How Many Limbs and Digits?


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.)

As a specific example, consider the five fingers on your hand. Your palm is the "body," and your digits are the "limbs." The only crucial difference between digits and limbs is that there is only a need for digits along roughly the outside half of the palm, whereas limbs are typically needed for animals all the way around. If you look at your hand, you will notice that your finger length is roughly the same as your palm diameter. Thus, your finger length, L, is roughly twice the radius of your palm, R, so that L=2R. The limb ratio is therefore, k = L/(L+R) = 2R/(2R + R) = 2R/3R =2/3. The predicted number of limbs from the equation N=(2*pi)/k is N=(2*pi)/(2/3), which becomes N = 3*pi = 9.42. But recall that hands are only selected to have digits on roughly one half of the hand's perimeter, not all the way around, and thus the expectation is actually half of this, or 4.71, which is about 5. We have five fingers, then, because given that the limb ratio k = 2/3, five fingers is the economical solution. This, however, depends on the limb ratio being what it is. Presumably k=2/3 for hands because there is some functional advantage to having fingers be roughly the same length as the palm's diameter, probably so that the fingers can properly enclose the palm. With this in mind, we can predict that, for any animal anywhere (Earthly or otherwise), if it has been selected to have hands capable of closing, then it will have approximately five digits. We may reasonably expect that if they develop a number system, it may well be base-10 like ours (supposing they have two hands).

For more information, contact Mark A. Changizi:
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This applet was written by Eric Bolz.
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