# How Many Limbs and Digits?

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Why do animals have the number of limbs or digits they do? For example, why do we have five fingers, and why do octopus have eight arms? A simple hypothesis has been put forward by Mark A. Changizi that appears to explain these cases, and more generally, seems to drive the number of limbs of some animals across at least seven phyla (a "phylum" is a large grouping of related animals). The hypothesis, called the "max-MST hypothesis", says that animals have been selected to have the maximum number of limbs subject to the constraint that the resulting network is minimally wired (in particular, it is a minimal spanning tree, or MST ). What does this mean? Imagine that you have a circular animal with limbs pointing outward around it (like a starfish). We may consider the animal's body to be a "node" in the network, and we may also consider the tips of the animal's limbs as nodes. The limbs themselves are the "edges" which link up between nodes in the network. A network may be "wired up" by having edges between nodes so that there is a path, possibly indirect, between any two nodes. Such a "wiring" is minimal if it requires the least wire of all possible wirings.

The hypothesis predicts a particular relationship between the body to limb ratio of an organism and its number of limbs. In particular, for circular bodies the hypothesis predicts that the number of limbs is approximately given by N=(2*pi)/k, where k is the limb ratio, and is L/(L+R), where L is the limb length and R the body radius. If the limbs are very very long compared to the body, then k=1 and we expect about six (2*pi) limbs. As the limbs become shorter and shorter compared to the body radius, k falls toward 0, and the number of limbs is expected to increase more and more. Examination of nearly 200 organisms across seven phyla shows very close conformance to this prediction. A paper on this appeared in Changizi (2001) "The economy of the shape of limbed animals." Biological Cybernetics 84: 23-29 [Winzipped PDF reprint]. Further development of these ideas are in his book, The Brain from 25,000 Feet: High Level Explorations of Brain Complexity, Perception, Induction and Vagueness (Kluwer, 2003) , and you can download the relevant section here.

On this page you can see how many limbs the max-MST hypothesis predicts as a function of the animal's body shape. You may play with the animal's length and with its limb length, and you will see how many limbs (and roughly how the animal might look) the hypothesis predicts. (The "smaller"-"bigger" axis just helps you adjust the size on the screen.)