19 Box-Kites, and so on.)
Unlike a standard fractal, however, such a Sky merits the prefix “meta”: for
each empty ET cell corresponds to a point in the usual fractal variety; and each
pair of filled ET cells, having (r,c) of one = (c,r) of the other), correspond to
diagonal-pairs in Assessor planes, orthogonal to all other such diagonal-pairs be-
longing to the other cells. Each empty ET cell, in other words, not only corre-
sponds to a point in the usual plane-confined fractal, but belongs to the comple-
ment of the filled cells’ infinite number of dimensions framing the Sky’s meta-
fractal.
We’ve one last thing to prove here. The French Windows Theorem shows us
the way the cell contents of the pairs of panes contained between the B = k+ 1
skybox’s corners are generated from those of the analogous pairings of quadrants
in the B = k skybox, by adding g to L-indices.
Theorem 13. For each half-square array of cells created by one or the other midline
(the French windows), each cell in the half-square parallel to that adjoining the
midline (one of the two shutters), but itself adjacent to the label-line delimiting
the former’s bounds, has content equal to g plus that of the cell on the same line
orthogonal to the midline, and at the same distance from it, as it is from the label-
line. All the empty long-diagonal cells then map to g (and are marked), or g+S
(and are unmarked). Filled cells in extensions of the label-lines bounding each
shutter are calculated similarly, but with reversed markings; all other cells in a
shutter have the same marks as their French-window counterparts.
Preamble. Note that there can be (as we shall see when we speak of hide/fill
involution) cells left empty for rule-based reasons other than P = R⊻C = 0 | S.
The shutter-based counterparts of such French-window cells, unlike those of long-
diagonal cells, remain empty.
Proof. The top and left (bottom and right) shutters are equivalent: one merely
switches row for column labels. Top/left and bottom/right shutter-sets are likewise
equivalent by the symmetry of strut-opposites. We hence make the case for the
left shutter only. But for the novelties posed by the initially blank cells and the
label lines (with the only real subtleties involving markings), the proof proceeds
in a manner very similar to Theorem 11: split into 3 cases, based on whether (1)
the L-index trip implied by the R,C,P values is a Zigzag; (2) u,v are both Vents;
or, (3) the edge signified by the cell content is the emanation of same-inner-signed
dyads (that is, one is a Vent, the other a Zigzag).
Case 1: Assume (u,v,w) a Zigzag L-trip in the French window’s contained
skybox; the general product in its shutter is
v − (G+ vopp)
(g+u) + (G+g+uopp)
−(G+g+wopp) +(g+w)
−(g+w) +(G+g+wopp)
(g+u) · v = P : (u,v,w)→ (g+w,v,g+u); hence, (−(g+w)).
(g+ u) · (G+ vopp) = P : (u,wopp,vopp) → (g+wopp,g+ u,vopp) → (G+
vopp,g+u,G+g+wopp); dyads’ opposite inner signs make (G+g+wopp) pos-
itive.
(G+g+uopp) · v = P : (uopp,wopp,v) → (g+wopp,g+uopp,v) → (G+g+
uopp,G+g+wopp,v); hence, (−(G+g+wopp)).
(G+g+uopp) · (G+ vopp) = P : (vopp,uopp,w) → (vopp,g+w,g+uopp) →
(G+g+uopp,g+w,G+vopp); dyads’ opposite inner signs make (g+w) positive.
Case 2: The product of two Vents must have negative edge-sign, hence nega-
tive inner sign in top dyad to lower dyad’s positive. The shutter product thus looks
like this:
(uopp)− (G+u)
(g+ vopp) + (G+g+ v)
+(G+g+wopp) +(g+w)
−(g+w) − (G+g+wopp)
(g+vopp) ·uopp = P : (vopp,uopp,w)→ (g+w,uopp,g+vopp); hence, (−(g+
(g+ vopp) · (G+ u) = P : (vopp,u,wopp) → (g+wopp,u,g+ vopp) → (G+
u,G+ g+wopp,g+ vopp); but dyads’ inner signs are opposite, so (−(G+ g+
wopp)).
(G+g+v) ·uopp = P : (uopp,wopp,v)→ (uopp,g+v,g+wopp)→ (uopp,G+
g+wopp,G+g+ v); hence, (+(G+g+wopp)).
(G+g+v) ·(G+u)= P : (u,v,w)→ (u,g+w,g+v)→ (G+g+v,g+w,G+
u); but dyads’ inner signs are opposite, so (+(g+w)).
Case 3: The product of Vent and Zigzag displays same inner sign in both
dyads; hence the following arithmetic holds:
(uopp)+(G+u)
(g+ v) + (G+g+ vopp)
+(G+g+w) +(g+wopp)
−(g+wopp) − (G+g+w)
As with the last case in Theorem 11, we omit the term-by-term calculations for
this last case, as they should seem “much of a muchness” by this point. What is
clear in all three cases is that index values of shutter cells have same markings
as their French-window counterparts, at least for all cells which have markings in
the contained skybox; but, in all cases, indices are augmented by g.
The assignment of marks to the shutter-cells linked to blank cells in French
windows is straightforward for Type I box-kites: since any containing skybox
must have g > S, and since g+ s has g as its strut opposite, then the First Vizier
tells us that any g must be a Vent. But then the R,C indices of the cell containing
g must belong to a Trefoil in such a box-kite; hence, one is a Vent, the other a
Zigzag, and g must be marked. Only if the R,C,P entry in the ET is necessar-
ily confined to a Type II box-kites will this not necessarily be so. But Part II’s
Appendix B made clear that Type II’s are generated by excluding g from their
L-indices: recall that, in the Pathions, for all S ¡ 8, all and only Type II box-kites
are created by placing one of the Sedenion Zigzag L-trips on the “Rule 0” circle
of the PSL(2,7) triangle with 8 in the middle (and hence excluded). This is a box-
kite in its own right (one of the 7 “Atlas” box-kites with S = 8); its 3 sides are
“Rule 2” triplets, and generate Type II box-kites when made into zigzag L-index
sets. Conversely, all Pathion box-kites containing an ’8’ in an L-index (dubbed
”strongboxes” in Appendix B) are Type I. Whether something peculiar might oc-
cur for large N (where there might be multiple powers of 2 playing roles in the
same box-kite) is a matter of marginal interest to present concerns, and will be left
as an open question for the present. We merely note that, by a similar argument,
and with the same restrictions assumed, g+S must be a Zigzag L-index, and R,C
either both be likewise (hence, g+S is unmarked); or, both are Vents in a Trefoil
(so g+S must be unmarked here too).
The last detail – reversal of label-line markings in their g-augmented shutter-
cell extensions – is demonstrated as follows, with the same caveat concerning
Type II box-kites assumed to apply. Such cells house DMZs (just swap u for g+u
in Theorem 11’s first setup – they form a Rule 1 trip – and compute). The LL
extension on top has row-label g; that along the bottom, the strut-opposite g+S.
Given trip (u,v,w), the shutter-cell index for R,C = (g,u) corresponds to French-
window index for R,C = (g,g+u). But (u,g,g+u) is a Trefoil, since g is a Vent.
So if u is one too, g+u isn’t; hence marks are reversed as claimed. �
3 Maximal High-Bit Singletons: (s,g)-Modularity for
16 < S ≤ 24
The Whorfian Sky, having but one high bit in its strut constant, is the simplest
possible meta-fractal – the first of an infinite number of such infinite-dimensional
zero-divisor-spanned spaces. We can consider the general case of such single-
ton high-bit recursiveness in two different, complementary ways. First, we can
supplement the just-concluded series of theorems and proofs with a calculational
interlude, where we consider the iterative embeddings of the Pathion Sand Man-
dalas in the infinite cascade of boxes-within-boxes that a Sky oversees. Then, we
can generalize what we saw in the Pathions to consider the phenomenology of
strut constants with singleton high-bits, which we take to be any bits representing
a power of 2 ≥ 3 if S contains low bits (is not a multiple of 8), else a power of 2
strictly greater than 3 otherwise. Per our earlier notation, g = G/2 is the highest
such singleton bit possible. We can think of its exponential increments – equiva-
lent to left-shifts in bit-string terms – as the side-effects of conjoint zero-padding
of N and S. This will be our second topic in this section.
Maintaining our use of S = 15 as exemplary, we have already seen that NSLs
come in quartets: a row and column are each headed by S mod g (henceforth, s)
and g, hence 7 and 8 in the Sand Mandalas. But each recursive embedding of the
current skybox in the next creates further quartets. Division down the midlines
to insert the indices new to the next CDP generation induces the Sand Mandala’s
adjoining strut-opposite sets of s and g lines (the pane-framing muntins) to be
displaced to the borders of the four corners and shutters, with the new skybox’s
g and g+ s now adjoining the old s and g to form new muntins, on the right and
left respectively, while g+g/2 (the old G+g) and its strut opposite form a third
muntin along the new midlines. Continuing this recursive nesting of skyboxes
generates 1, 3, 7, · · ·, 2B+1 −1 row-and-column muntin pairs involving multiples
of 8 and their supplementings by s, where (recalling earlier notation) B = 0 for the
inner skybox, and increments by 1 with each further nesting. Put another way, we
then have a muntin number µ = (2N−4 −1), or 4µ NSL’s in all.
The ET for given N has (2N−1 −2) cells in each row and column. But NSLs
divvy them up into boxes, so that each line is crossed by 2µ others, with the 0,
2 or 4 cells in their overlap also belonging to diagonals. The number of cells in
the overlap-free segments of the lines, or ω , is then just 4µ · (2N−1 − 2− 2µ) =
24µ(µ +1): an integer number of Box-Kites. For our S = 15 case, the minimized
line shuffling makes this obvious: all boxes are 6 x 6, with 2-cell-thick boundaries
(the muntins separating the panes), with µ boundaries, and (µ + 1) overlap-free
cells per each row or column, per each quartet of lines.
The contribution from diagonals, or δ , is a little more difficult, but straight-
forward in our case of interest: 4 sets of 1,2,3, · · · ,µ boxes are spanned by mov-
ing along one empty long diagonal before encountering the other, with each box
contributing 6, and each overlap zone between adjacent boxes adding 2. Hence,
δ = 24 · (2N−3 − 1)(2N−3 − 2)/6 – a formula familiar from associative-triplet
counting: it also contributes an integer number of Box-Kites. The one-liner we
want, then, is this:
BKN, 8<S<16 = ω +δ = (2N−4)(2N−4 −1) + (2N−3 −1)(2N−3 −2)/6
For N = 4,5,6,7,8,9,10, this formula gives 0,3,19,91,395, 1643,6699. Add
4N−4 to each – the immediate side-effect of the offing of all four Rule 0 candidate
trips of the Sedenion Box-Kite exploded into the Sand Mandala that begins the
recursion – and one gets “déjà vu all over again”: 1, 7, 35, 155, 651, 2667, 10795
– the full set of Box-Kites for S ≤ 8.
It would be nice if such numbers showed up in unsuspected places, having
nothing to do with ZDs. Such a candidate context does, in fact, present itself, in Ed
Pegg’s regular MAA column on “Math Games” focusing on “Tournament Dice.”
[7] He asks us, “What dice make a non-transitive four player game, so that if three
dice are chosen, a fourth die in the set beats all three? How many dice are needed
for a five player non-transitive game, or more?” The low solution of 3 explicitly
involves PSL(2,7); the next solution of 19 entails calculations that look a lot like
those involved in computing row and column headers in ETs. No solutions to the
dice-selecting game beyond 19 are known. The above formulae, though, suggest
the next should be 91. Here, ZDs have no apparent role save as dummies, like the
infinity of complex dimensions in a Fourier-series convergence problem, tossed
out the window once the solution is in hand. Can a number-theory fractal, with
intrinsically structured cell content (something other, non-meta, fractals lack) be
of service in this case – and, if not in this particular problem, in others like it?
Now let’s consider the more general situation, where the singleton high-bit
can be progressively left-shifted. Reverting to the use of the simplest case as
exemplary, use S = g+1 = 9 in the Pathions, then do tandem left-shifts to pro-
duce this sequence: N = 6, S = g+1 = 17; N = 7, S = g+1 = 33; · · · ; N = K,
S = g+1 = 2K−2 + 1. A simple rule governs these ratchetings: in all cases, the
number of filled cells = 6 · (2N−1 − 4), since there are two sets of parallel sides
which are filled but for long-diagonal intersections, and two sets of g and 1 entries
distributed one per row along orthogonals to the empty long diagonals. Hence,
for the series just given, we have cell counts of 72, 168, · · · , 6 · (2N−1 − 4) for
BKN, S = 3, 7, · · · , 2
N−3 − 1, for g < S < g + 8 = G in the Pathions, and all
g < S ≤ g+8 in the Chingons, 27-ions, and general 2N-ions, in that order.
Algorithmically, the situation is just as easy to see: the splitting of dyads,
sending U- and L- indices to strut-opposite Assessors, while incorporating the S
and G of the current CDP generation as strut-opposites in the next, continues. For
S = 17 in the Chingons, there are now 2N−3 −1 = 7, not 3, Box-Kites sharing the
new g = 16 (at B) and S mod g = 1 (at E) in our running example. The U- indices
of the Sand Mandala Assessors for S = g+1 = 9 are now L-indices, and so on:
every integer < G and 6= S gets to be an L-index of one of the 30(= 2N−1 − 2)
Assessors, as 16 and S mod g = 1 appear in each of the 7 Box-Kites, with each
other eligible integer appearing once only in one of the 7 ·4= 28 available L-index
slots.
As an aside, in all 7 cases, writing the smallest Zigzag L-index at a mandates
all the Trefoil trips be “precessed” – a phenomenon also observed in the S = 8
Pathion case, as tabulated on p. 14 of [8]. For Zigzag L-index set (2,16,18),
for instance, (a,d,e) = (2,3,1) instead of (1,2,3); ( f ,c,e) = (19,18,1) not
(1,19,18); and ( f ,d,b) = (19,3,16). But otherwise, there are no surprises: for
N = 7, there are (27−3 − 1) = 15 Box-Kites, with all 62(= 2N−1 − 2) available
cells in the rows and columns linked to labels g and S mod g being filled, and so
Note that this formulation obtains for any and all S > 8 where the maximum
high-bit (that is, g) is included in its bitstring: for, with g at B and S mod g at
E, whichever R,C label is not one of these suffices to completely determine the
remaining Assessor L-indices, so that no other bits in S play a role in determining
any of them. Meanwhile, cell contents P containing either g or S mod g, but
created by XORing of row and column labels equal to neither, are arrayed in off-
diagonal pairs, forming disjoint sets parallel or perpendicular to the two empty
ones. If we write S mod g with a lower-case s, then we could call the rule in
play here (s,g)-modularity. Using the vertical pipe for logical or, and recalling the
special handling required by the 8-bit when S is a multiple of 8 (which we signify
with the asterisk suffixed to “mod”), we can shorthand its workings this way:
Theorem 14. For a 2N-ion inner skybox whose strut constant S has a singleton
high-bit which is maximal (that is, equal to g = G/2 = 2N−2), the recipe for its
filled cells can be condensed thus:
R | C| P = g | S mod∗ g
Under recursion, the recipe needs to be modified so as to include not just the
inner-skybox g and S mod∗ g (henceforth, simply lowercase s), but all integer
multiples k of g less than the G of the outermost skybox, plus their strut opposites
k ·g+ s.
Proof. The theorem merely boils down the computational arguments of prior para-
graphs in this section, then applies the last section’s recursive procedures to them.
The first claim of the proof is identical to what we’ve already seen for Sand Man-
dalas, with zero-padding injected into the argument. The second claim merely
assumes the area quadrupling based on midline splitting, with the side-effects al-
ready discussed. No formal proof, then, is called for beyond these points. �
Remarks. Using the computations from two paragraphs prior to the theorem’s
statement, we can readily calculate the box-kite count for any skybox, no matter
how deeply nested: recall the formula 6 · (2N−1 − 4) for BKN, S = 2
N−3 − 1. It
then becomes a straightforward matter to calculate, as well, the limiting ratio of
this count to the maximal full count possible for the ET as N → ∞, with each cell
approaching a point in a standard 2-D fractal. Hence, for any S with a singleton
high-bit in evidence, there exists a Sky containing all recursive redoublings of its
inner skybox, and computations like those just considered can further be used to
specify fractal dimensions and the like. (Such computations, however, will not
concern us.) Finally, recall that, by spectrographic equivalence, all such compu-
tations will lead to the same results for each S value in the same spectral band or
octave.
4 Hide/Fill Involution: Further-Right High-Bits with
24 < S < 32.
Recall that, in the Sand Mandala flip-book, each increment of S moved the two
sets of orthogonal parallel lines one cell closer toward their opposite numbers:
while S = 9 had two filled-in rows and columns forming a square missing its cor-
ners, the progression culminating in S = 15 showed a cross-hairs configuration:
the parallel lines of cells now abutted each other in 2-ply horizontal and vertical
arrays. The same basic progression is on display in the Chingons, starting with
S = 17. But now the number of strut-opposite cell pairs in each row and column
is 15, not 7, so the cross-hairs pattern can’t arise until S = 31. Yet it never arises
in quite the manner expected, as something quite singular transpires just after flip-
ping past the ET in the middle, for S = 24. Here, rows and columns labeled 8 and
16 constrain a square of empty cells in the center · · · quickly followed by an ET
which seems to continue the expected trajectory – except that almost all the non-
long-diagonal cells left empty in its predecessor ETs are now inexplicably filled.
More, there is a method to the “almost all” as well: for we now see not 2, but 4
rows and columns, all being blanked out while those labeled with g and S mod g
are being filled in.
This is an inevitable side effect of a second high-bit in S: we call this phe-
nomenon, first appearing in the Chingons, hide/fill involution. There are 4, not 2,
line-pairs, because S and G, modulo a lower power of 2 (because devolving upon
a prior CDP generation’s g), offer twice the possibilities: for S = 25, S mod 16 is
now 9, but S mod 8 can result in either 1 or 17 as well – with correlated multiples
of 8 (8 proper, and 24) defining the other two pairings. All cells with R |C | P
equal to one of these 4 values, but for the handful already set to “on” by the first
high-bit, will now be set to “off,” while all other non-long-diagonal cells set to
“off” in the Pathion Sand Mandalas are suddenly “on.” What results for each
Chingon ET with 24 < S < 32 is an ensemble comprised of 23 Box-Kites. (For
the flip-book, see Slides 40 – 54.) Why does this happen? The logic is as straight-
forward as the effect can seem mysterious, and is akin, for good reason, to the
involutory effect on trip orientation induced by Rule 2 addings of G to 2 of the
trip’s 3 indices.
In order to grasp it, we need only to consider another pair of abstract calcula-
tion setups, of the sort we’ve seen already many times. The first is the core of the
Two-Bit Theorem, which we state and prove as follows:
Theorem 15. 2N-ion dyads making DMZs before augmenting S with a new high-bit
no longer do so after the fact.
Proof. Suppose the high-bit in the bitstring representation of S is 2K, K < (N −
1). Suppose further that, for some L-index trip (u,v,w), the Assessors U and V
are DMZ’s, with their dyads having same inner signs. (This last assumption is
strictly to ease calculations, and not substantive: we could, as earlier, use one
or more binary variables of the sg type to cover all cases explicitly, including
Type I vs. Type II box-kites. To keep things simple, we assume Type I in what
follows.) We then have (u+ u ·X)(v+ v · X) = (u+U)(v+V ) = 0. But now
suppose, without changing N, we add a bit somewhere further to the left to S, so
that S < (2K = L) < G. The augmented strut constant now equals SL = S+L.
One of our L-indices, say v, belongs to a Vent Assessor thanks to the assumed
inner signing; hence, by Rule 2 and the Third Vizier, (V,v,X)→ (X +L,v,V +L).
Its DMZ partner u, meanwhile, must thereby be a Zigzag L-index, which means
(u,U,X)→ (u,X +L,U +L). We claim the truth of the following arithmetic:
v + (V +L)
u + (U +L)
+(W + L) +w
+ w − (W +L)
NOT ZERO (+w’s don’t cancel)
The left bottom product is given. The product to its right is derived as follows:
since u is a Zigzag L-index, the Trefoil U-trip (u,V,W) has the same orientation as
(u,v,w), so that Rule 2 → (u,W +L,V +L), implying the negative result shown.
The left product on the top line, though, has terms derived from a Trefoil U-trip
lacking a Zigzag L-index, so that only after Rule 2 reversal are the letters arrayed
in Zigzag L-trip order: (U +L,v,W +L). Ergo, +(W +L). Similarly for the top
right: Rule 2 reversal “straightens out” the Trefoil U-trip, to give (U+L,V +L,w);
therefore, (+w) results. If we explicitly covered further cases by using an sg
variable, we would be faced with a Theorem 2 situation: one or the other product
pair cancels, but not both. �
Remark. The prototype for the phenomenon this theorem covers is the “explo-
sion” of a Sedenion box-kite into a trio of interconnected ones in a Pathion sand
mandala, with the S of the latter = the X of the former. As part of this process, 4 of
the expected 7 are “hidden” box-kites (HBKs), with no DMZs along their edges.
These have zigzag L-trips which are precisely the L-trips of the 4 Sedenion Sails.
Here, an empirical observation which will spur more formal investigations in a
sequel study: for the 3 HBKs based on trefoil L-trips, exactly 1 strut has reversed
orientation (a different one in each of them), with the orientation of the triangular
side whose midpoint it ends in also being reversed. For the HBK based on the
zigzag L-trip, all 3 struts are reversed, so that the flow along the sides is exactly
the reverse of that shown in the “Rule 0” circle. (Hence, all possible flow patterns
along struts are covered, with only those entailing 0 or 2 reversals corresponding
to functional box-kites: our Type I and Type II designations.) It is not hard to show
that this zigzag-based HBK has another surprising property: the 8 units defined
by its own zigzag’s Assessors plus X and the real unit form a ZD-free copy of the
Octonions. This is also true when the analogous Type II situation is explored, al-
beit for a slightly different reason: in the former case, all 3 Catamaran “twistings”
take the zigzag edges to other HBKs; in the latter, though, the pair of Assessors in
some other Type II box-kite reached by “twisting” – (a,B) and (A,b), say, if the
edge be that joining Assessors A and B, with strut-constant copp = d – are strut
opposites, and hence also bereft of ZDs. The general picture seems to mirror this
concrete case, and will be studied in “Voyage by Catamaran” with this expecta-
tion: the bit-twiddling logic that generates meta-fractal “Skies” also underwrites
a means for jumping between ZD-free Octonion clones in an infinite number of
HBKs housed in a Sky. Given recent interest in pure “E8” models giving a privi-
leged place to the basis of zero-divisor theory, namely “G2” projections (viz., A.
Garrett Lisi’s “An Exceptionally Simple Theory of Everything”); a parallel vogue
for many-worlds approaches; and, the well-known correspondence between 8-D
closest-packing patterns, the loop of the 240 unit Octonions which Coxeter dis-
covered, and E8 algebras – given all this, tracking the logic of the links across
such Octonionic “brambles” might prove of great interest to many researchers.
Now, we still haven’t explained the flipside of this off-switch effect, to which
prior CDP generation Box-Kites – appropriately zero-padded to become Box-
Kites in the current generation until the new high-bit is added to the strut-constant
– are subjected. How is it that previously empty cells not associated with the sec-
ond high-bit’s blanked-out R, C, P values are now full? The answer is simple, and
is framed in the Hat-Trick Theorem this way.
Theorem 16. Cells in an ET which represent DMZ edges of some 2N-ion Box-
Kites for some fixed S, and which are offed in turn upon augmenting of S by a
new leftmost bit, are turned on once more if S is augmented by yet another new
leftmost bit.
Proof. We begin an induction based upon the simplest case (which the Chingons
are the first 2N-ions to provide): consider Box-Kites with S ≤ 8. If a high-bit
be appended to S, then the associated Box-Kites are offed. However, if another
high-bit be affixed, these dormant Box-Kites are re-awakened – the second half of
hide/fill involution. We simply assume an L-index set (u,v,w) underwriting a Sail
in the ET for the pre-augmented S, with Assessors (u,U) and (v,V ). Then, we
introduce a more leftified bit 2Q = M, where pre-augmented S < L < M < G, then
compute the term-by term products of (u+(U +L+M)) and (v+ sg · (V +L+
M)), using the usual methods. And as these methods tell us that two applications
of Rule 2 have the same effect as none in such a setup, we have no more to prove.
Corollary. The induction just invoked makes it clear that strut constants equal to
multiples of 8 not powers of 2 are included in the same spectral band as all other
integers larger than the prior multiple. The promissory note issued in the second
paragraph of Part II’s concluding section, on 64-D Spectrography, can now be
deemed redeemed.
In the Chingons, high-bits L and M are necessarily adjacent in the bitstring for
S < G = 32; but in the general 2N-ion case, N large, zero-padding guarantees that
things will work in just the same manner, with only one difference: the recursive
creation of “harmonics” of relatively small-g (s,g)-modular R,C,P values will
propagate to further levels, thereby effecting overall Box-Kite counts.
In general terms, we have echoes of the formula given for (s,g)-modular cal-
culations, but with this signal difference: there will be one such rule for each
high-bit 2H in S, where residues of S modulo 2H will generate their own near-
solid lines of rows and columns, be they hidden or filled. Likewise for multiples
of 2H <G which are not covered by prior rules, and multiples of 2H supplemented
by the bit-specific residue (regardless of whether 2H itself is available for treat-
ment by this bit-specific rule). In the simplest, no-zero-padding instances, all even
multiples are excluded, as they will have occurred already in prior rules for higher
bits, and fills or hides, once fixed by a higher bit’s rule, cannot be overridden.
Cases with some zero-padding are not so simple. Consider this two-bit in-
stance, S = 73,N = 8: the fill-bit is 64, the hide-bit is just 8, so that only 9 and 64
generate NSLs of filled values; all other multiples of 8, and their supplementing
by 1 (including 65) are NSLs of hidden values. Now look at a variation on this
example, with the single high-bit of zero-padding removed – i. e., S = 41,N = 8.
Here, the fill-bit is 32, and its multiples 64 and 96, as well as their supplements by
S modulo 32 = 9, or 9 and 73 and 105, label NSLs of filled values; but all other
multiples of 8, plus all multiples of 8 supplemented by 1 not equal to 9 or 73 or
105, label NSLs of hidden values. Cases with multiple fill and hide bits, with or
without additional zero-padding, are obviously even more complicated to handle
explicitly on a case-by-case basis, but the logic framing the rules remain simple;
hence, even such messy cases are programmatically easy to handle.
Hide/fill involution means, then, that the first, third, and any further odd-
numbered high-bits (counting from the left) will generate “fill” rules, whereas
all the even-numbered high-bits generate “hide” rules – with all cells not touched
by a rule being either hidden (if the total number of high-bits B is odd) or filled (B
is even).
Two further examples should make the workings of this protocol more clear.
First, the Chingon test case of S = 25: for (R | C | P = 9 | 16), all the ET cells are
filled; however, for (R | C | P = 1 | 8 | 17 | 24), ET cells not already filled by the
first rule (and, as visual inspection of Slide 48 indicates, there are only 8 cells in
the entire 840-cell ET already filled by the prior rule which the current rule would
like to operate on) are hidden from view. Because the 16- and 8- bits are the only
high-bits, the count of same is even, meaning all remaining ET cells not covered
by these 2 rules are filled.
We get 23 for Box-Kite count as follows. First, the 16-bit rule gives us 7 Box-
Kites, per earlier arguments; the 8-bit rule, which gives 3 filled Box-Kites in the
Pathions, recursively propagates to cover 19 hidden Box-Kites in the Chingons,
according to the formula produced last section. But hide/fill involution says that,
of the 35 maximum possible Box-Kites in a Chingon ET, 35− 19 = 16 are now
made visible. As none of these have the Pathion G = 16 as an L-index, and all the
7 Box-Kites from the 16-bit rule do, we therefore have a grand total of 7+16= 23
Box-Kites in the S = 25 ET, as claimed (and as cell-counting on the cited Slide
will corroborate).
The concluding Slides 76–78 present a trio of color-coded “histological slices”
of the hiding and filling sequence (beginning with the blanking of the long diago-
nals) for the simplest 3-high-bit case, N = 7,S = 57. Here, the first fill rule works
on 25 and 32; the first hide rule, on 9, 16, 41, and 48; the second fill rule, on 1, 8,
17, 24, 33, 40, 49, and 56; and the rest of the cells, since the count of high-bits is
odd, are left blank.
We do not give an explicit algorithmic method here, however, for computing
the number of Box-Kites contained in this 3,720-cell ET. Such recursiveness is
best handled programmatically, rather than by cranking out an explicit (hence,
long and tedious) formula, meant for working out by a time-consuming hand cal-
culation. What we can do, instead, is conclude with a brief finale, embodying
all our results in the simple “recipe theory” promised originally, and offer some
reflections on future directions.
5 Fundamental Theorem of Zero-Divisor Algebra
All of the prior arguments constitute steps sufficient to demonstrate the Funda-
mental Theorem of Zero-Divisor Algebra. Like the role played by its Gaussian
predecessor in the legitimizing of another “new kind of [complex] number the-
ory,” its simultaneous simplicity and generality open out on extensive new vistas
at once alien and inviting. The Theorem proper can be subdivided into a Proposi-
tion concerning all integers, and a “Recipe Theory” pragmatics for preparing and
“cooking” the meta-fractal entities whose existence the proposition asserts, but
cannot tell us how to construct.
Proposition: Any integer K > 8 not a power of 2 can uniquely be associated with
a Strut Constant S of ZD ensembles, whose inner skybox resides in the 2N-ions
with 2N−2 < K < 2N−1. The bitstring representation of S completely determines
an infinite-dimensional analog of a standard plane-confined fractal, with each of
the latter’s points associated with an empty cell in the infinite Emanation Table,
with all non-empty cells comprised wholly of mutually orthogonal primitive zero-
divisors, one line of same per cell.
Preparation: Prepare each suitable S by producing its bitstring representation,
then determining the number of high-bits it contains: if S is a multiple of 8, right-
shift 4 times; otherwise, right-shift 3 times. Then count the number B of 1’s
in the shortened bitstring that results. For this set {B} of B elements, construct
two same-sized arrays, whose indices range from 1 to B: the array {i} which
indexes the left-to-right counting order of the elements of {B}; and, the array
{P} which indexes the powers of 2 of the same element in the same left-to-right
order. (Example: if K = 613, the inner skybox is contained in the 211-ions; as
the number is not a multiple of 8, the bistring representation 1001100101 is right-
shifted thrice to yield the substring of high-bits 1001100; B = 3, and for 1 ≤ i ≤
3, P1 = 9, P2 = 6;P3 = 5.)
Cookbook Instructions:
[0] For a given strut-constant S, compute the high-bit count B and bitstring
arrays {i} and {P}, per preparation instructions.
[1] Create a square spreadsheet-cell array, of edge-length 2I , where I ≥G/2= g
of the inner skybox for S, with the Sky as the limit when I → ∞.
[2] Fill in the labels along all four edges, with those running along the right
(bottom) borders identical to those running along the left (top), except in
reversed left-right (top-bottom) order. Refer to those along the top as col-
umn numbers C, and those along the left edge, as row numbers R, setting
candidate contents of any cell (r,c) to R⊻C = P.
[3] Paint all cells along the long diagonals of the spreadsheet just constructed
a color indicating BLANK, so that all cells with R =C (running down from
upper left corner) else R⊻C = S (running down from upper right) have their
P-values hidden.
[4] For 1 ≤ i ≤ B, consider for painting only those cells in the spreadsheet
created in [1] with R | C | P = m · 2γ | m · 2γ + σ , where γ = Pi,σ =
S mod∗ 2γ , and m is any integer ≥ 0 (with m= 0 only producing a legitimate
candidate for the right-hand’s second option, as an XOR of 0 indicates a
long-diagonal cell).
[5] If a candidate cell has already been painted by a prior application of these
instructions to a prior value of i, leave it as is. Otherwise, paint it with R⊻C
if i = odd, else paint it BLANK.
[6] Loop to [4] after incrementing i. If i < B, proceed until this step, then
reloop, reincrement, and retest for i = B. When this last condition is met,
proceed to the next step.
[7] If B is odd, paint all cells not already painted, BLANK; for B even, paint
them with R⊻C.
In these pseudocode instructions, no attention is given to edge-mark gener-
ation, performance optimization, or other embellishments. Recursive expansion
beyond the chosen limits of the 2N-ion starting point is also not addressed. (Just
keep all painted cells as is, then redouble until the expanded size desired is at-
tained; compute appropriate insertions to the label lines, then paint all new cells
according to the same recipe.) What should be clear, though, is any optimization
cannot fail to be qualitatively more efficient than the code in the appendix to [9],
which computes on a cell-by-cell basis. For S > 8, N > 4, we’ve reached the
onramp to the Metafractal Superhighway: new kinds of efficiency, synergy, con-
nectedness, and so on, would seem to more than compensate for the increase in
dimension.
It is well-known that Chaotic attractors are built up from fractals; hence, our
results make it quite thinkable to consider Chaos Theory from the vantage of pure
Number · · · and hence the switch from one mode of Chaos to another as a bitstring-
driven – or, put differently, a cellular automaton-type – process, of Wolfram’s
Class 4 complexity. Such switching is of the utmost importance in coming to
terms with the most complex finite systems known: human brains. The late Fran-
cisco Varela, both a leading visionary in neurological research and its computer
modeling, and a long-time follower of Madhyamika Buddhism who’d collabo-
rated with the Dalai Lama in his “Tibetan Buddhists talk with brain scientists”
dialogues [10], pointed to just the sorts of problems being addressed here as the
next frontier. In a review essay he co-authored in 2001 just before his death [11,
p. 237], we read these concluding thoughts on the theme of what lies “Beyond
Synchrony” in the brain’s workings:
The transient nature of coherence is central to the entire idea of large-
scale synchrony, as it underscores the fact that the system does not be-
have dynamically as having stable attractors [e.g., Chaos], but rather
metastable patterns – a succession of self-limiting recurrent patterns.
In the brain, there is no “settling down” but an ongoing change marked
only by transient coordination among populations, as the attractor it-
self changes owing to activity-dependent changes and modulations of
synaptic connections.
Varela and Jean Petitot (whose work was the focus of the intermezzo conclud-
ing Part I, in which semiotically inspired context the Three Viziers were intro-
duced) were long-time collaborators, as evidenced in the last volume on Naturaliz-
ing Phenomenology [12] which they co-edited. It is only natural then to re-inscribe
the theme of mathematizing semiotics into the current context: Petitot offers sepa-
rate studies, at the “atomic” level where Greimas’ “Semiotic Square” resides; and
at the large-scale and architectural, where one must place Lévi-Strauss’s “Canon-
ical Law of Myths.” But the pressing problem is finding a smooth approach that
lets one slide the same modeling methodology from the one scale to the other:
a fractal-based “scale-free network” approach, in other words. What makes this
distinct from the problem we just saw Varela consider is the focus on the structure,
rather than dynamics, of transient coherence – a focus, then, in the last analysis, on
a characterization of database architecture that can at once accommodate meta-
chaotic transiency and structural linguists’ cascades of “double articulations.”
Starting at least with C. S. Peirce over a century ago, and receiving more
recent elaboration in the hands of J. M. Dunn and the research into the “Semantic
Web” devolving from his work, data structures which include metadata at the
same level as the data proper have led to a focus on “triadic logic,” as perhaps best
exemplified in the recent work of Edward L. Robertson. [13] His exploration of
a natural triadic-to-triadic query language deriving from Datalog, which he calls
Trilog, is not (unlike our Skies) intrinsically recursive. But his analysis depends
upon recursive arguments built atop it, and his key constructs are strongly resonant
with our own (explicitly recursive) ones. We focus on just a few to make the point,
with the aim of provoking interest in fusing approaches, rather than in proving any
particular results.
The still-standard technology of relational databases based on SQL statements
(most broadly marketed under the Oracle label) was itself derived from Peirce’s
triadic thinking: the creator of the relational formalism, Edgar F. “Ted” Codd, was
a PhD student of Peirce editor and scholar Arthur W. Burks. Codd’s triadic “re-
lations,” as Robertson notes (and as Peirce first recognized, he tells us, in 1885),
are “the minimal, and thus most uniform” representations “where metadata, that
is data about data, is treated uniformly with regular data.” In Codd’s hands (and in
those of his market-oriented imitators in the SQL arena), metadata was “relegated
to an essentially syntactic role” [13, p. 1] – a role quite appropriate to the appli-
cations and technological limitations of the 1970’s, but inadequate for the huge
and/or highly dynamic schemata that are increasingly proving critical in bioinfor-
matics, satellite data interpretation, Google server-farm harvesting, and so on. As
Robertson sums up the situation motivating his own work,
Heterogeneous situations, where diverse schemata represent semanti-
cally similar data, illustrate the problems which arise when one per-
son’s semantics is another’s syntax – the physical “data dependence”
that relational technology was designed to avoid has been replaced by
a structural data dependence. Hence we see the need to [use] a simple,
uniform relational representation where the data/metadata distinction
is not frozen in syntax. [13, pp. 1-2]
As in relational database theory and practice, the forming and exploiting of
inner and outer joins between variously keyed tables of data is seminal to Robert-
son’s approach as well as Codd’s. And while the RDF formalism of the Semantic
Web (the representational mechanism for describing structures as well as contents
of web artifacts on the World Wide Web) is likewise explicitly triadic, there has,
to date, been no formal mechanism put in place for manipulating information in
RDF format. Hence, “there is no natural way to restrict output of these mecha-
nisms to triples, except by fiat” [13, p. 4], much less any sophisticated rule-based
apparatus like Codd’s “normal forms” for querying and tabulating such data. It is
no surprise, then, that Robertson’s “fundamental operation on triadic relations is
a particular three-way join which takes explicit advantage of the triadic structure
of its operands.” This triadic join, meanwhile, “results in another triadic relation,
thus providing the closure required of an algebra.” [13, p. 6]
Parsing Robertson’s compact symbolic expressions into something close to
standard English, the trijoin of three triadic relations R, S, T is defined as some
(a,b,c) selected from the universe of possibilities (x,y,z), such that (a,x,z) ∈ R,
(x,b,y) ∈ S, and (z,y,c) ∈ T . This relation, he argues, is the most fundamental
of all the operators he defines. When supplemented with a few constant relations
(analogs of Tarski’s “infinite constants” embodied in the four binary relations of
universality of all pairs, identity of all equal pairs, diversity of all unequal pairs,
and the empty set), it can express all the standard monotonic operators (thereby
excluding, among his primitives, only the relative complement).
How does this compare with our ZD setup, and the workings of Skies? For one
thing, Infinite constants, of a type akin to Tarski’s, are embodied in the fact that
any full meta-fractal requires the use of an infinite G, which sits atop an endless
cascade of singleton leftmost bits, determining for any given S an indefinite tower
of ZDs. One of the core operators massaging Robertson’s triads is the flip, which
fixes one component of a relation while interchanging the other two · · · but our
Rule 2 is just the recursive analog of this, allowing one to move up and down
towers of values with great flexibilty (allowing, as well, on and off switching
effecting whole ensembles). The integer triads upon which our entire apparatus
depends are a gift of nature, not dictated “by fiat,” and give us a natural basis for
generating and tracking unique IDs with which to “tag” and “unpack” data (with
“storage” provided free of charge by the empty spaces of our meta-fractals: the
“atoms” of Semiotic Squares have four long-diagonal slots each, one per each of
the “controls” Petitot’s Catastrophe Theory reading calls for, and so on.)
Finally, consider two dual constructions that are the core of our own triadic
number theory: if the (a,b,c) of last paragraph, for instance, be taken as a Zigzag’s
L-index set, then the other trio of triples correlates quite exactly with the Zigzag
U-trips. And this 3-to-1 relation, recall, exactly parallels that between the 3 Tre-
foil, and 1 Zigzag, Sails defining a Box-Kite, with this very parallel forming the
support for the recursion that ultimately lifts us up into a Sky. We can indeed
make this comparison to Robinson’s formalism exceedingly explicit: if his X, Y,
Z be considered the angular nodes of PSL(2,7) situated at the 12 o’clock apex and
the right and left corners respectively, then his (a,b,c) correspond exactly to our
own Rule 0 trip’s same-lettered indices!
Here, we would point out that these two threads of reflection – on underwriting
Chaos with cellular-automaton-tied Number Theory, and designing new kinds of
database architectures – are hardly unrelated. It should be recalled that two years
prior to his revolutionary 1970 paper on relational databases [14], Codd published
a pioneering book on cellular automata [15]. It is also worth noting that one
of the earliest technologies to be spawned by fractals arose in the arena of data
compression of images, as epitomized in the work of Michael Barnsley and his
Iterative Systems company. The immediate focus of the author’s own commercial
efforts is on fusing meta-fractal mathematics with the context-sensitive adaptive-
parsing “Meta-S” technology of business associate Quinn Tyler Jackson. [16] And
as that focus, tautologically, is not mathematical per se, we pass it by and leave it,
like so many other themes just touched on here, for later work.
References
[1] Robert P. C. de Marrais, “Placeholder Substructures I: The Road From NKS
to Scale-Free Networks is Paved with Zero Divisors,” Complex Systems, 17
(2007), 125-142; arXiv:math.RA/0703745
[2] Robert P. C. de Marrais, “Placeholder Substructures II: Meta-Fractals, Made
of Box-Kites, Fill Infinite-Dimensional Skies,” arXiv:0704.0026 [math.RA]
[3] Robert P. C. de Marrais, “The 42 Assessors and the Box-Kites They Fly,”
arXiv:math.GM/0011260
[4] Robert P. C. de Marrais, “The Marriage of Nothing and All: Zero-Divisor
Box-Kites in a ‘TOE’ Sky,” in Proceedings of the 26th International Col-
loquium on Group Theoretical Methods in Physics, The Graduate Center
of the City University of New York, June 26-30, 2006, forthcoming from
Springer–Verlag.
[5] Robert P. C. de Marrais, “Placeholder Substructures: The Road from NKS
to Small-World, Scale-Free Networks Is Paved with Zero-Divisors,” http://
wolframscience.com/conference/2006/ presentations/materials/demarrais.ppt
(Note: the author’s surname is listed under “M,” not “D.”)
[6] Benoit Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and
Company, San Francisco, 1983)
[7] Ed Pegg, Jr., “Tournament Dice,” Math Games column for July 11, 2005, on
the MAA website at http://www.maa.org/editorial/ mathgames/mathgames
_07_11_05.html
[8] Robert P. C. de Marrais, “The ‘Something From Nothing’ Insertion Point,”
http://www.wolframscience.com/conference/2004/ presentations/
materials/rdemarrais.pdf
[9] Robert P. C. de Marrais, “Presto! Digitization,” arXiv:math.RA/0603281
[10] Francisco Varela, editor, Sleeping, Dreaming, and Dying: An Exploration
of Consciousness with the Dalai Lama (Wisdom Publications: Boston, 1997).
[11] F. J. Varela, J.-P. Lachauz, E. Rodrigues and J. Martinerie, “The brainweb:
phase synchronization and large-scale integration,” Nature Reviews Neuro-
science, 2 (2001), pp. 229-239.
http://arxiv.org/abs/math/0703745
http://arxiv.org/abs/0704.0026
http://arxiv.org/abs/math/0011260
http://www.maa.org/editorial/
http://www.wolframscience.com/conference/2004/
http://arxiv.org/abs/math/0603281
[12] Jean Petitot, Francisco J. Varela, Bernard Pachoud and Jean-Michel Roy,
Naturalizing Phenomenology: Issues in Contemporary Phenomenology and
Cognitive Science (Stanford University Press: Stanford, 1999)
[13] Edward L. Robertson, “An Algebra for Triadic Relations,” Technical Re-
port No. 606, Computer Science Department, Indiana University, Bloom-
ington IN 47404-4101, January 2005; online at http://www.cs.indiana.edu/
pub/techreports/TR606.pdf
[14] E. F. Codd, The Relational Model for Database Management: Version 2
(Addison-Wesley: Reading MA, 1990) is the great visionary’s most recent
and comprehensive statement.
[15] E. F. Codd, Cellular Automata (Academic Press: New York, 1968)
[16] Quinn Tyler Jackson, Adapting to Babel – Adaptivity and Context-Sensiti-
vity in Parsing: From anbncn to RNA (Ibis Publishing: P.O. Box3083, Ply-
mouth MA 02361, 2006; for purchasing information, contact Thothic Tech-
nology Partners, LLC, at their website, www.thothic.com).
http://www.cs.indiana.edu/
	The Argument So Far
	8 <S < 16, N  : Recursive Balloon Rides in the Whorfian Sky
	Maximal High-Bit Singletons: (s,g)-Modularity for 16 < S 24
	Hide/Fill Involution: Further-Right High-Bits with 24 < S < 32.
	Fundamental Theorem of Zero-Divisor Algebra
