Cw: no mention of Cthulhu
The last post discussed why 3 is a terrible number of legs for a species to have, contra the Idirans in the prominent sci-fi novel Consider Phlebas. The physicist’s obvious next question is how we generalize expected number of legs to higher dimensions.
(Caveat 1: if inclined, stop and think about this for a bit now; untouched puzzles don’t come around all that often, especially this more-conceptual and less-mathematical kind. If you are the opposite and want only the answer, the abstract is at the bottom.)
(Caveat 2: there are usually reasons why life could not exist with very different physical parameters than we have, called the fine-tuning paradox in physics and heavily debated. This extends to number of spatial dimensions. So don’t assume life could actually exist in these other dimensions: this is a purely theoretical exercise.)
On the surface of a planet in 3-dimensional space, gravity pulls in one dimension through a 2-dimensional plane (the ground). A 2-d plane is defined by 3 points (a 1-d line defined by 2 points, and a 0-d point defined by 1 point, so we can say that an N-dimensional plane is defined by N+1 points). For an animal to stand straight up, it needs to keep itself on the defined plane of the ground, which means 3 points of contact. This is pretty close to the definition of stability, but we need an extra leg to move (for reasons explained in the Idiran post). For D-dimensional space, if we still imagine gravity as pointing in a single direction (which physically it should), we then have defined a plane with D-1 dimensions needing D legs for stability and D+1 legs on the most prevalent animals.
If you are checking me, you’ll notice that this gives 4 legs for our world, a prescient model given the number of earthly quadrupeds. However, strange life forms abound on earth: snakes have 0 legs, sea creatures have as many as they want to, some creatures—I kid you not—fly through gas, and humans themselves are bipedal. For simplicity, I generally ignore these. Gas- and liquid-dwellers have numerous forms, snakes I deal with in the comments, and humans are strange. We have freed up two of our legs to become arms, and might conjecture that the first “intelligent” or “environment-changing” organism would for this reason have (some?) arms and thus (a few?) less legs than the norm for the dimension. However, note that humans do touch their heels to the ground to establish third and fourth points of contact. Now try standing on tiptoe.