# How Alan Turing Cracked Nature’s Code

What does the formation of galaxies have to do with oceanic migratory patterns and embryos? The answer may surprise you: Alan Turing.
If you’re interested in math, you probably already know of Alan Turing, and if you don’t, you should. Alan Turing was an ingenious mathematician, computer scientist, and cryptist. He is without doubt the forefather to computers, artificial intelligence, algorithms, and theoretical computer science, and he utilized his cryptology skills during World War II to help the Allies defeat the Nazis in many key battles, including the longest continuous military campaign of the war, the Battle of the Atlantic. This could easily be a manifesto on Alan Turing’s mainstream contributions, but the focus of this article will be on perhaps one of the most interesting—although shrouded—codes he worked on: that of nature.

Shortly before his death, Alan Turing was working on finding sufficient basis for a mathematical model that comes up again and again in nature. His research was fruitful, but he died soon thereafter and his ideas were more or less lost up until recently. Turing’s focus was reaction-diffusion systems, and if that sounds scary, don’t worry—they’re not as bad as they may sound.

# Reaction-Diffusion Systems

Reaction-diffusion systems model the interactions of often naturally produced systems, in which a multitude of things exist and where almost always the interactions of substances lead to the creation of new substances, or the change of previous ones.

In his last paper—written 3 years before his passing—The Chemical Basis of Morphogenesis, Turing detailed research into the interaction between molecules which he referred to as morphogens and his theorizations of possible mathematical explanations for natural phenomena, namely reaction-diffusion systems.

# Morphogenesis

Morphogenesis is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of developmental biology along with the control of tissue growth and patterning of cellular differentiation.

He went into great detail on how patterns, especially those we observe in nature, arise from initial homogenic states.

These patterns have since become known as ‘Turing patterns’ and they’ve been noted in anything from fish scale pigmentation to photoisomerization in dye-doped crystalline structures (a change in crystalline structure due to the light-induced transition of a molecule to an isomer). He noted in the paper the apparent appearance of these patterns in the tentacle patterns on Hydra(small oceanic invertebrates), spiral-patterned leaves, gastrulation(the process of cells moving to the interior of an embryo), and phyllotaxis(the arrangement of leaves on a stem). Perhaps most famously, he noted the now-commonly-known pattern of fibonacci numbers in sunflowers, a phenomenon first observed by one of the most important astronomers of the 17th century, Johannes Kepler.

# Where are we now?

Research groups are picking up where the Math Goliath left off and they’re investigating where exactly these patterns, modeled by the reaction-diffusion theory of morphogenesis, appear. What they’ve found is absolutely incredible; these patterns arise from the macro to the micro, from the formation of entire galaxies to oceanic migratory patterns to the formation of embryonic structures.

One of the most recent and fascinating applications of research into these prevalent patterns is in whale sharks in the Maldives, a small archipelagic country in South Asia. A short film called “A Natural Code” documents the Maldives Whale Shark Research Programme (MWSRP), a charity that carries out the research there. Their focus is the conservation and protection efforts of their non-mammalian neighbors, whale sharks. The beauty of this technique is that the beautiful spot patterns on these sharks can and has been modeled using Turing patterns. The MWSRP leverages this and the fact that each whale shark has a unique spot pattern to track individual sharks. All they have to do is get photos of them from all sides and insert them into their database, in effect creating and storing shark fingerprints.

The benefits of such an approach is that it is harmless to the shark, there is no need for any sort of physical tracker, it lets researchers track the sharks for the entirety of their life, these spot patterns remain constant for whale sharks’ adult life, and it’s entirely safe—these sharks, while menacingly giant, are absolutely harmless to people. This means MWSRP can have volunteers help them out with their work, I myself am hoping to join them this summer for a two week volunteer session, and you should check them out too! These patterns don’t only model the behavior of microcosms though; these Turing patterns have been found in the macro of the macrocosms, galaxies. Once again, the rationale comes down to morphogens and their interactions in a state of equilibrium. In short, there are two entities; one is an activator and the other an inhibitor. The activator encourages production of itself whereas the inhibitor inhibits production of the activator. However, if the two are at equal rates, in effect keeping one another in check, then according to Turing, there are a certain set of circumstances in which the equilibrium may tip to one side, and under these circumstances it can continue to tip more and more to that side. This enables one of the two entities to ‘take over’, which is how patterns like spirals, stripes, spots, rings, and so many more are formed. This process is theorized to be the way galaxies take form, which explains the arisal of Turing patterns in these magnificent gargantuan spirals.

# What next?

Turing’s work, even in the last years of his life, was groundbreaking. Before him, and perhaps without him, few would think to ask about or investigate mathematics in nature, especially on such a broad scale. His work largely dictates a magnificent portion of our modern lives: the incredibly complex silicone-chipped machines that we rely on for so much of our professional and personal lives, the algorithms that run them, the cybersecurity that protects them from malicious attacks, and so much more. Beyond that, he clearly has had a significant impact into biomathematics, impacting not just the lives of members of our species but so many beyond us, providing a fantastic mathematical insight into the intricacy and beauty of life. Perhaps our generation and those beyond ours can continue to marvel at the absolute brilliance of the man, and build on the incredible work he has inspired.

# Bibliography

1. S. (2016, July 14). Can one mathematical model explain all patterns in nature? Retrieved April 25, 2021, from https://youtu.be/F1hX$$\_$$nzTlgU

2. Conservation through research and COMMUNITY MOBILISATION. (n.d.). Retrieved April 25, 2021, from https://maldiveswhalesharkresearch.org/

3. Turing, A. (2004). The chemical basis of morphogenesis (1952). The Essential Turing.
doi:10.1093/oso/9780198250791.003.0022

4. Nicolis, G., & Wit, A. (n.d.). Reaction-diffusion systems. Retrieved April 25, 2021, from
http://www.scholarpedia.org/article/Reaction-diffusion$$\_$$systems

5. ComplexityExplorer. (2019, August 02). Origins of life: Introduction - pattern formation in chemical systems - reaction diffusion systems. Retrieved April 25, 2021, from

6. Reaction diffusion: A visual explanation. (2019, June 24). Retrieved April 25, 2021, from

7. Sunflowers and Fibonacci: Models of efficiency. (2014, June 05). Retrieved April 25, 2021, from
https://thatsmaths.com/2014/06/05/sunflowers-and-fibonacci-models-of-efficiency/

8. A century of Turing. (2012, November 30). Retrieved April 25, 2021, from
https://www.nature.com/articles/nphys2512

9. Kuttler, C. (n.d.). Reaction-Diffusion equations with applications. Retrieved from https://www-m6.ma.tum.de/ kuttler/script$$\_$$reaktdiff.pdf

10. Ghosh, A., & Kriete, A. (2006). Reaction-Diffusion system. Retrieved April 25, 2021, from https://www.sciencedirect.com/topics/mathematics/reaction-diffusion-system

11. DNewsChannel, S. (2021, March 29). Scientists find our world could be ruled by secret patterns | a natural code. Retrieved April 25, 2021, from https://www.youtube.com/watch?v=9c$$\_$$8ksn8YaM

12. Klein, J. (2018, May 08). How the father of computer science decoded nature’s mysterious patterns. Retrieved April 25, 2021, from https://www.nytimes.com/2018/05/08/science/alan-turing-desalination.html

13. Smolin, L. (1996, December 03). Galactic disks as reaction-diffusion systems. Retrieved April 25, 2021, from https://arxiv.org/abs/astro-ph/9612033

14. Davenport, M. (2014). NASA viz: Galaxy formation. Retrieved April 25, 2021, from https://svs.gsfc.nasa.gov/11534

15. Nozakura, T., & Ikeuchi, S. (1984). Formation of dissipative structures in galaxies. The Astrophysical Journal, 279, 40. doi:10.1086/161863