# The Quadrupedal Conjecture

Last summer, I asked myself several related questions concerning the locomotion of quadrupeds:

1) What if the cheetah had more than four legs?
2) Why don’t hexapods gallop?
3) Have we ever had F1 cars with more than 4 wheels?

I found all these related questions interesting but after more reflection I summarised my problem mathematically:

Could four legs be optimal for rapid linear and quasi-linear locomotion on rough planar surfaces?

After searching for papers on this subject and finding none, I decided to name this problem ‘The Quadrupedal Conjecture’ and it may be informally described as saying that four legs allow a creature to travel fastest on a nearly-flat surface. However, I thought it might be interesting to tackle a simpler problem first which we may call the ‘Little Polypedal Conjecture’:

Can we show that given a polyped with $2 n$ legs having pre-defined mass and energy, that there exists $N$ such that for $2 N$ limbs or greater, its maximum linear velocity on a rough planar surface would be reduced?

I believe that there is an elegant solution to the above problem and I think that the quadrupedal conjecture has an elegant solution as well. But it is by no means guaranteed that a solution to such a problem would be simple. Finding the right degree of abstraction would be one of the main challenges as there are many different ways of approaching this problem with varying degrees of realism.

First, I must say that this is not primarily a problem in biology although this question has attracted the attention of biologists on the biology stackexchange. Second, I think this is a question that would heavily involve mathematical physics although this question has been controversial on the physics stackexchange. Further, I think that the solution to this problem would be affected by mass scaling. By this I mean that the optimal number of legs would probably vary with the range of masses available to the polyped.

Finally, I think that this question is highly relevant to roboticists who build legged robots as a thorough investigation of this question would probably lead to better models for polypedal locomotion.

That’s all I can reasonably say for now.

Note: I thought I’d add a touch of melodrama to this blog post as a friend of mine told me that my blog posts can be a bit dry.