Symmetry in the Markings of Animals

Palindromes such as “madam”, “racecar” or “was it a car or a cat I saw” are having reversal symmetry with respect to the letters that compose such words or sentences. Beyond symmetry of words, the symmetry of structures such as Taj Mahal is crafted in wonder and awe. A goat died hungry because it couldn’t decide which way to go, as the grass was terribly symmetric on either side. Symmetry is full of appeal and is often surprising. William Blake wrote in “The Tyger”:

Tyger Tyger, burning bright,

In the forests of the night;

What immortal hand or eye,

Could frame thy fearful symmetry?

Blake used fearful symmetry as a staunch reminder of absence of symmetry or symbiosis in a universal sense, which is clear from his another poem “The Lamb”, where his views go to the other extreme. In fact if the poem 'The Tyger' is read in conjunction with the poem 'The Lamb' (from 'Songs of Innocence'), a contrary notion develops which causes aspersions and doubts on the word symmetry, which is more of an innocence versus ferocity. 'The Tyger' thus reminds us of side by side existence of innocence and ferocity, and Blake wants to see the hand that created a meek animal such as lamb and such a ferocious animal as Tyger. Thus Blake probably was impressed more by beauty and asymmetry than by the ordered pattern.

However, the patterns on the skin of animals are indeed presenting some remarkable geometric features and have regularity and order. The markings of animals may mean different things to different professionals, e.g., to an artist, there are patterns that seem to convey messages of beauty and joy, while to a common man there are demarcations in the colours that seamlessly connect in somewhat haphazard way, still resulting in gorgeous patterns. Is there any logic or mathematics on which these patterns dwell? As someone has said, “What would life be without arithmetic, but a scene of horrors”.

Alan Turing, a computational mathematician was the first person to discern mathematics behind the markings of the animals. Turing found a surprising similarity in the animal markings, that all such markings could be produced by similar kind of equations. These kind of equations tell about chemicals as they mix with each other (over a surface or through a solid medium) and diffusion of colours that follows chemical mix. Sometimes these are referred to as reaction-diffusion equations. These equations provide a first hand insight into mathematical basis that seems to control pattern formations on animals

Turings patterns have been obtained for Emperor angelfish (Pomacanthus imperator). The parallel stripes run seventy five percent of the body with yellow and purple linings. Markings on seashells have also been obtained by Turing equations. The markings on seashells are diverse and breathtaking beauty that may bring element of skepticism in accepting that Turing equations are so powerful.However, many irregular and complicated patterns have been obtained through computer simulations by several researchers e.g., Hans Meinhardt with amazing accuracy.

If we look at overall morphology of animal markings, two aspects need to be separated, one is shape and other is pattern. The morphological evolution that occurs during growth of an organism is sometimes known as morphogenesis. Cellular rearrangements are generally linked to shape. Turings hypothesis takes activity of genes as fixed( which is not in tune with reality) and therefore has some innate shortcomings for predicting the markings in animals.

There may be pattern making genes (which become active and passive) which may be revealed by genome. However, Turing was interested more into generic way of looking at patterns. Through the concept of morphogens (prepatterns), which react and diffuse through tissues. Patterns are generated by symmetry breaking. A slight perturbation in otherwise uniform state (of distribution of chemicals) would grow quite rapidly. This does not result in a chaotic patchwork of chemicals, however, the patchy work is connected by process of diffusion that results in definite patterns which may be similar to spots and stripes.

Real-life example of Turing patterns are Bee-Zee(B-Z) reactions with wonderful concentric rings and rotating spirals. Using Turings theory, Jim Murray obtained dappling on giraffes and stripes on zebras and lions. Such computer simulated patterns are called Turing patterns. Discovery of DNA coupled with genetics posed challenges to Turing hypothesis,(e.g. fruit fly Drosophila has stripe patterns not in accordance with Turing model prediction.).

Brian Goodwin modified Turing model in which predicts breaking symmetry when creature develops. So symmetry keeps changing. Goodwin model has been successfully applied to Acetabularia, a single-celled marine alga for pattern prediction.

Finally I would say that there is a great play of mathematics in the markings of animals. Some of it now revealed by new connections invoked between zoology and mathematics. Benoit Mandelbrot has said, ”Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line.”