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 onedimensional 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 yellowgrey
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 onedimensional arrangement of cells. This is sensible in
the case of the seashells but one can also develop the method for
two or threedimensional 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 [1719]. 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.
References
1. Meinhardt H (1984) Models for positional signaling, the threefold
subdivision of segments and the pigmentation patterns of molluscs.
J Embryol Exp Morphol [Suppl] 83:289311
2. Meinhardt H, Klingler M (1987) A model for pattern formation
on the shells of molluscs. J Theor Biol 126:6389
3. Gierer A, Meinhardt H (1972) A theory of biological pattern formation.
Kybernetik 12:3039
4. Meinhardt H (1987) Bildung geordneter Strukturen bei der Entwicklung
höherer Organismen. In: Küppers BO (ed) Ordnung aus dem Chaos.
Piper, Munich, pp 215241
5. Wolfram S (1984) Universality and complexity in cellular automata.
Physica 10D:135
6. Toffoli T, Margolus N (1987) Cellular automata machines  a new
environment for modelling. MIT Press, Cambridge
7. Plath PJ (1989) Modelling of heterogeneously catalyzed reactions
by cellular automata of dimension between one and two. Optimal structures
in heterogeneous reaction systems. Springer, Berlin Heidelberg New
York (Springer series in synergetics, vol 44)
8.Schwietering J, Plath PJ. Wachsende Muster. Submitted to Wissenschaft
und Fortschritt, AkademieVerlag, Berlin (submitted for publication)
9. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence.
Springer, Berlin Heidelberg New York (Spnnger series in synergetics,
vol 19)
10.Krinky VI (1984) Selforganization autowaves and structures far
from equilibrium. Spnnger, Berlin Heidelberg New York (Springer
series in synergetics, vo128)
11. Rebbi C (1979) Solitonen. Spektrum Wiss 4:6378
12. Eilenberger G (1983) Solitons  mathematical methods for physicists.
Springer, Berlin Heidelberg New York (Springer series in solidstate
sciences, vol.19)
13.Mandelbrodt BB (1987) Die Fraktale Geometrie der Natur. Akademie,
Berlin
14. Walls JG (1978) Cone shells  a synopsis of the living conidae.
TFH, Hong Kong
15. Wilson BR, Gillett K (1971) Australian shells. Tuttle, Rutland
16. Peitgen HO, Richter PH (1986) The beauty of fractals  images
of complex dynamical systems. Spnnger, Berlin Heidelberg New York
17. Gerhardt M (1987) Mathematische Modellierung der Dynamik der
heterogen katalysierten Oxidation von Kohlenmonoxid: Numerische
Behandlung eines diskreten mathematischen Modells von über Diffusion
miteinander gekoppelter Speicher. Thesis, University of Bremen
18. Gerhardt M, Schuster H (1989) A cellular automaton describing
the formation of spatially ordered structures in chemical systems.
Physica D36:209221
19. Dewdney AK (1989) Wellen aus der Computer Retorte. Spektrum
Wiss [Sonderh] 8:3841
