The Algorithmic Beauty of Sea Shells (eBook)

Preface 6
Contents 10
1 Shell patterns - a natural picture book to study dynamic systems and biological pattern formation 14
1.1 Dynamic systems everywhere 14
1.2 Pattern formation 15
1.3 Dynamic systems are difficult to predict 16
1.4 Pattern formation in biology 17
1.5 Most shell patterns preserve a faithful time record 18
1.6 Elementary patterns: Lines perpendicular, parallel and oblique to the direction of growth 19
1.7 Oblique lines 21
1.8 Relief-like patterns follow the same rules 22
1.9 Many open questions and some hints 23
1.10 The hard problem: complex patterns 28
1.11 Earlier attempts to understand shell patterns 30
2 Pattern formation by local self-enhancement and long range inhibition 32
2.1 The activator -- inhibitor scheme 32
2.2 Stable patterns require a rapid antagonistic reaction 33
2.3 Periodic patterns in space 34
2.4 The width of stripes and the role of saturation 38
2.5 Early fixation of a pattern 40
2.6 The activator - depleted substrate scheme 42
2.7 The influence of growth 43
2.8 Inhibition via destruction of the activator 45
2.9 Autocatalysis by an inhibition of an inhibition 46
2.10 Formation of graded concentration profiles 48
2.11 Pattern formation in two dimensions 51
3 Oscillations and traveling waves 54
3.1 The coupling between the oscillators by diffusion 57
3.2 The width of bands and interbands 60
3.3 Oblique lines: traveling waves in an excitable medium 60
3.4 Traveling waves require a pace-maker region 62
4 Superposition of stable and periodic patterns 66
4.1 The formation of undulating lines and the partial synchronization of cells by activator diffusion 67
4.2 Reducing wave termination with a longer activation period 71
4.3 Interconnecting wavy lines and the formation of arches 71
4.4 Hidden waves 73
4.5 Pattern on the shell of Nautilus pompilius 74
4.6 Stabilizing an otherwise oscillating pattern by diffusion 75
4.7 Combinations of oscillating and nonoscillating patterns 76
4.8 Rows of patches parallel to the direction of growth 76
4.9 The possible role of a central oscillator 79
4.10 Conclusion 81
5 Crossings, meshwork of oblique lines and staggered dots: the combined action of two antagonists 84
5.1 Displacement of stable maxima or enforced de-synchronization by a second antagonist 84
5.2 Pattern variability 86
5.3 Global pattern rearrangements 87
5.4 Traces of the additional inhibition: oblique lines initiated or terminated out of phase 89
5.5 Crossings and branching 92
l5.6 Changing the wave speed before and during collisions 95
5.7 Parallel and oblique rows of staggered dots 97
5.8 Conclusion 102
6 Branch initiation by global control 104
6.1 Branch formation: the trigger of backwards waves 104
6.2 Simultaneous pattern change in distant regions 106
6.3 No Oliva shell is like another 111
6.4 The influence of parameters 112
l6.5 Alternative mechanisms 113
6.6 A very different pattern generated by the same interaction 114
7 The big problem: two or more time-dependent patterns that interfere with each other 118
7.1 Inherent similarities in complex patterns 118
7.2 White nonpigmented drop-like pattern on a pigmented background 121
7.3 Evidence of a sudden extinguishing reaction 123
7.4 Resolving an old problem with the separate extinguishing reaction 124
7.5 The next step in complexity: an additional stabilizing pattern 125
7.6 Branch formation by a temporary stabilization 129
7.7 Intimate coupling of an enhancing and an extinguishing reaction 132
7.8 Extinguishing that results from a depletion of resources due to an enhancing reaction 134
7.9 Related patterns reveal unsolved problems 136
7.10 Apparently different patterns can be simulated by closely related models 139
7.11 Conclusion 141
8 Triangles 144
8.1 The crossing solution through the backdoor 145
8.2 Triangle versus branch formation 148
8.3 The involvement of three inhibitory reactions 152
l8.4 Breakdown as a failure of the enhancing reaction 156
8.5 Conclusion 158
9 Parallel lines with tongues 160
9.1 Survival using a precondition pattern 160
9.2 Tongue formation: refresh comes too late 163
9.3 Variations on a common theme 170
l9.4 Conus textile: tongues and branches on the same shell 172
9.5 Missing elements, missing links 175
10 Shell models in three dimensions 180
10.1 Mathematical descriptions of shell shape: a brief history 180
10.2 Elements of shell shape 181
10.3 The helico-spiral 182
10.4 The generating curve 184
10.5 Incorporating the generating curve into the model 184
10.6 Modeling the sculpture on shell surfaces 187
10.7 Shells with patterns 192
11 The computer programs 200
11.1 Introductory remarks 200
11.2 Using the program 200
11.3 GUIDED TOURS 203
11.4 Implementation of the interactions 203
11.5 Numerical instabilities that may cause errors 205
11.6 Compilers and versions 206
11.7 Parameters used in the program 207
12 Pattern formation in the development of higher organisms 218
12.1 Hydra, a versatile model system 221
12.2 Tissue polarity and graded competence 224
12.3 How to avoid periodic structures during growth 225
12.4 How to generate structures at a distance: head and foot of hydra 227
12.5 Induction of adjacent structures 228
12.6 The evolution of the main body axes 229
12.7 Gene activation under the control of a morphogen gradient 232
12.8 Position-dependent activation of several genes 234
12.9 A problem that the mollusks don't have: the initiation of legs and wings 237
12.10 Conclusion 241
13 Pattern formation in development in which shell-related mechanisms are implicated 244
13.1 Arrangement of leaves and staggered dots on shells - two similar patterns 244
13.2 Veins and nerves: the formation of net-like structures 248
13.3 Chemotactic orientation of cell polarity 252
13.4 Highly dynamic effects in preparing cell division in budding yeast 255
13.5 Out-of-phase oscillations in E.coli bacteria for center-finding to determine the plane of cell division 257
13.6 Dictyostelium: traveling waves at the border to multicellular organisms 258
13.7 Feather patterns 260
13.8 Color patterns of feathers 261
l13.9 Barbs of flight feathers are separated by traveling waves of local signals 263
13.10 Nerve conduction as a traveling wave phenomenon 264
13.11 Activation and extinguishing waves in blood coagulation 265
References 267
Index 276