These endeavors are inspired by the famous work of Alan Turing in the early 1950s on a theory of biological pattern formation, now often referred to as Turing patterns or reaction-diffusion patterns (a good explanation can be found here). An example is the formation of spots or stripes on an animal's coat. According to the theory, pigment cells randomly decide to be coloured or not coloured. The coloured cells produce two diffusible substances which spread in the coat, an "activator" and an "inhibitor". The activator promotes cells to become coloured, and the inhibitor discourages them. The activator is short range, because it diffuses slowly and/or is destroyed quickly, while the inhibitor is longer range because it diffuses more quickly or is more stable. A coloured spot is stable as the coloured cells encourage their neighbours to be coloured whilst telling the surrounding area to stay colourless.
[from left to right: tp8, tp2] I implemented such a system as a computer program and sure enough, it developed simple spots and stripes. I had seen more complicated patterns, such as the patterns on goannas, which seemed to have structure on several length scales. I modified my program to have interacting turing instabilities at two different length scales, and got more complicated patterns such as the pair of above examples.
The next obvious step was to extend the system to coupled Turing instabilities at more length scales, from the large to the small, with results like the above bonemusic_475. An unexpected feature of the resulting patterns is that they look like electron microscope images. This may be an example of a "frustrated system" where each Turing instability is "trying" to form a pattern at a certain length scale which is incompatible with the other Turing instabilities. Animations of the process show that the system doesn't settle into a stable state, unlike simple Turing patterns, but continuously moves between states of high entropy. A further embellishment was to impose radial symmetry, resulting in the MSRSTP series (examples of which are pictured on the next page).
[Details from The 84th plate from Ernst Haeckel's Kunstformen der Natur (1904), depicting diatoms (Bacillariophyceae) / image: Wikipedia Commons] They remind me of electron microscope images of diatoms. There is an interesting contrast between the seemingly organic nature of the patterns and the strict symmetry. The symmetry is simply imposed by averaging the concentrations of "morphogens" at each point with its counterpart points at 1/n, 2/n... (n-1)/n pi radians around the circle. A different method of producing pattern at multiple levels resulted in the csym series. Initially a large scale Turing instability results in large patches of black and white, then Turing instabilities of smaller and smaller scale are applied, resulting in subdivision of the patches. This could plausibly occur in embryos, with the growth of the embryo continuously changing the effective scale of the Turing instability. In this set the imposition of radial symmetry is turned off at a certain stage, resulting in a more natural looking almost-symmetry. |
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