Can we construct an easy mono-tile ?
Shigeki Akiyama, Niigata Univ.
22 December 2011 at RIMS

– Typeset by FoilTEX –A tile is a compact set which is the closure of its interior.
Consider a ﬁnite set of tiles A as an alphabet. A tiling Tof Rd is a collection of tiles which covers Rd without interior
intersections, and each tile is congruent to an element of A
under rigid motion of Rd. A tiling T has a period p ∈ Rd
when the tiling exactly matches with its translation by p ∈ Rd.
A tiling T is non periodic if the only period of T is 0. A setof tiles A is called aperiodic if A generates a tiling, but all
tilings generated by A are non periodic. A well known example
of aperiodic tiles is due to Penrose which consists of two kindsof tiles: kites and darts with matching rules as in Figure 1:– Typeset by FoilTEX –1

Figure 1: Penrose Tile

that the circular markings must match at the boundary like:– Typeset by FoilTEX –2

Figure 2: Penrose tiling– Typeset by FoilTEX –3

To show that Penrose tiles are aperiodic, we have to show
two things.
• They admit a tiling.
• Each tiling generated by kite and dart has no period.
Both are shown through self-similarity. First we show that
there is a substitution rule which make a tile larger and thensubstitute to a larger patch. Second we prove the uniquecomposition property. However both of them are quite nontrivial.
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(a) Kite(b) Dart

Figure 3: Substitution ruleFix a point on the kite. Then the points on the kite of
the same orientation forms a Delone set, the distance tothe nearest point is bounded from below, and there is no– Typeset by FoilTEX –5

big hole without a point. de Bruijn [5] showed that Penrose
tiling is really a good model of quasi-crystal by showing thatsuch Delone set is understood by cut and projection. Moreprecisely this Delone set is produced by the projection of 5-dimlattice points which lies in some irrational band. The cut andprojection from 2-dim to 1-dim is described in Figure 4.

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Figure 4: Cut and Projection– Typeset by FoilTEX –7Tiling dynamical system and diﬀraction

Tiling dynamical system is a topological dynamics
generated by the orbit closure of translation of a single tiling.
This can be viewed as a generalization of symbolic substitution
dynamical system.
To study diﬀraction of such Delone sets, tiling dynamicalsystem is studied in detail. Under certain condition, thisdynamics becomes uniquely ergodic and admit a spectral study.
It is known that pure pointedness of dynamical spectrum isequal to those of diﬀraction spectrum.
– Typeset by FoilTEX –8Figure 5: Penrose Diﬀraction
From this diﬀraction, we can imagine the 5-dim structure.
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R. Ammann, an amateur mathematician working at postoﬃce, was strongly motivated by Penrose tiling when he readthe article in Scientiﬁc American. He found other type ofaperiodic tiles (c.f. [7, 3, 11]). Here is one of them with two
diﬀerent markings:– Typeset by FoilTEX –

10

1

1
1Τ

1

Τ21Τ
1Τ
Τ

Τ
1

Τ

Τ

Figure 6: Ammann Tile
The black markings must form complete ovals at theboundary:– Typeset by FoilTEX –11

– Typeset by FoilTEX –Figure 7: Ammann Tiling

12

The second striking marking by segments must be continued
to a straight line or a dashed line, now called Ammann Bars.
1

11Τ

1

Τ2

1Τ
1Τ
Τ

Τ
1

Τ

Τ

Figure 8: Ammann Tile 2– Typeset by FoilTEX –13

– Typeset by FoilTEX –Figure 9: Ammann Bar

14

We guess that these lines came from higher dimensionallattice structure, which shows Ammann’s strong mathematicaltalent. As far as I know, it is still a mysterious construction tothis date. The proof of aperiodicity of Ammann tiles is again
by self-similarity in the same course as Penrose tiling. None ofthem is trivial.

In this talk, we propose a variant of Ammann tiles which
provides us the simplest marking, as far as I know:– Typeset by FoilTEX –15c1c2
1

11c1c11c2c1c2c
11c1c1c11c2

Figure 10: Similar Ammann tiles A and B
having slightly modiﬁed shapes on which GrünbaumShephard [7] claimed:– Typeset by FoilTEX –16‘However it does not seem possible to devise a simple
matching condition for these tiles involving markings which
are such that the similarity between the tiles extends tomarkings as well.’and proposed a pretty complicated matching rules on theedges to enforce√ aperiodicity. Here c = 1.272 . . . is a squareroot of (1 + 5)/2. The marking on the edges means thatedges are cut into smaller segments at those points. Our only
matching rule is edge to edge: the edges exactly match fromone end to the other in the tiling.
In fact, this work is motivated by our former work [2] to ﬁnd– Typeset by FoilTEX –17

a self-similar dissection of a give sets. The shapes seem to be
independently found in his similar dissection puzzle of Scherer[10]. He called it ‘Golden Bee’, which is a polygon composedof two similar copies of itself (see Figure 15).

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Tiling property

We show that two tiles in Figure 10 admit a tiling. First we
discard the marking and consider only shapes. Our substitution
rule makes c-times larger A and B. Then A is divided intoA and B as in Figure 15 and B becomes A. The results aredepicted in Figure 11.
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Figure 11: Super-tiles cA and cB: broken lines indicate 4 ghost
markings and the former boundary
Starting from the tile A, we successively obtain a larger
patch by this substitution rule like Figure 12.
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Figure 12: Inﬂation subdivisionThis gives a tiling of the plane by A and B, which is called
a ﬁxed point of the substitution rule:– Typeset by FoilTEX –21

Figure 13: A patch of tiling by A and B– Typeset by FoilTEX –22

We may view this ﬁxed point as a super-tiling by c3A andc3B without markings. Then c3A is the composition of three
A’s and two B’s, and c3B is a composition of two A’s and one
B. Boundaries of A and B give natural markings of super-tiles
c3A and c3B as black points in Figure 14. Conﬁrm that they
are exactly located at the c3 scaled markings of A and B.

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Figure 14: Markings from c3A and c3B
As the tiling by c3A and c3B surely exists, we know that
there exists a tiling by A and B with the markings scaled downby ratio 1/c3.

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Aperiodicity

Now we show the aperiodicity of two tiles in Figure 10
by proving unique composition property, that is, there is aunique super-tiling structure. Firstly, it is plain to see that
there are no way to tile R2 only by A and the dent of B canbe ﬁlled only by A as in Figure 15.
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Figure 15: Filling the dent of B
As we already see that this composition of A and B is theinﬂated image of A by ratio c. We call this super-tile A′. The
remaining A which does not ﬁll the dent of B is seen as asuper-tile B ′, the inﬂated image of B by ratio c. We have toshow that additional marking can be removed. In other word,– Typeset by FoilTEX –26

if we could show that the tiles in Figure 11 can be identiﬁed
with cA and cB including markings, we are done.
Now we meet the main diﬃculty. We have to remove the reddashed markings indicated in Figure 11. However, the eﬀectof removal is involved. So we treat them as Ghost markings,
meaning that we can think that they exist or not exist as wewish.
So a ghost marking can be used to cut the edge but we canalso neglect it. Under this treatise, the set of tilings by twotiles in Figure 11 is surely larger than those without such ghostmarkings. Then we show that ghost markings do no harm by– Typeset by FoilTEX –27

showing that the patches which essentially use ghost markings,
do not grow into a tiling. We have 20 cases to check out.

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Figure 16: Illegal patches of the 1-st kind

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Figure 17: Illegal patches of the 2nd kind

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Diﬀraction property

Similar to Penrose tiling, D. Frettlöh [6] showed that theDelone set of Ammann tiling is given by cut and projection.
Thus Ammann tiling gives a pure diﬀractive point set.
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Figure 18: Diﬀraction pattern of Ammann tiling– Typeset by FoilTEX –32

Questions• Is there an aperiodic mono-tile ? Solved: Socolar-Taylor[12].

Figure 19: Socolar-Taylor’s aperiodic monotile– Typeset by FoilTEX –33

• Is there an edge to edge aperiodic mono-tile ? Ourconstruction suggests some possibility using rep-tile, thatis, a tile consisting of same-sized similar copies of itself.

Figure 20: Rep-tiles• Related question: Must all rep-tile be a fundamental region– Typeset by FoilTEX –34

of a crystallographic group ?
• Tiling Question Substitution rule of tiles {T1, . . . , Tm} iswritten as:∪QTj =Ti + Dijiwhere Q is an expanding matrix and Dij are ﬁnite setsof translations. By Lagarias-Wang’s duality, the associatedDelone set satisﬁes∪
QΛj + Dij .
Λi =jGiven expanding Q, which digit set Dij gives rise to a tiling– Typeset by FoilTEX –35

?
• For which Q and Dij , the associated Delone set dynamics
contains pure parts ? We know a lot.: E.BombieriJ.E.Taylor [4], B. Solomyak, J.-Y. Lee [13, 9]• Characterize pure diﬀractive substitution tilings. Diﬃcult:
Overlap coincidence by Solomyak [13], Algorithm by
Akiyama-Lee [1]• Pure diﬀractive substitution tiling is given by cut andprojection ? Mostly solved: Lee [8]– Typeset by FoilTEX –36

References[1] S. Akiyama and J.-Y. Lee, Algorithm for determining purepointedness of self-aﬃne tilings, Adv. Math. 226 (2011),
no. 4, 2855–2883.
[2] S. Akiyama, J. Luo, R. Okazaki, W. Steiner, and
J. Thuswaldner, Similar dissection of sets, Geom. Dedicata
150 (2011), 233–247.
[3] R. Ammann, B. Grünbaum, and G. C. Shephard, Aperiodictiles, Discrete Comput. Geom. 8 (1992), no. 1, 1–25.
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[4] E. Bombieri and J. E. Taylor, Quasicrystals, tilings, andalgebraic number theory: some preliminary connections,Contemp. Math., vol. 64, Amer. Math. Soc., Providence,RI, 1987, pp. 241–264.
[5] N. G. de Bruijn, Algebraic theory of Penrose’s nonperiodic
tilings of the plane. I, II, Nederl. Akad. Wetensch. Indag.
Math. 43 (1981), no. 1, 39–52, 53–66.
[6] D. Frettlöh, Duality of model sets generated bysubstitutions, Rev. Roumaine Math. Pures Appl. 50(2005), no. 5-6, 619–639.
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[7] B. Grünbaum and G. C. Shephard, Tilings and patterns,
W. H. Freeman and Company, New York, 1987.
[8] J.-Y. Lee, Substitution Delone sets with pure point
spectrum are inter-model sets, J. Geom. Phys. 57 (2007),
no. 11, 2263–2285.
[9] J.-Y. Lee and B. Solomyak, Pisot family substitution
tilings, discrete spectrum and the Meyer property, Discr.
Conti. Dynam. Sys. 32 (2012), no. 3, 935–959.
[10] K. Scherer, A puzzling journey to the reptiles and relatedanimals, Private Publication, 1987.
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[11] M. Senechal, The mysterious Mr. Ammann, Math.
Intelligencer 26 (2004), no. 4, 10–21.
[12] J. E. S. Socolar and J. M. Taylor, An aperiodic hexagonaltile, Journal of Combinatorial Theory 18 (2011), 2207–2231.
[13] B. Solomyak, Dynamics of self-similar tilings, Ergodic
Theory Dynam. Systems 17 (1997), no. 3, 695–738.

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