Pergamon

0960-0779(94) 00249-5

Chaos, Solitons & Fractals Vol. 6, pp. 105-112, 1995 Elsevier Science Ltd

Printed in Great Britain 0960-0779/95 $9.50 + .00

Divisibility and Cellular Automata

C. CRESPO CRESPO, Ch. PONTEVILLE and V.W. de SPINAl)EL

Departamento de Matematica

Facultad de Ciencias Exactas y Naturales

Universidad de Buenos Aires

Abstract - Cellular automata (CA) are perfect feedback machines which change the state of their

cells step by step. In a certain sense, Pascal's triangle was the first CA and there is a strong

connection between Pascal's triangle and the fractal pattern formation known as Sierpinski

gasket.

Generalizing divisibility properties of the coefficients of Pascal's triangle, binomial arrays as well

as gaussian arrays are evaluated mod p. In these arrays, two fractal geometric characteristics are

evident: a) self-similarity and b) non integer dimension.

The conclusions at which we arrive,as well as the conjectures we propose, are important facts to

take into account when modelling real experiments like catalytic oxidation reactions in Chemistry,

where the remarkable resemblance of the graph: number of entries in the k ~ row of the Pascal's

triangle which are not divisible by 2 vs k and the measurement of the chemical reaction rate as a

function of time, provides the reason to model a catalytic converter by a one-dimensional CA [4].

INTRODUCTION

Spatio-temporal chaos is usually investigated in simple model systems, such as CA [15, [16]. To run a CA

we need to know the initial state of its cells and a set of rules that describe how the state of a cell in a new layer is

obtained from the states of cells from the preceding layer.s. There is a whole class of CA closely related to

Pascal's triangle and its divisibility schemes. To avoid too large numbers inthe entries of Pascals's triangle, the

array is modified to display divisibility properties, writing the entries mod p The simplest example is the Pascal

array mod 2, that is obtained using the recursing formula

(~) = ( 7 51) + I n 7 1 )

and the addition rules mod 2

1 + 1 = 0 = 0 + 0 = 0 ; 1 + 0 = 1

Pascal's triangle rood 2 appears in the analysis of structures generated by CA. These systems have been

106 C. CRESPO CRESPO et al.

studied as mathematical models for natural processes which exhibit the phenomenon of "self-organized criticality"

SOC (see e.g. the sand-pile ~,utomaton devised by Bak et al, [1], where the state &the system is specified by an

array of integers representing the height difference between neighboring plateaus).

BINOMIAL ARRAYS mod p (prime)

To characterise the geometry of the binomial arrays, we shall use the notation introduced by M. Sved

[14]. The entry (~) mod p will be indicated [~]. A "principal cell" is the array of the I~1 entries for 0 _< n < p g

"principal cluster of order k" is the array of the [~[ entries for 0 g n < pk .A "cell" is a triangular array, similar to

the principal cell. The single entry in the first row of the cell is the"head" while the last row is the "base". In the

same way, "clusters" are similar to the principal cluster of the same order. The "zero-holes" are arrays consisting of zero entries exclusively (Fig. 1).

In the k ~ row of Pascal's triangle mod 2 (Fig. 2), the quantity of entries different from zero is 2 n, being

nthe quantity of I that appear in the binary expression of k. This property may be generalized (see Fig. 3) in the following way:

FOr p triangle are not divisible by p, may any prim~ the quantiw of entrles in !the which

be ~eulated in the following way: : [ . . . . . Wri te k in base p~sum I toeve, y digit and multi,,, ply .... the . . . . . numbers so obtained.

Detailed proofs &this result, as well as of the rest of this paper, are formally given in [3].

BINOMIAL ARRAYS rood pn

In calculating Pascal's arrays mod pn we notice a similar structure to the case mod p (prime): there

appear clusters mixed with zero-holes. Analyzing binomial arrays rood 2, 4 (22), 8 (23) and comparing with

binomial arrays rood 3 and 9 (32), it is noticed that there are entries divisible by pb and also by pb-~, while some

of them are divisible by pb but not by p ~ . This property is generalized [3], in the following way:

BINOMIAL ARRAYS mod p.q

Analyzing the arrays, we may arrive to the conclusion that the entry I ~ I = 0 in the array rood p.q iff

[ ~] - 0 (rood p) and [ ~] -- 0 (rood q). This proposition explains why in binomial arrays that are not rood p

(prime) or mad pn, there are zero-holes that do not correspond to the symmetry superimposed by the cluster structure.

It is proved [3], that zero holes in binomial arrays rood p.q (both primes), are obtained multiplying zero

holes corresponding to its two factors: that is why there is no more symmetry on cluster structures (Fig. 5).

Divisibi l i ty and cellular au tomata 107

Li,~i 1 0 0 !

i i 0 1 I I 2 i 12 1

I D 0 2 0 0 1 I I O i 2 0 i 1

I 2 1 2 i 2 1 2 1 , 4 ~ 0 0 0 0 0 0 0 1

I i 0 0 0 0 0 0 0 i i ~' 2 I 0 0 0 0 0 0 1 2 1

I 0 0 i 0 0 0 0 0 , ~ 0 0 i i 0 ~ I 0 0 0 0 ~ ~ 0 i I

i~ : i:,,ooo£i~:', I ~ o i o o ~ o o ~ o ~ o o ~

i ~ 6 2 2 0 1 1 0 i ! 0 2 2 0 I ~ I 2 i i I 2 i 2 I i 2 I 2 1 2 i 2 !

i 0 ~ 0 0 0 0 0 0 ~ 0 0 0 C' 0 0 0 ~ i I 0 0 0 0 0 0 0 4 ~ i ~ 0 0 0 0 0 0 0 1 i

I 2 1 0 0 0 0 O 0 ~ L . 0 0 0 0 0 0 I 2 1 i 0 0 1 0 0 0 0 0 ~ 0 0 2 O 0 0 0 0 1 0 0 1

i i 0 ! I 0 0 0 0 2 2 0 2 2 0 0 0 0 ! ! O I i ; 2 i i 2 I 0 0 0 2 1 2 2 I 2 0 0 0 i 2 I I 2 ! 0 0 2 n 0 i 0 0 2 O 0 1 0 0 2 O O I 0 0 2 0 0 !

i 0 2 2 0 1 t 0 2 2 O i 1 i 0220 i I 02201

Fig. 1 - Binomial Array mod 3

I

l!

i11i

Ii,0~!i i 0 1 ~ I 0 ! 1 1 1 1 1 1 1 1

i 0 i 0 0 0 0 0 I 0 i i I i i O 0 0 ~ l i i Z

i ~ F - E ~ 1 0 0 ~xi 00 ~ i

I O i ~" I ~0 01 '~ i 01 ! I i ! i I i i i I [ i I i i

i i] 0 lj 0 0 0 0 0 ~ O 0 0 0 " ~ i i i ,O 00 O ~ 0 0 0 0 0 B 00 {I i i

i 0 1 0 0 0 0 0 O 0 O 0 0 0 0 0 I r,~ i ii

121

i£91i 1 2 1 1 2 1

5i 2 2 iZ 1 2 1 2 1 2 1 2 1

i i 0 0 0 0 0 0 0 i i 1 2 1 0 0 0 0 0 0 1 2 1

11 ! 1 0 0 0 , 1 1 1!

i i 22 ~' I I ' I 1 ~ 22 '~, i i 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1

i i B 0 0 0 0 0 0 212 0 0 0 0 00 i I ~ o o o o o o 2 2 o o o o , 2 ~

i 2 i ! 2 1 ~ o ,"2 i 2 2 i 2'.~ ~ '~x i 2 i i 2 1

i i l i o o o o o o o o o o o o i ili ! ~-~, i o o o o o o o o o o o I ~-o"~ i

! i ' ~ i i ~ o o o o o o o o o t i ~d I i 101 ~ I 0 i 0 O 0 0 0 0 0 0 Q/'i 01 101

~ 0 0 0 - ~ - ~ , ~ ~ o o o o o ~:'~ r - ~ - - ~ ~ ~. 1 i\@ 0 0 0 0 O, I 1,0 0 0 0 0 0 ~ * ' , i ~ 0 0 0 0 ~ i ~.

Fig. 2 - Binomial Array rood 2

Fig 3 - Binomial Array mod 3

108 C. CRESPO CRESPO e t al.

i

: i

i 2 l

! 5 : I

• i

1 7 5 3 3 5 7 1

I ~ 6 6 ~ B f i I

: ~.,W?W~v~., - 2 i

i 3 : 5',,2 :. 6 2 5 7 3 1

:

2 3 i 7 I 7 5 3 5 3 3 5 3 5 7 1 7 1

'":, o. 5, I

1 7 ~. 3 7 i 3 5 ,7. ~ 2 ~ 8 2 ~ Z 5 3 1 7 3 5 7 1

, 1

Fig. 4 - Binomial Arrays m o d 2. 22 and 23

t _

" " ~ " i 2 Z :

~ 2 1 l 1 i ' : "2 ~ i i

2 2 I i • t 2 : : 2 i

i ~ 2 " 3 _ i 2 : : i S i 2 2 :

2 7: ! i 3 3 a J i I ~ : ~ ~ i 2 2

i

: / . i 5 "2 / o j,, 2

c

: J

}

Fig. 5 - Binomial Arrays m o d 2.3, 3 and 2

Fi- 6 - Gauss ian Array rood 5 for q = 2

D i v i s i b i l i t y a n d ce l lu l a r a u t o m a t a 109

i

1

1

I 22

i i 3 L

: I 10

I i i 22i l

i830,:,01

1 1 3 3 3 3 1 2

i i :

1 :, I i

i ~ 2 D 1

J 1 2 2 1 i

i ~ 5 ~ 3 D 1 i i 3 3 3 3 1

2

1 2 2 2 2 I 2 ~ 3 ~

3 3 ~ 3 ~ 3 , 1

3 3 3 3 1 1

2 2 2 2 2 2

i

1

2

2 2 0

1

Gi 311

~i:r i

~ 1 1 i 1

1 0 2 ~ I

I 1 2 2 1 1

i ~ 3 8 3 0 1

1 3 3 3 3 1 i

Fig. 7 - Gauss ian Array m o d 4 for q = 3

1 ! !

i t : I : i I 'y; i~ i i

! i

i 1 i i i 1 i i

1

i

I l

i

i

i $

i ,

~771 !

i Ti il : i I ! i i i i i i ! I

1 7 7 1 1 7 7 1 L ~

i i

W: i i

i

u

: i , 7 i !

7 ? ! 799 7 i 7 7 i

i 3 i : $ I i

i .-; : : 1

I 7 7 I ~- ! i 3 } i I : i 7 7 i

i I ~ ~ 9 ~ ! 9 9 ! i i

:, 7 I 3 i

? 7 ~ 339'I 77 i 9 3 3 9 1 7 7 1 1

m ! 1

1

Ll!lllli

1 !

i Z i 7

Fig. 8 - Gauss ian Array m o d 10 for q = 5

1 i

7 7 i

i

1 !

Fig. 9 - Gauss ian An'ay mod 10 for q = 2

Divisibility and cellular automata 111

GAUSSIAN COEFFICIENTS

Gaussian coefficients are defined, for q ;~ 1, in the following way

... (q" - l ) ( q * " - 1 ) . , . ( q "''+' I I

l lq ( q - l)(q" . I)... (q" - 1) I

If0 < r < n: [ 0n]q = 1 and i f r < 0 or else r > n, then [ n]r q = 0. With this definition, it is possible to derive a

recursive expression for arrays, similar to the gaussian arrays for natural numbers

II = 1 ,:I1, + q'l°;'lq I a) Casep and q coprimes: Let a be the gaussian o fq rood p,that is, the smallest positive exponent for which q - 1 (rood p). Due to the reeursion formula, in a -file there appears a zero hole. The first zero hole has

zero entries. Applying Fermat's theorem, it is possible to prove [3] that the heads of the cells and of the clusters are the binomial coefficients rood p (Figs. 6 and 7).

b) Case p and q not coprime~. An important property of these arrays is the following: An array of gaussian coefficients of q rood p, has no zero entries iff q and p are not coprimes (Figs. 8 and 9).

FRACTAL CHARACTERISTICS

Fractal geometric characteristics [13] of binomial and gaussian arrays are put in evidence by the self- similarity obtained in limiting patterns.

If the number of inverted triangles with base length i is indicated T i , then it is easy to see that for Pascal's triangle rood 2 is Ti/2 = 3 T i . The fractal dimension of this self-similar figure can be calculated as in the ease of Sierpinski's gasket

D = l o g 2 3 ~ - 1,59

For Pascal's triangles reduced rood p (prime), the fractal dimension may be calculated by a similar method [ 15] p

Dp = l o g p ~ i = l + i o g p ( p + 1 ) / 2 i = l

For large p:

1 D = l i m Dp ~ 2 -

p --. ** log 2 P

CONJECTURES

a) In the gaussian array o f q mod p, with p and q not coprimes, there exist "zeros holes" o fw elements, with w depending on p and q

112 C. CRESPO CRESPO et al.

b) The structure o f the "zero-holes" o f the gaussian arrays o f q mod p (with p and q coprimes), coincides in some cases with the one for "not zero holes" o f arrays with p and q not coprimes.

c) The entries o f the gaussian arrays o f q mod p with p and q not coprimes, correspond to a proper subset o f the set Ip-i \{0}.

C O N C L U S I O N S

Refen'ing to the binomial arrays mod p, the entries o f the main cell have sufficient information for constructing an array as big as one may want.

For two different primes, the arrays are essentially different. Binomial arrays rood p are interrelated because zero holes become smaller as the exponent grows.

Gaussian arrays o f q rood p, with p and q coprimes, are a "central enlargement" in ~ (gaussian o f q rood p) o f the binomial arrays.

I f the conjectures are proved in the future, it would be an advance in this interesting research subject which relates Number Theory with Fractal Geometry and, as Peitgen et al. [10] remark: "Fractal Geometry is the Geometry o f Chaos"

REFERENCES

1. P. Ball Ch. Tang, K. Wiesenfeld, Sclf-organizedcriticality, Phys. Rev. A, 38, 1, 364-374 (1988). 2. A. W. Broomhead, Pascal (mod p), Math. Gazzete, 56, 267-271 (1972). 3. C. Crespo Crespo. Ch. Ponteville, V. W. de Spinadel. Binomial and Gaussian arrays, to appear. 4. A. W. M. Dress. M. Gerhardt, N. I. Jaeger, P. J. Plath, H. Schuster. Some proposals concerning the mathematical modelling of oscillating heterogeneous catalytic reactions on metal surfaces. In L. Rensing and N. I. Jaeger (eds.), Temporal Order. Springer- Verlag, Berlin (1984). 5. N. J. Fine. Binomial coefficients modulo a prime. Amer. Math. Monthly, 54, 589-592 (1947). 6. E. Garcia, Geometria computacional- lntroducci6n a la Geometrla Fractal. Elem. de Matem., vol. VI, 22 (1991). 7.K. IL Maclean, Divisibility properties of binomial coefficients. Math. Ga~'zete, 58, 17-24 (1974). 8. B. MandelbroL Los objetosfractales. Tusquets ed., Barcelona (1987). 9. B. Mandelbrot. Thefractal Geometry o f Nature, W. H. Freeman and Co., N¢~' York. 1977, 1982. 1983. 10. H.-O. Peitgen, H.Jtmrgeus, D. Saupe. Chaos andFractals, Springer-Verlag (1992). 11. L. Santal6, Conjuntosfractales. Elem. de Matem., vol. VI, 23 (1992). 12. M. IL Schroeder. Number Theory in Science and Communication, Springer-Verlag, Berlin'(1986). 13. V. Spinadel, Geometria Fractal, Editorial Nueva Libreria (1993). 14. M. Sved, Divisibility - with Visibility, The Mathem. lntellig., 10, 2, 56-64 (1988). 15. S. Wolfram. Geometry of binomial coefficients, Amer. Math. ~lonthly, 94, 566-571 (1984). 16. S. Wolfram, Universality and Complexity in Cellular Automata, Physica D, 10, 1-35 (1984).