# Why don’t mammals have more than 4 limbs?

-a cheetah in action captured by Mark Dumont

Approximately one month ago I wondered why there were no mammals that had more than 4 limbs. I wondered whether they were somehow optimal for the challenges faced by mammals. I was thinking about the Cheetah in particular: would it run faster if it had an additional pair of limbs?

Unsure about how to proceed I posted the question on the biology StackExchange where it generated some interest. The accepted answer by a zoology researcher could be summarized in this way:

“So basically, terrestrial vertebrates have four legs because they evolved from a fish ancestor that had four members that were possibly used as legs (that could “easily” evolved into legs). The explanation is as simple and basic as that. You can have a look to the diversity of terrestrial vertebrates here (click on the branches).”

Unsatisfied with this answer, I asked another zoologist from Cambridge University, Jenny Clack, who also happens to be an expert on tetrapod evolution and she gave me a similar answer:

“If four appendages are optimal, it’s because they were optimal for swimming before tetrapods became terrestrial. Clearly four are not always optimal for tetrapods since many of them lose them, but it would take a major Hox gene duplication – probably deleterious if not fatal – for tetrapods to get more.”

It appears that nobody has tried to do computer simulations or analyses of any kind to determine whether 6 legs may be better or worse for mammals that have particular constraints due to their habitats and functional purposes. After more research I have come to this conclusion. But, after attempting to transform the question into a falsifiable hypothesis(Are 4 legs optimal for certain/all mammals?) I have realised that there are quickly serious difficulties.

Even if we considered a special case of this question: “Are 4 legs optimal for a 100 kg carnivorous savannah mammal?” there are serious challenges. To begin with, there are many important factors that aren’t well understood:
a) type of vascular system used
b) type of respiratory system
c) other variables
d) dynamic relationship between these components

Given the limited understanding of leading researchers of the biological components of land mammals, I realised that doing significant analysis of this problem is basically impossible.

This is the first time I’ve encountered a scientifically interesting problem that can’t lend itself to any kind of reasonable analysis.

# Soap bubbles and the isoperimetric problem

During a moment of reflection last summer, I momentarily thought about the shape of balloons and bubbles in general. I wondered why of all the possible shapes a balloon could have, why the sphere? If there was a reason why the bubble converged to the shape of a sphere I wanted to know it. But, I forgot about this question amid other concerns until the thought occurred to me once again a couple weeks ago…I realized that it might have something to do with the isoperimetric inequality which in 2-dimensions states that for a closed curve $C$, if the length of its perimeter is a constant then the circle maximizes its area.

This time I posed the question on the Physics StackExchange. A person by the name of ‘Michiel’ gave a sufficiently detailed answer which I shall summarize here:

For a bubble to be spherical, the surface tension has to dominate other forces per unit length. This means that other factors such as gravity and the size of the balloon must be significantly less important. Now, if surface tension is indeed dominant then the pressure is mainly determined by the Laplace pressure jump across the bubble interface($\Delta P_{c}$)

$\gamma$ is the surface tension between the gas and liquid.

$\kappa = 1/R_{1} + 1/R_{2}$ is the local curvature of the surface.

$\Delta P_{c} = \gamma(1/R_{1} + 1/R_{2})$

The equilibrium situation is the case where the surface tension is dominant
and the bubble pressure is equal everywhere.   If that is not the case we get a
flow from one place to the other. $\gamma$ is constant so $\kappa$ must be constant over the whole interface, and a sphere is the only closed shape for which this is possible.

Now, I was not entirely satisfied with this answer because I hoped to find
a connection with the isoperimetric problem. But, I found one answer that
mentioned that systems try to minimize their potential energy…This led me
to the following great insight into the nature of this problem:

If we assume that the bubble tries to minimize its potential energy then, for a given surface area, it will try to offer the maximum volume for the gas particles contained within it. This is equivalent to finding the minimum surface for a given volume.

Taking the isoperimetric inequality into account, the fact that a bubble
immersed in a fluid(air/gas) converges to the shape of a sphere isn’t surprising.