Math in the wild

If the spots were mobile towers,
the polygons represent the area covered

What is common to mobile towers, giraffes and cracked clay? If you do not already know about it, you might like to play around with the Java Applet here (or this) or see a particularly beautiful visualization of a particular computational method here. You can also look up more about Voronoi tesselations. The connections are well known to most folks with an interest in  "recreational mathematicians"  and although I have been aware of this from my school days, search engine results still surprise me and its great to see lots of new content. This museum/art exhibit seems particulary interesting as a way of introducing such topics  (the exhibit, strangely enough, is named after the thesis of Dr. Theodore Kaczynski - better known as the Unabomber! Incidentally, his manifesto is eminently readable and worthy of examination in our troubled times, quite unlike his thesis which was said to be accessible to only about at best 10 mathematicians when it was published )

Cracking clay

Coming back to giraffes. We usually think that every cell on our body has the same DNA  but this is just the abridged version. There can be major variations and these are large enough sources of error for one to worry about a lot of research. We used to be taught in "genetics 101" about chimerism, but I have recently learnt about the nuanced differences between that and "mosaicism". The patterns by which cells differ, multiply and communicate with each other can be thought of as driving some of the intricate patterns that one can find in nature.  In Siamese cats, for instance, it has been shown that a heat-sensitive mutant of Tyrosinase which leads to a failure in melanin synthesis causing albinism in the warm parts of the body and leaving the cooler nose and extremities black. An increasing number of works are melding ideas from genetics and computer science and some of these demonstrations include the simulation of mammalian coat pattern, such as those of giraffe. The earliest ideas on the mathematical, physical and chemical principles for biological patterns were illustrated  by pioneers like D'Arcy Thompson and Alan Turing. Turing is of course better known for his work in the theory of computability.

Clonal mosaic models to simulate coat patterns (Walter et al. 1998)

Working on the intersections of multiple fields often produces interesting results, and while these are largely ignored by professional biologists, they are becoming increasingly visible, in cartoon animation. Some of the techniques used in computer graphics and to a limited amount, biological studies are becoming more accessible on the Internet. One of these is http://algorithmicbotany.org/ with a PDF version of the book "Algorithmic beauty of plants" on simulating plant forms by Przemyslaw Prusinkiewicz and other related works including the simulation of seashell patterns by Hans Meinhardt (his book "The Algorithmic beauty of Seashells" is into its third edition but is not publicly available online, but a related paper can be found here)

Prum & Williamson (2002)

An interesting 2003 paper on the evolutionary patterns of eyespots in Satyrine butterflies is accompanied by a nice animation that gives a visual way of looking at patterns and correlating them with molecular phylogenies. (Flash animation) Another study has looked at the simulation of feather patterns. Unfortunately, nobody has made an accessible Java Applet to illustrate this idea although for  overall structure there is a nice applet here.

Silhouettes generated using a software toy

Some years ago I considered the idea of simulating the silhouettes of birds. Although rather skeletal and comical, it is something that those with an understanding of bird identification can toy with. You can find the Windows compatible program here. The illustration alongside has a collage of some of the shapes that can be generated using the program. While it is mostly a toy, it might help in sharpening one's idea of how the positions and relative bone lengths alter our perception of shape and how just the relative lengthening or shortening of feathers can give pointed, or rounded wings and graduated, wedge-shaped or forked tails .

Postscript: Unfortunately, nothing that I have written above will change the life of   millions of school children who complain about boring mathematics classes.

Image credits: The Great Wave (public domain); Cracks  in clay (Hannes Grobe, Wikimedia Commons)

Further reading

* Arbesman, S. , Enthoven, L. & Monteiro, A. (2003) Ancient Wings: animating the evolution of butterfly wing patterns. BioSystems, 71:289 - 295  (paper)
* DL Imes, LA Geary, RA Grahn, and LA Lyons (2006) Albinism in the domestic cat (Felis catus) is associated with a tyrosinase (TYR) mutation. Anim Genet. 37(2):175–178.
* Ting-Xin Jiang; R B Widelitz; Wei-Min Shen; P. Will; Da-Yu Wu; Chih-Min Lin; Han-Sung Jung and Cheng-Ming Chuong (2004) Integument pattern formation involves genetic and epigenetic controls: feather arrays simulated by digital hormone models. Int. J. Dev. Biol. 48: 117-136 (2004)
* Shigeru Kondo (2002) The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin. Genes to Cells 7:535–541
* Iwashita M, Watanabe M, Ishii M, Chen T, Johnson SL, et al. (2006) Pigment Pattern in jaguar/obelix Zebrafish Is Caused by a Kir7.1 Mutation: Implications for the Regulation of Melanosome Movement. PLoS Genet 2(11): e197. 

* Plant studio - an interesting software project http://www.kurtz-fernhout.com/summary_plantstudio.html

* Richard O Prum and Scott Williamson (2002) Reaction-diffusion models of within-feather pigmentation patterning. Proc Biol Sci. 269(1493): 781–792.
* Applets - fern leaf -  tree - tree

The Great Wave off Kanagawa by Hokusai (1831)
A fractal wave