CHAPTER 35

Radiolaria: validating

the Turing theory

bernard richards

I

n his 1952 paper ‘The chemical basis of morphogenesis’ Turing postulated his now

famous Morphogenesis Equation. He claimed that his theory would explain why plants

and animals took the shapes they did. When I joined him, Turing suggested that I

might solve his equation in three dimensions, a new problem. After many manipulations

using rather sophisticated mathematics and one of the first factory-produced computers in

the UK, I derived a series of solutions to Turing’s equation. I showed that these solutions

explained the shapes of specimens of the marine creatures known as Radiolaria, and that

they corresponded very closely to the actual spiny shapes of real radiolarians. My work pro-

vided further evidence for Turing’s theory of morphogenesis, and in particular for his belief

that the external shapes exhibited by Radiolaria can be explained by his reaction–diffusion

mechanism.

Introduction

While working in the Computing Machine Laboratory at the University of Manchester in the

early 1950s, Alan Turing reignited the interests he had had in both botany and biology from

his early youth. During his school-days he was more interested in the structure of the flowers

on the school sports field than in the games played there (see Fig. 1.3). It is known that during

the Second World War he discussed the problem of phyllotaxis (the arrangement of leaves and

florets in plants), and then at Manchester he had some conversations with Claude Wardlaw, the

Professor of Botany in the University. Turing was keen to take forward the work that D’Arcy

Thompson had published in On Growth and Form in 1917.^1 In his now-famous paper of 1952

Turing solved his own ‘Equation of Morphogenesis’ in two dimensions, and demonstrated a

solution that could explain the ‘dappling’—the black-and-white patterns—on cows.^2

The next step was for me to solve Turing’s equation in three dimensions. The two- dimensional

case concerns only surface features of organisms, such as dappling, spots, and stripes, whereas