Frozen Temporal Pattern in Growing System
J. Schwietering1 and P.J. Plath2    Hämatol. Bluttransf. Vol.35

Fig. 4. a The pigmentation of a top shell of the family Trochidae. b Coloured pattern of the number of prepigments produced by the onedimensional vector automaton. The rule is shown at the bottom

automaton showing interference with each other. There is another very interesting behaviour (see Fig. 3 a): a wave consisting of two dispersing one-dimensional excited states periodically creates an excited block in between. The end points of the blocks become the starting points of two new waves, which will behave like the original wave. By this means, the wave is creating a fractal core inside itself. Figure 3b shows the pattern of a feather which greatly resembles this fractal core wave. Starting not just with one special cell but with a few of them, which may~be distributed randomly, a universe of different and complex patterns can be produced. Among these patterns there are classes which strongly resemble the frozen pattern ofthe seashells (see Fig. 4). What is really astonishing is the major role of the fractals among the pattern in the seashells (see Fig. 5). For example, the Cymbiolacca shell exhibits a pattern of brown triangles of different sizes. The way the triangles are interlocked is typical for patterns of penetrating fractal Sierpinsky gaskets [13].

Fig. 5. a A Cymbiolacca shell (family Volutidae) with a fractal pigment pattern. b An automaton, the pattern of which resembles the main elements of Cymbiolacca pigmentation

Sometimes one can observe showers of small triangles, while on other positions the sides of large triangles run through pale yellow-grey parts of the shell. A book of seashell patterns look like a zoo of fractal patterns and their combinations, which can be observed especially on cone shells [14, 15]. Another typical fractal pattern can be seen in the shell of Conus princeps (Fig. 6). Simulating this pattern with our automaton machine, it can be classified by the type ofinterpenetrating core waves which have been mentioned above. automaton, the pattern of which resembles the main elements of Cymbiolacca pigmentation

Concluding Remarks

The regularities and the irregularities in the pattern of the shells can be reproduced by the automaton model if fractal patterns interact starting from different positions at time t = O. What is so fascinating about the fractal patterns? Fractals are strongly related to the occurrence of deterministic chaos [16], which does not mean that one loses all regularities but only the simple symmetries such as translational or rotational symmetry. If all these symmetries break down in a

Fig.6. a Conus prinzceps (Linnaeus 1758), Sinaloa, Mexico. b The pattern of an automaton which resembles the main elements of the pigmentation of the Conus princeps

state of chaos, one very characteristic symmetry survives: the dilatation symmetry, which is mostly disregarded. In the pattern of the seashells it is precisely this dilatation symmetry which plays the major role, since all the other symmetries vanished. The growth mechanism of the pattern proposed above is based on a strictly ordered one-dimensional arrangement of cells. This is sensible in the case of the seashells but one can also develop the method for two- or three-dimensional processes to explain other phenomena in pure chemical or living systems. Even reaction in fluids can be modelled in such a way, if the cells stay together for a suitable period of time [17-19]. It would be of great interest to look for the spreading of the patterns of excited cells, even in flowing systems. Acknowledgement. The photograph in Fig. 7a is reproduced with kind permission of T.F.H. Publication Inc. Ltd., The British Crown Colony Hong Kong, to whom we are indebted.


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