Expectations on Fractal Sets
David H. Bailey
http://www.davidhbailey.com
Lawrence Berkeley Natl. Lab. (retired)Computer Science Dept., University of California, Davis
Co-authors:
Jonathan M. Borwein (CARMA, University of Newcastle, Australia)
Richard E. Crandall (deceased)
Michael G. Rose (CARMA, University of Newcastle, Australia)

14 December 2014

Background and motivation

This study arose when the late number theorist Richard Crandall (deceased December2012) was asked by some Stanford researchers to analyze some recent data on mousebrain synapses.
I Approximately one million synapses were analyzed.
I Synapses were found to be distributed in a fractal manner.
I The fractal dimension was significantly less than three, which would be expected

if the points had been randomly distributed through a cubic volume.
I In other words, the expected pair-wise distances do not follow the statistics ofrandom distributions.

The PSLQ integer relation algorithm
Let (xn ) be a given vector of real numbers. An integer relation algorithm either finds
integers (an ) such thata1 x1 + a2 x2 + · · · + an xn = 0(to within the “epsilon” of the arithmetic being used), or else finds bounds within
which no relation can exist.
The “PSLQ” algorithm of mathematician-sculptor Helaman Ferguson is the mostwidely used integer relation algorithm.
Integer relation detection requires very high precision (at least n × d digits, where d isthe size in digits of the largest ak ), both in the input data and in the operation of thealgorithm.
1. H.R.P. Ferguson, D.H. Bailey and S. Arno, “Analysis of PSLQ, An Integer Relation Finding Algorithm,” Mathematics of Computation, vol.
68, no. 225 (Jan 1999), pg. 351–369.
2. D.H. Bailey and D.J. Broadhurst, “Parallel Integer Relation Detection: Techniques and Applications,” Mathematics of Computation, vol. 70,
no. 236 (Oct 2000), pg. 1719–1736.

Applications of PSLQ in number theory and experimental mathematics
Methodology:
1. Compute various mathematical entities (limits, infinite series sums, definiteintegrals, etc.) to high precision, typically 100–10,000 digits.
2. Use PSLQ to recognize these numerical values in terms of well-known
mathematical constants.
3. When results are found experimentally, seek formal mathematical proofs of thediscovered relations.
Many results have recently been found using this methodology, both in puremathematics and in mathematical physics.

Earlier study of box integrals
The following integrals appear in numerous applications:
Z 1
Z 1
s/2
Bn (s) :=···
r12 + · · · + rn2
dR00Z 1Z 1
s/2
∆n (s) :=···(r1 − q1 )2 + · · · + (rn − qn )2dRdQ
00I Bn (1) is average distance of a random point from the origin.
I ∆n (1) is average distance between two random points.
I Bn (−n + 2) is average electrostatic potential in an n-cube whose origin has a unitcharge.
I ∆n (−n + 2) is average electrostatic energy between two points in a uniform

n-cube of charged “jellium.”
High-precision numerical values were computed for these values, then PSLQ wasemployed to recognize the resulting values.
D. H. Bailey, J. M. Borwein and R.E. Crandall, “Box integrals,” Journal of Computational and Applied Mathematics, vol. 206 (2007), pg. 196–208.

Sample evaluations of box integralsnany
1222222333333seven s ≥ 0s 6= −1-4-3-1
13s 6= −2-5-4-2-1
13Bn (s)rational, e.g., : B2 (2) = 2/3
1s+1
1− π8−4√− 2√2log(1+ 2)√√
1
13√ 2 + 3 log(1 + √2)732)
5 2 + 20 log(1 +21s 32F,−;
;
−1
12+s
√2 21 21−√
6 3 − 12 π− 32 2 arctan √12√√
−3G + 32 π log(1 + 2) + 3 √Ti2(3 − 2 2)
1
32 + 3√ √− 4 π 1+ 2 log
113−π+log2 + √3 
4√24221735 3 − 60 π − 20 log 2 +Here F is hypergeometric function; G is Catalan; Ti is Lewin’s inverse-tan function.

Sample evaluations of box integrals, continued
ns

4-54-34-2

4-1

4

15-35-25-1

51Bn (s)  √− 8 arctan √18√
4 G− 12 Ti2(3 − 2 2)√2π log 2 + 3 − 2 G − π8 √√2 log 3 − 23 G + 2 Ti2 3 − 2 2 − 8 arctan √18 √√ G
32− 10+ 10Ti2 3 − 2 2 + log 3 − 7102 arctan √185 √ √ √
1 251+ 5√1− 110
G−
10log
2−
3θ
−π+5log−
3arctan
982215
−10 Cl2 31 θ + 31 π + 10 Cl2 13 θ − 16π
20+ 10Cl2 θ + 61 π + 20
Cl2 θ + 43 π − 10Cl2 θ + 53 π −Cl2 θ + 11π
33336
118B (−6)
B5 (−4) + 25 π log(3) +10 G 
3 53  10√Ti2 3 − −√
√210110
11+ 55− 27 G + 3 log 2 − 3 θ + 48 π + 5 log−3arctan √115
22 10101111π6+ 3 20Cl2 3 θ + 43 π − 103 Cl2 3 θ −1015− 9 Cl2 θ+ 6 π +Cl2 θ + 3 π − 3 Cl2 θ + 3 π + 20Cl2 θ + 11π 396√√√√1− 77
G + 79 log 2 − 3 θ + 360
π 2 + 16 5 + 10log 1+2 5 − 43 3 arctan √115 +813
7117112 3 θ + 3 π − 9 Cl2 3 θ − 6 π 9 Cl
711447514− 27 Cl2 θ + 6 π − 27 Cl2 θ + 3 π + 27 Cl2 θ + 3 π + 27Cl2 θ + 11π
6String-generated Cantor sets
Fix positive integers n and p, and consider the unit n-cube [0, 1]n . Let P = P1 P2 . . . Pp
denote a period string.
The String-Generated Cantor Set (SCS), denoted Cn (P), is the set of all admissiblex ∈ [0, 1]n , wherex admissible ⇐⇒ U(ck ) ≤ Pk∀k ∈ N(1)with notational periodicity assumed: Pp+k := Pk for all k ≥ 1.
Examples:I C1 (0) is the classical “middle-thirds-removed” Cantor set on [0, 1], as a pointx ∈ C1 (0) is defined to be admissible iff its ternary expansion is entirely devoid of1’s (i.e. U(ck ) ≤ 0).
I Cn (n) is the full unit n-cube [0, 1]n , as all points x ∈ [0, 1]n are admissible (every
ternary vector is allowed for every column ck ).
D. H. Bailey, J. M. Borwein, R. E. Crandall, M. G. Rose, “Expectations on fractal sets,” Appl. Math. and Comp., vol. 220 (1 Sep 2013), pg. 695-721.

Some string-generated Cantor sets

Image due to R. Dickau (2008).

Graphical illustration of expectations for string-generated Cantor sets

Evaluations of expectations for string-generated Cantor sets
We have been successful in finding closed-form values for many of these expectations.
Surprisingly, all of them are rational numbers, a counter-intuitive result, since for
non-Cantor sets they are much more complicated.
We also verified these results by numerically computations. Computing these values
was non-trivial.
Possible approaches:
1. Utilize a uniform (0, 1) pseudorandom number generator to generate pairs ofn-tuples at random, then check each pair of n-tuples so generated to see if it is“admissible” for the given set Cn (P). This simple scheme can be used effectively
in cases where the average “density” of the fractal set is not too small.
2. (Employed in results below.) Construct a table of all admissible columns ck foreach of the components of P, and then, for each of a large number of trials, andfor each ternary column 1 ≤ k ≤ 20 of a single trial, pseudorandomly select anadmissible column from the appropriate table. While coding is much more
complicated, each pair of n-tuples is guaranteed to be admissible.

Sample results for one dimension
δ(C1 (P))
P

B(2, C1 (P))

DecimalRational

00.630930

380001

0.723197

7379
19680

0010

0.723197

2457
6560

0100

0.723197

2433
6560

1000

0.723197

2217
6560

0111

0.907732

7289
19680

1011

0.907732

6641
19680

1101

0.907732

6569
19680

1110

0.907732

2187
6560

1

1.000000

13Decimal∆(2, C1 (P))
Numeric

Rational

Decimal

Numeric

0.375000

0.375013

140.250000

0.250009

0.374949

0.374941

2459
9840

0.249898

0.249904

0.374543

0.374543

817
3280

0.249085

0.249080

0.370884

0.370887793
3280

0.241768

0.241765

0.337957

0.337937577
3280

0.175915

0.175928

0.370376

0.370350

2369
9840

0.2407520.240757

0.337449

0.337440

1721
9840

0.174898

0.174903

0.333791

0.333765

1649
9840

0.167581

0.167574

0.333384

0.333386547
3280

0.166768

0.166774

0.333333

0.333333

160.166667

0.166671

Sample results for two dimensions
δ(C2 (P))
P

B(2, C2 (P))

DecimalRational

01.261863401

1.577324

119
160

10

1.577324

111
160

02

1.630930

89
120

20

1.630930

27
40

11.892789

11
16

12

1.946395

329
480

21

1.946395

107
160

2.000000

23

2

Decimal∆(2, C2 (P))
Numeric

Rational

Decimal

Numeric

0.750000

0.750012

120.500000

0.5000360.743750

0.743744

3980

0.487500

0.487478

0.693750

0.693716

3180

0.387500

0.387514

0.741667

0.741664

29
60

0.483333

0.483325

0.675000

0.674983

7200.350000

0.349996

0.687500

0.687486

380.375000

0.374998

0.685417

0.68541489
2400.370833

0.370830

0.668750

0.668743

2780

0.337500

0.337494

0.666646130.333333

0.333335

0.666667

Sample results for three dimensionsPδ(C3 (P))

B(2, C3 (P))

DecimalRational

01.892789

98

01

2.309811

447
400

102.309811

423
400

02

2.429218

579
520

20

2.429218

531
520

03

2.446395

89
80

30

2.446395

81
80

23

2.982824

1049
1040

32

2.982824

1041
1040

33.000000

1Decimal∆(2, C3 (P))
Numeric

Rational

Decimal

Numeric

1.125000

1.124983

340.750000

0.749994

1.117500

1.117460

147
200

0.735000

0.734995

1.057500

1.057481

123
200

0.615000

0.614989

1.113462

1.113436

189
260

0.726923

0.726925

1.021154

1.021104141260

0.542308

0.542313

1.112500

1.112453

2940

0.725000

0.725009

1.012500

1.012502

21
40

0.525000

0.525011

1.008654

1.008633

269
520

0.517308

0.517303

1.000962

1.000925

261
520

0.501923

0.501918

1.000000

0.999960

120.500000

0.499983

Ongoing work
Computed value for expectation of Sierpinski gasket triangle:
0.6180082171582247074177418624555167834492481641438960879796572765289
4992781724125962846495857367069910611...
What is this number? We are trying hard to identify it.
Computing this numerical value required 160 CPU-hours run time and critically relied
on a self-avoiding random walk approach due to Nathan Clisby.
Stay tuned!

For additional details
These and many more results are presented in:
I D. H. Bailey, J. M. Borwein, R. E. Crandall, M. G. Rose, “Expectations on fractalsets,” Appl. Math. and Comp., vol. 220 (1 Sep 2013), pg. 695-721.
Preprint version is available at
http://www.davidhbailey.com/dhbpapers/fracboxes.pdf
This talk is available at:
http://www.davidhbailey.com/dhbtalks/dhb-fractal-boxes.pdf