PII: 0898-1221(89)90165-X Computers Math. Applic. Vol. 17, No. 1-3, pp. 321-336, 1989 0097-4943/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press pie M A T H E M A T I C S A N D B E A U T Y - - V I I I T E S S E L A T I O N A U T O M A T A D E R I V E D F R O M A S I N G L E D E F E C T C. A. PICKOVER IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. Abstract--To help characterize complicated physical and mathematical structures and phenomena, computers with graphics can be used to produce visual representations with a spectrum of perspectives. In this paper, unusual "tesselation automata" (TA) are presented which grow according to certain symmetrical recursive rules. TA are a class of simple mathematical systems which exhibit complex behavior and which are becoming important as models for a variety of physical processes. This paper differs from others in that its focuses on symmetrical TA derived from a single defect, and reader involvement is encouraged by giving "recipes" for the various chaotic forms which represent a visually striking and intricate class of shapes. I N T R O D U C T I O N "Some people can read a musical score and in their minds hear the music . . . . Others can see, in their mind's eye, great beauty and structure in certain mathematical functions . . . . Lesser folk, like me, need to hear music played and see numbers rendered to appreciate their structures." P. B. SCHROEDER T o d a y , there are several scientific fields d e v o t e d to the s t u d y o f h o w c o m p l i c a t e d b e h a v i o r c a n arise in systems f r o m simple rules a n d h o w m i n u t e changes in the i n p u t o f n o n l i n e a r systems c a n lead to large differences in the o u t p u t ; such fields include c h a o s a n d tesselation a u t o m a t a ( T A ) theory. I n this p a p e r , I briefly discuss s o m e empirical results o b t a i n e d b y e x p e r i m e n t a t i o n with a p a r t i c u l a r class o f s y m m e t r i c a l T A . S o m e o f the resulting p a t t e r n s are reminiscent o f the p l a n a r o r n a m e n t s o f a v a r i e t y o f cultures ( o r n a m e n t s with a r e p e a t i n g m o t i f in a t least two n o n p a r a l l e l directions). " T e s s e l a t i o n a u t o m a t a " are a class o f simple m a t h e m a t i c a l systems which are b e c o m i n g i m p o r t a n t as m o d e l s f o r a v a r i e t y o f physical processes. R e f e r r e d to variously as "cellular a u t o m a t a " , " h o m o g e n e o u s s t r u c t u r e s " , "cellular s t r u c t u r e s " a n d " i t e r a t i v e a r r a y s " , they h a v e been applied to a n d r e i n t r o d u c e d f o r a wide v a r i e t y o f p u r p o s e s [1-4]. T h e t e r m " t e s s e l a t i o n " is used in this p a p e r f o r the following reasons: when a floor is c o v e r e d with tiles, a s y m m e t r i c a l a n d repetitive p a t t e r n is o f t e n f o r m e d - - s t r a i g h t edges being m o r e c o m m o n t h e n c u r v e d ones. Such a division o f a p l a n e into p o l y g o n s , r e g u l a r o r irregular, is called a " t e s s e l a t i o n " - - a n d I h a v e c h o s e n " t e s s e l a t i o n " here to e m p h a s i z e these g e o m e t r i c aspects o f t e n f o u n d in the figures in this p a p e r . U s u a l l y T A consist o f a grid o f cells which c a n exist in two states, occupied o r unoccupied. T h e o c c u p a n c y o f o n e cell is d e t e r m i n e d f r o m a simple m a t h e m a t i c a l analysis o f the o c c u p a n c y o f n e i g h b o r cells. O n e p o p u l a r set o f rules is set f o r t h in w h a t has b e c o m e k n o w n as the g a m e o f " L I F E " [2]. T h o u g h the rules g o v e r n i n g the c r e a t i o n o f T A are simple, the p a t t e r n s they p r o d u c e a r e very c o m p l i c a t e d a n d s o m e t i m e s seem a l m o s t r a n d o m , like a t u r b u l e n t fluid flow o r the o u t p u t o f a c r y p t o g r a p h i c system. T h e t e r m " c h a o s " is often used to describe the c o m p l i c a t e d b e h a v i o r o f n o n l i n e a r systems, a n d T A are useful in describing certain aspects o f d y n a m i c a l systems exhibiting irregular ( " c h a o t i c " ) b e h a v i o r [5, 6]. O t h e r simple a l g o r i t h m s studied b y the a u t h o r which p r o d u c e interesting a n d c o m p l i c a t e d b e h a v i o r are described in Ref. [7]. A p a r t f r o m their curious m a t h e m a t i c a l properties, m a n y n o n l i n e a r m a p s n o w h a v e a n i m m e n s e a t t r a c t i o n to physicists, because o f the role they p l a y in u n d e r s t a n d i n g certain p h a s e transitions a n d o t h e r c h a o t i c n a t u r a l p h e n o m e n o n [5]. T h e p r e s e n t p a p e r is n u m b e r eight in a " M a t h e m a t i c s a n d B e a u t y " series [7] which presents aesthetically a p p e a l i n g a n d m a t h e m a t i c a l l y interesting p a t t e r n s derived f r o m simple functions. T h e resulting pictures should b e o f interest to a r a n g e o f scientists as well as h o m e - c o m p u t e r artists. 321 322 C.A. PICKOVER M O T I V A T I O N One goal o f this paper is to demonstrate and emphasize the role o f recursive algorithms in generating complex forms a n d to show the reader h o w to create such shapes using a computer. A n o t h e r goal is to demonstrate how research in simple mathematical formulas can reveal an inexhaustible new reservoir o f magnificent shapes and images. Indeed, structures produced by these equations include shapes o f startling intricacy. The graphics experiments presented, with the variety o f accompanying parameters, are good ways to show the complexity o f the behavior. This paper differs from others in t h a t its focuses on TA derived from a single defect (explained below) using symmetrical rules, a n d that reader involvement is encouraged by giving "recipes" for the various chaotic forms which represent a visually striking and intricate class o f shapes. M E T H O D A N D O B S E R V A T I O N S T A are mathematical idealizations o f physical systems in which space a n d time are discrete [1]. Here I present unusual patterns exhibited by figures " g r o w i n g " according to certain recursive rules. The growth occurs in a plane subdivided into regular square tiles. Note, in particular, that with the rules o f growth in this paper, the figures will continue increasing in size indefinitely as time progresses. In each o f my cases, the starting configuration is only 1 occupied square, which can be t h o u g h t o f a single defect (or perturbation) in a lattice o f all 0s, represented by: 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 (1) TA Type I This is the simplest system to set up, yet the behavior is still interesting. Given the n t h generation, I define the (n + 1)th as follows. A square o f the next generation is formed if it is orthogonaUy contiguous to one and only one square o f the current generation. Starting with the pattern in equation (1) for n = 1 pattern for n = 2 would be: iioooo 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 (2) Figure 1 indicates the results at n = 200. This T A is similar to that described in Refs [2, 4]. N o t e t h a t no " d e a t h ' s " o f squares occur (i.e. no 1 ~ 0 transitions can occur; deaths are employed in m a n y C A experiments [2]). N o t e also t h a t on the four perpendicular axes [which go t h r o u g h (0, 0)], all the squares will be present. These are the stems from which branching occurs. TA Type 2. Time dependence o f rules A. " M o d 2 " TA. Given the n t h generation, I define the (n + 1)th as follows. A square o f the next generation is formed if: 1. It is orthogonally contiguous to one and only one square o f the current generation for even n (i.e. n m o d 2 = 0). 2, It is contiguous to one and only one square o f the current generation, where the local n e i g h b o r h o o d is b o t h o r t h o g o n a l and diagonal, for odd n (n m o d 2 = 1). Mathematics and beauty--VIII 323 In other words, for (n m o d 2 = 0) if ~ Corth = 1 ~ CO.= 1, where F o r (n m o d 2 = 1) co,, = I t , j ÷ , . cij_ ,. C,+ ,j. C,_ ,j]. (3) (4) (5) if ~ Corth_diag ---~ 1 ~ C/j = l, where Cort,~i~ = [Cij+ ,, Cij_ ,, C,+ ,j, Ci_ ,j CI+ ,j+,, Ci_ ,j_ l, Ci_ t..,+ ,, Ci+ ,j_ ,]. (6) Notice the discrete symmetrical planes running through these TA. F o r example, see the planes in Figs 2 a n d 3. We can use this observation to get a visual idea o f resultant patterns, for large n, in a multi-defect system (see TA Type 3). B. " M o d 6" TA. Given the n th generation, I define the (n + 1)th in a same m a n n e r as for Type 2A, except that n rood 6 = 0 vs n m o d 6 :/: 0 determines the temporal evolution o f the pattern (Fig. 4). TA Type 3. Contests between defects M o r e t h a n one initial defect can be placed on a large infinite lattice. We can let them each grow and finally merge (and compete) according to a set o f rules. Figure 5 is a TA o f Type 2A, a n d it shows three defects after just a few generations (this figure is magnified relative to others). Figure 6 shows the growth for large n. To help see the n u m e r o u s s y m m e t r y planes a n d to get an idea a b o u t the shape o f the figure as it evolves, the reader can draw the primary radiating symmetrical discrete planes [see Type 2A] for example, see Fig. 6. F o r a recent fascinating article on competition o f TA rules, see Ref. [8] which models biological p h e n o m e n a o f competition a n d selection by TA "subrule competition". Type 4. Defects in a centered rectangular lattice In this type o f TA, a single defect is placed in a lattice o f the form: I 0" 0 1 1 0 0 1 1 0 (7) F0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 This is k n o w n as a "centered rectangular lattice" [9]. In some experiments, two different b a c k g r o u n d lattices with adjacent boundaries are used, and the defect propagates from its beginning point in the centered rectangular lattice through the interface into the second lattice (8) defined by: - 0 1 0 1 0 - 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 (known simply as a " r e c t a n g u l a r lattice"). A d d i n g a defect to these two-phase systems bears some similarity to seeding supersaturated solutions and watching the crystallization process grow and " h i t " the b o u n d a r y o f a solution with a different composition. In the examples in this paper, the two phases are also reminiscent o f m e t a l - m e t a l interfaces--such a silicon 100 (centered rec- tangular) and c h r o m i u m (rectangular). N o t e that with no defect present, the rules described have no effect on either lattice! Only when the defect is placed in the lattice does a n y growth occur. 324 C . A . PICKOVER Fig. 1. T A T y p e 1 " g r o w i n g " for 200 generations, s t a r t i n g w i t h a single seen in t h e c e n t e r o f this figure. Mathematics and b e a u t y - - V I I I 325 Fig. 2. TA Type 2A. The TA presented here has a time dependency to its rules o f growth. 326 C . A . PICKOVER Fig. 3. S a m e as Fig. 2, b u t p l o t t e d as its negative. Mathematics and beauty--VIII 327 Fig. 4. TA Type 2B, with time dependent growth. Fig. 5. Multi-defect system composed of three initial seeds of TA Type 2 (figure is magnified relative to Fig. 6). CAMWA 17-1/3--U 328 C . A . PICKOVER Fig. 6. Same as Fig. 5, except c o m p u t e d for longer time. Fig. 7. T A Type 4A defect which has been growing f r o m a center position in a centered rectangular lattice (seen as a diffuse grey b a c k g r o u n d at this resolution). W i t h o u t the presence o f the defect, the rules have no effect o n the lattice. Mathematics and b e a u t y - - V I I 1 329 The rules for growth o f the defects are as follows (note that deaths o f cells can occur in these systems): A. TA Type 4,4. if ~ C = 3 ^ if C ~ , j = O ~ C i j = I , (9) if ~ C = 3 ^ if C~,j=I--*Ci, j = O , (10) where if if Type 5. Larger local neighborhood c = ICe.j+,, c i . j _ , , c,+,j, c,._,~., C + , j + , , c,. ,j_,]. ( l l ) The symbol A denotes a logical " a n d " . Figure 7 shows a defect which has been growing from a central position in a centered rectangular lattice (which is seen as a diffuse grey background at this resolution). Figures 8(a)-(d) show the propagation o f the defect through a two-phase b o u n d a r y . Note that the p r o p a g a t i o n behavior is visually different once in the second layer. F o r example, notice that the growth in the b o t t o m layer appears to be constrained to planes 0 ° and 60 ° with respect to the lattice. The introduction o f " g e r m " cells appears to be useful in simulating real nucleation processes. An interesting paper in the literature describes solid-solid phase transformations o f shape m e m o r y alloys, such as C u - Z n - A I , using a 1-D cellular a u t o m a t a approach [10]. In this investigation, each cell represents several h u n d r e d atoms. The search for muitiphase systems, such as the ones in this paper, which are unaffected by a rule system until a defect is added, remains a provocative avenue o f future research. B. TA Type 4B. This case (see Fig. 9) is the same as the subset 4A, except that if ~ C = 3 / x C ~ , j = I ~ C ~ , j = O (12) if ~ C = 3 ^ C~,j = 0 --. C~,j = 1 (13) C # 3 A C i , j = 1 ---~Ci, j = 1 (14) C ~ 3 ^ Cj, j = 0 ~ Ci, j = 0. (15) In TA Types 1-4, the neighborhood was defined as being within one cell o f the center cell under consideration. In this system, the local neighborhood is larger. The rule is as follows: if ~ C = 0(mod 2) ---, Cij = 0, (16) if ~ C # O ( m o d 2 ) - - , C i , j = 1, (17) where C = [C/- 2j+2, Ci+2./+2, Cij+ ,, Ci_ ,.j, Ci+ ,j. Cij_ I, C~-2j-2, C,'+ 2./- 2]. (18) Figures 10(a)--(e) show the evolution o f a two-state b a c k g r o u n d defined by the lattices in equations (7) a n d (8) for several different snapshots in time. Unlike Type 4, the background without a defect is disturbed by this rule-set. Notice the visually unusual behavior o f this system with both symmetry and stochasticity present. Also note the interesting growth o f the two defects which have been placed next to each other in the top layer. S U M M A R Y A N D C O N C L U S I O N S "Blindness to the aesthetic element in mathematics is widespread and can account for a feeling that mathematics is dry as dust, as exciting as a telephone took . . . . O n the contrary, appreciation o f this element makes the subject live in a wonderful m a n n e r and b u r n as n o other creation o f the h u m a n mind seems to d o . " P. J. DAvis and R. HERscr~ A m o n g the methods available for the characterization o f complicated artistic, mathematical and natural phenomena, computers with graphics are emerging as an i m p o r t a n t tool (for several papers by this author, see Ref. [11]). In natural phenomena, there are examples o f complicated and ordered !!iii iiiiiiiiiil;{!iiii!;ii!i~ ~:i~!~ii!ii~!i?iiiiiii;~ii~?i?i!ii~:;ii!iii~!iii~i~i;~!i!i~!ii~i?!?ii!;ii!i~!ii!~!i!!!!iii~!i!!i3?i~i~,!;~i~!?ii~:}~i~3ii~ 4!iii ii;i!!i~;!!iiii!~!ii!~ii~!~i~!!!!~;~iii!~i!~!~iiii~!!!1~!i~ii!iii~i!~~!iiii~!iii!!ii~i~ii!i!!i~i~i~i!i!!iii!iiii~!!i~!ii!!ii~itiì~i!)i!i~ ~ 7 ` ~ ; . ; @ ~ . ; ~ ; ~ . ~ ° 2 2 ~ ; : ~ ? 2 ` . ' : ~ ` ~ ; ~ . ; . 2 2 , ~ ` ~ 2 ~ : ~ ; ~ . ~ . ~ ` 1 ; , 2 ~ ; . ~ . ~ , 2 ; ~ ; ~ ; ~ ` ; ' ~ ° ~ @ ; . . : ~ ; ~ 2 4 - ~ 2 @ 2 ~ ; < ~ 2 ~ @ 2 ~ < ~ , ~ - , °>;°2" ~,. ,~.,.:-' Iltlillllil!l!!i l!lillt!llilttlllllli[ J , , ,,:: i " i Ii1 i~illlllll~lJ ~ li !I II " ~ . . . . . . . . . "~ li~ I t,, !!!!!!!!! III Fig. 8(a). Magnified picture of the beginning of propagation of a Type 4A defect through a two-phase system. The top phase is a centered rectangular lattice, while the bottom phase is a rectangular lattice. Fig. 8(b). Same as Fig. 8(a) except less magnified and computed for 60 generations. The defect has just "broken through" the boundary. 330 Mathematics and b e a u t y - - V I I I 331 Fig. 8(c). Same as Fig. 8(b), for 80 generations. Fig. 8(d). Same as Fig. 8(c), for 100 generations. 332 C.A. PICKOVEP, Fig. 9(a). Propagation for TA. Type 4B in a two-phase system (n = 120). structures arising spontaneously from " d i s o r d e r e d " states and examples include: snowflakes, patterns o f flow in turbulent fluids, and biological systems. As W o l f r a m points out [1], TA are sufficiently simple to allow detailed mathematical analysis, yet sufficiently complex to exhibit a wide variety o f complicated phenomena, and they can perhaps serve as models for some real processes in nature. In contrast to previous systems where mathematical and aesthetic beauty relies on the use o f imaginary numbers [12], there calculations use integers--which also facilitates their study with p r o g r a m m i n g languages having no complex d a t a types on small personal computers. The forms in this paper contain b o t h symmetry a n d stochasticity, and the richness o f resultant forms contrasts with the simplicity o f the generating formula. R u n n i n g T A at high speeds on a computer lets observers actually see the process o f growth. T A portraits contain a beauty and complexity which corresponds to behavior which mathemati- cians were n o t able to fully appreciate before the age o f computer graphics. This complexity makes it difficult to objectively characterize structures such as these, and, therefore, it is useful to develop graphics systems which allow the maps to be followed in a qualitative and quantitative way. The T A graphics p r o g r a m allows the researcher to display patterns for a specified length o f time and for different rule systems. Some o f these figures contain what is known as n o n s t a n d a r d scaling symmetry, also called dilation symmetry, i.e. invariance under changes of scale (for a classification o f the various forms o f self-similarity symmetries, see the second reference in R e f [12]). F o r example, if we look at any Mathematics and beauty--VIII 333 Fig. 9(b). Propagation for TA Type 4B in a two-phase system (n --200). one o f the geometric motifs we notice that the same basic shape is f o u n d at a n o t h e r place in a n o t h e r size. Dilation symmetry is sometimes expressed by the formula (r ~ a t ) . Thus, an expanded piece o f some TA can be moved in such a way as to make it coincide with the entire TA, and this operation can be performed in an infinite number o f ways. Other more trivial symmetries in the figures include the bilaterial symmetries and the various rotation axes and other mirror planes in the TA. Note the dilation symmetry has been discovered and applied in different kinds o f p h e n o m e n o n in condensed matter physics, diffusion, polymer growth and percolation clusters. One example given by K a d a n o f f [13] is petroleum-bearing rock layers. These typically contain fluid-filled pores o f m a n y sizes, which, as K a d a n o f f points out, might be effectively understood as 2-D fractal networks k n o w n as gaskets [13], and I would add that T A m a y also serve as visual a n d physical models for these types o f structures. These figures m a y also have a practical importance in that they can provide models for materials scientists to build entirely new structures with entirely new properties [14]. F o r example, G o r d o n e t al. [14] have created wire networks on the micron size scale similar to some o f these figures with repeating triangles. The area o f their smallest triangle was 1.38 +_ 0.01 # m 2, and they have investigated m a n y unusual properties o f their superconducting network in a magnetic field (see their paper for details). F r o m an artistic standpoint, T A provide a vast and deep reservoir from which artists can draw. The c o m p u t e r is a machine which, when guided by an artist, can render images o f captivating power and beauty. New "recipes", such as those outlined here, interact with such traditional elements as 334 C, A. PICKOVErt Fig. 10(a). E v o l u t i o n o f a two-state b a c k g r o u n d defined by t h e lattices in e q u a t i o n s (7) a n d (8). T w o adjacent defects have been placed in t h e t o p layer, a n d the result for two generations (n = 2) is shown. .l•r•,•.•.•'•l:.•:l,•.-"• -- ~ . . . I IIII Fig. 10(b). Figure after 8 generations (n = 8). ?i ?i Fig. 10(c). n = 20. Fig. 10(d). n = 40. Fig. 10(e). n = 80. N o t e that t h e patterns, previously well ordered, appears to be o n the r o u t e to " c h a o s " . Mathematics and beauty--VIII 335 form, shading and color to produce futuristic images and effects. The recipes function as the artist's helper, quickly taking care o f much o f the repetitive and sometimes tedious detail. By creating an environment o f advanced computer graphics, artists with access to computers will gradually change our perception o f art. Also from a purely artistic standpoint, some o f the figures in this paper are reminiscent o f Persian carpet designs [15], ceramic tile mosaics [15], Peruvian striped fabrics [15], brick patterns from certain Mosques [16], and the symmetry in Moorish ornamental patterns [17]: A v _ ~ & A w . . ~ & A v . . v A A ~ . . v A A t _ v & & ~ A & " ~ A Y A A k v / ~ A & v A A- A v A , • k v A A ,A ~kAA r ~ A v v v , ,%T.VAV.v~ v & A v v,,, A v . T A & v . . v & & v . . v & & ~ m v & A v v~ t',~ k--AV,,'A--AV,: • ~ V ~ , t ~ w r • i , v ~ ~ ,-AV~ fk~ • v , . v • yAv. v . r A y - v yAv_v_v, Av...~V,A$ ~ • . v ~ v . v , r k v " .v~ ~,v. y , ~ A*..*A A*_vA ,," ~..•~v. v y A v . • .v~v. • .v•v. • .v~v. • • . • • ~ k , ~ v A ~ _ v _ vAv,,,~ .vAv..,v..V'Av V...g,.VAV_.V_VAV. ~k,Y..,vA Av..y.,dkA v,,dP'A A v , dP'& A Y l v A ~ w L V A k ~ I Scheme 1 T h e i d e a o f i n v e s t i g a t i n g t h e o r n a m e n t s a n d d e c o r a t i o n s o f v a r i o u s c u l t u r e s b y c o n s i d e r a t i o n o f t h e i r s y m m e t r y g r o u p s a p p e a r s t o h a v e o r i g i n a t e d w i t h P o l y a [15, 18]. T h i s a r t i s t i c r e s e m - b l a n c e is d u e t o t h e c o m p l i c a t e d s y m m e t r i e s p r o d u c e d b y t h e a l g o r i t h m , a n d i t is s u g g e s t e d t h a t t h e r e a d e r e x p l o r e t h e v a r i o u s p a r a m e t e r s t o a c h i e v e a r t i s t i c c o n t r o l o f t h e v i s u a l effect m o s t d e s i r e d . I n s u m m a r y , a l l t h e T A s h o w n h e r e h a v e a n i n f i n i t e v a r i e t y o f s h a p e s , a n d a l t h o u g h t h e e q u a t i o n s s e e m t o d i s p l a y w h a t m i g h t b e c a l l e d " b i z a r r e " b e h a v i o r , t h e r e n e v e r t h e l e s s s e e m s t o b e a l i m i t e d r e p e r t o r y o f r e c u r r e n t p a t t e r n s . A r e p o r t s u c h a s t h i s c a n o n l y b e v i e w e d a s i n t r o d u c t o r y . H o w e v e r , it is h o p e d t h a t t h e t e c h n i q u e s , e q u a t i o n s , a n d s y s t e m s will p r o v i d e a u s e f u l t o o l a n d s t i m u l a t e f u t u r e s t u d i e s in t h e g r a p h i c c h a r a c t e r i z a t i o n o f t h e m o r p h o l o g i c a l l y r i c h s t r u c t u r e s p r o d u c e d b y r e l a t i v e l y s i m p l e g e n e r a t i n g f o r m u l a . Acknowledgement--I owe a special debt of gratitude to Charles Bennett for introducing me to TA Type 4B. R E F E R E N C E S 1. S. Wolfram, Statistical mechanics of cellular automata. Re•. mod. Phy. 55, 601-644 (1983). 2. W, Poundstone, The Recursive Universe. Morrow, New York (1985). 3. S. Levy, The portable universe: getting to the heart of the matter with cellular automata. Whole Earth Re•. 49, 42-48 (1985). 4. R. Schrandt and S. Ulam, On recursively defined geometrical objects and patterns of growth, In Essays on Cellular Automata (Ed. A. Burks). 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Kristallogr. 60, 278-282 (1924). A P P E N D I X Recipe for Picture Computation In order to encourage reader involvement, the following pseudocode is given. Typical parameter constants are given within the code. Readers are encouraged to modify the equations to create a variety o f patterns o f their own design. Initially, the C array is 0 for all its elements, except f o r a value of I placed in its center. For the program below, a temporary array, Ctemp, is used to save the new results o f each generation. The routine below would be called n = 200 times in a typical simulation. ALGORITHM: TA GENERATION (TYPE 2A) INPUT: I DEFECT, Centered in the C array OUTPUT: TA PATTERN TYPICAL PARAMETER VALUES: Size = 400 N is the generation counter - goes from I to 200 do i = 2 to size-l; (* X - direction *) do j = 2 to size-l; (. Y - direction .) if C(i,j) = 0 then do; (. Test for vacancy .) if mod(n,21 = 0 then (. Test for even number .) sum = C(i,j+1)+C1i,j-1)+C(i+1,j)+C(i-l,j); else sum = C(i,j+1)+C(i,j-1)+C(i+1,j)+C(i-l,j) + C(I+I,j+I) + C(i-l,j-1}+ C(I-I,j+I)+C(i+I,j-I); if sum = I then Ctemp(i,j) = 1; end; (* End j loop .) end; {* End i loop ") Program 1