The Algorithmic Beauty of Sea Shells

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Springer Science & Business Media, Aug 2, 2009 - Science - 269 pages
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The pigment patterns on tropical shells are of great beauty and diversity. Their mixture of regularity and irregularity is fascinating. A particular pattern seems to follow particular rules but these rules allow variations. No two shells are identical. The motionless patterns appear to be static, and, indeed, they consist of calci?ed material. However, as will be shown in this book, the underlying mechanism that generates this beauty is eminently dynamic. It has much in common with other dynamic systems that generate patterns, such as a wind-sand system that forms large dunes, or rain and erosion that form complex rami?ed river systems. On other shells the underlying mechanism has much in common with waves such as those commonly observed in the spread of an epidemic. A mollusk can only enlarge its shell at the shell margin. In most cases, only at this margin are new elements of the pigmentation pattern added. Therefore, the shell pattern preserves the record of a process that took place over time in a narrow zone at the growing edge. A certain point on the shell represents a certain moment in its history. Like a time machine one can go into the past or the future just by turning the shell back and forth. Having this complete historical record opens the possibility of decoding the generic principles behind this beauty.
 

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Contents

Shell patterns a natural picture book to study dynamic systems and biological pattern formation
1
12 Pattern formation
2
13 Dynamic systems are difficult to predict
3
14 Pattern formation in biology
4
15 Most shell patterns preserve a faithful time record
5
Lines perpendicular parallel and oblique to the direction of growth
6
17 Oblique lines
8
18 Relieflike patterns follow the same rules
9
76 Branch formation by a temporary stabilization
116
77 Intimate coupling of an enhancing and an extinguishing reaction
119
78 Extinguishing that results from a depletion of resources due to an enhancing reaction
121
79 Related patterns reveal unsolved problems
123
710 Apparently different patterns can be simulated by closely related models
126
711 Conclusion
128
Triangles
131
81 The crossing solution through the backdoor
132

19 Many open questions and some hints
10
complex patterns
15
111 Earlier attempts to understand shell patterns
17
Pattern formation by local selfenhancement and long range inhibition
19
22 Stable patterns require a rapid antagonistic reaction
20
23 Periodic patterns in space
21
24 The width of stripes and the role of saturation
25
25 Early fixation of a pattern
27
26 The activator depleted substrate scheme
29
27 The influence of growth
30
28 Inhibition via destruction of the activator
32
29 Autocatalysis by an inhibition of an inhibition
33
210 Formation of graded concentration profiles
35
211 Pattern formation in two dimensions
38
Oscillations and traveling waves
41
31 The coupling between the oscillators by diffusion
44
32 The width of bands and interbands
47
34 Traveling waves require a pacemaker region
49
Superposition of stable and periodic patterns
53
41 The formation of undulating lines and the partial synchronization of cells by activator diffusion
54
42 Reducing wave termination with a longer activation period
58
44 Hidden waves
60
45 Pattern on the shell of Nautilus pompilius
61
46 Stabilizing an otherwise oscillating pattern by diffusion
62
47 Combinations of oscillating and nonoscillating patterns
63
49 The possible role of a central oscillator
66
410 Conclusion
68
Crossings meshwork of oblique lines and staggered dots the combined action of two antagonists
71
52 Pattern variability
73
53 Global pattern rearrangements
74
oblique lines initiated or terminated out of phase
76
55 Crossings and branching
79
56 Changing the wave speed before and during collisions
82
57 Parallel and oblique rows of staggered dots
84
58 Conclusion
89
Branch initiation by global control
91
62 Simultaneous pattern change in distant regions
93
63 No Oliva shell is like another
98
64 The influence of parameters
99
65 Alternative mechanisms
100
66 A very different pattern generated by the same interaction
101
The big problem two or more timedependent patterns that interfere with each other
105
72 White nonpigmented droplike pattern on a pigmented background
108
73 Evidence of a sudden extinguishing reaction
110
74 Resolving an old problem with the separate extinguishing reaction
111
an additional stabilizing pattern
112
82 Triangle versus branch formation
135
83 The involvement of three inhibitory reactions
139
84 Breakdown as a failure of the enhancing reaction
143
85 Conclusion
145
Parallel lines with tongues
147
refresh comes too late
150
93 Variations on a common theme
157
tongues and branches on the same shell
159
95 Missing elements missing links
162
Shell models in three dimensions
167
102 Elements of shell shape
168
103 The helicospiral
169
104 The generating curve
171
106 Modeling the sculpture on shell surfaces
174
107 Shells with patterns
179
The computer programs
187
113 GUIDED TOURS
190
115 Numerical instabilities that may cause errors
192
116 Compilers and versions
193
117 Parameters used in the program
194
Pattern formation in the development of higher organisms
205
121 Hydra a versatile model system
208
122 Tissue polarity and graded competence
211
123 How to avoid periodic structures during growth
212
head and foot of hydra
214
125 Induction of adjacent structures
215
126 The evolution of the main body axes
216
127 Gene activation under the control of a morphogen gradient
219
128 Positiondependent activation of several genes
221
the initiation of legs and wings
224
1210 Conclusion
228
Pattern formation in development in which shellrelated mechanisms are implicated
231
the formation of netlike structures
235
133 Chemotactic orientation of cell polarity
239
134 Highly dynamic effects in preparing cell division in budding yeast
242
135 Outofphase oscillations in Ecoli bacteria for centerfinding to determine the plane of cell division
244
traveling waves at the border to multicellular organisms
245
137 Feather patterns
247
138 Color patterns of feathers
248
139 Barbs of flight feathers are separated by traveling waves of local signals
250
1310 Nerve conduction as a traveling wave phenomenon
251
1311 Activation and extinguishing waves in blood coagulation
252
References
255
Index
264
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Meinhardt-Max-Plank-Institut fur Entwicklungsbiologie, Tubingen, Germany

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