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: changizi@changizi.com

Or visit www.changizi.com

This applet was
written by Eric Bolz.

View the Source code for this applet.