Legs in more than 3 dimensions

Importance: 2
Confidence level: 6
Mathematical: 6
Length: 2100 words

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.

For our primary model of D+1 legs, second order corrections are now in order. We somewhat expect bilateral symmetry, in which case we always get an even number of legs and end up with D+1+(D+1 MOD2). We might also consider that at higher dimensions, animals would have enough legs that they would want several extra for moving, to preclude them from taking forever to move all 11 legs one at a time while walking. We’ll keep this in mind, but generally stick with D+1 for simplicity.

However, there are further considerations that play a role: we noticed earlier that radial symmetry is more prevalent in seagoing animals that can move in all 3 dimensions and have gravity “screened” off by the fluid around them. A possibility is that, in 4-D space and above where the defined plane is already greater than or equal to 3 dimensions, animals might have more radial symmetry and thus have fairly varying numbers of legs. The naïve argument fails because, despite the plane you can move in being 3 or more dimensions, you still need legs supporting you in the direction that matters. However, other symmetry arguments might bring weight to bear. A good thought experiment here is to, instead of going upward in dimension, to go down: in 2 dimensions, your planet becomes a circle, so an ocean will be 2-d and land will be 1-d (incidentally, the changing laws of gravity and electromagnetism make stars unstable here, so no 2-d life in the near future). Now any animal on land has a confusing relationship with bilateral symmetry: it has only 2 dimensions to live in, with gravity pointing in one of them. It has two options for bilateral symmetry then. If gravity points down, it could have symmetry from right to left with legs on bottom, in which case it can move either direction, but now probably needs two mouths (in 2-d space, the digestive tract would also cut through the organism, a whole new problem in itself). The other option is to have bilateral symmetry between up and down, effectively forcing two legs on the bottom and two on the top, with a mouth at one end and then flipping its whole body in order to point left instead of right. It could get around this with simple radial symmetry, like a Paramecium with cilia surrounding its body. In sea, bilateral symmetry becomes more reasonable, because bilateral “fish” could turn around by swimming upward and downward. Radial symmetry seems usually used by organisms with their mouths facing downward in the sea, like starfish and jellyfish, but this is impossible in 2-d.

Given that in 2-d bilateral symmetry seems to partially dominate in the sea while neither works well on land, while in 3-d radial is only in the sea but bilateral dominates both land and sea, what should we expect in 4-d? Since there are no turning issues past 2-d, I would expect bilateral symmetry to outcompete radial symmetry once again in 4-d. Further, the foray into 2-d suggested that jumping up a dimension would lead to new symmetry possibilities. It is difficult to imagine, but we can extrapolate some. A 3-d bilateral organism has legs down in the z-axis (asymmetric), a digestive tract pointing in the y-direction (asymmetric), with the bilateral symmetry a reflection in the x-axis. A radially symmetric organism has legs and digestive tract both down in the z-axis, and the x-y plane radially symmetric. In 4-d, we imagine a “bilateral” animal to have legs down in the z-axis, the digestive tract along the y-axis, and the symmetry to be in the w-x plane. However, we now have a whole plane for symmetry, and one begins to get creative. We could have it symmetric in the w and x axes (double bilateral symmetry), or make it radially symmetric in this plane (imagine putting the 2-d w-x plane into polar coordinates). An animal with radial symmetry here could look kind of like a 4-d worm with legs. I raise you all-in, Dune.

It is hard to visualize radial symmetry in this sense, and I’m hard-pressed to say how this would make the legs look. However, it seems to work mathematically, so I’ll run with it. In this case, for every two extra free dimensions, you could add either radial symmetry or two instances of bilateral symmetry. But… what about three free dimensions?

In our own 3-d space, we only have things like green algae that are spherically-symmetric. But this generates the generalization: bilateral symmetry is 1-d, radial symmetry is 2-d, spherical symmetry is 3-d, and N-spherical symmetry is N-d. Asymmetric dimensions reduce our “free dimension” count, as the legs and digestive tract reduced our count above. Then we can fill out our 4-d organism pantheon: asymmetric (0), bilateral (1), double bilateral (2), radial (2), triple bilateral (3), biradial (3), spherical (3), quadruple bilateral (4), bispherical (4), double bilateral plus radial (4), double radial (4), and 4-spherical (4). This counts to 12 possible symmetries (1+1+2+3+5), and looking like a Fibonacci sequence. However, when we get to 5 dimensions, we end up with quintuple bilateral (5), triple bilateral plus radial (5), double bilateral plus spherical (5), bilateral plus 4-spherical (5), bilateral plus double radial (5), radial plus spherical (5), and 5-spherical (5); there are 7 of these new symmetries, bringing us to (1+1+2+3+5+7), and we can recognize this as the partition sequence (the total number of partitions of D, or unique combinations of positive integers that sum to D), which is of course exactly how we were counting the symmetries anyways.

To constrain our blossoming imaginations, we should note a few expectations. Firstly, only the lamest organisms are completely asymmetric, and radial organisms tend to be outperformed even in their main no-effective-gravity environments due to their poor motility. Further, we only need an even number of legs if we have a bilateral dimension, so our 4-d legworm could have the optimal D+1 (5) legs in this case if I’m visualizing it right (actually not sure on this one). We also know that in 3-d, where radial, biradial, and spherical symmetries are only employed by the simplest organisms, all dominant animals have two asymmetric dimensions for legs and digestion and all the rest of the dimensions (1) bilateral. Given this, it seems almost certain that in higher dimensions we would see D-2 dimensions be symmetric; in other words, for the partition sequence p(n), though there are ∑n=0Dp(n) possible symmetries, we only expect to see p(D-2) symmetry combinations or fewer employed by dominant animals. There will be an additional p(D-1)+p(D) combinations that are rarely used (the higher-D analogues to spherical, biradial, and radial in our world). The dominant p(D-2) combinations are probably culled down to one or two that dominate the landscape, because, as Stuart Russell puts it in an abridged form of one of my favorite quotations,

“A system that is optimizing a function of n variables, where the objective depends on a subset of size k<n, will often set the remaining unconstrained variables to extreme values…”

Then we can, in our refined approximation, expect that organisms with two asymmetric dimensions and either D-2 bilateral symmetries or D-2-spherical symmetry will be the most prevalent (note that D-2-spherical symmetry in 3-d is just bilateral symmetry).

We can refine this a little further with extra biological considerations. An animal with a bilateral dimension can house the eyes and ears, both of which need two copies so as to judge distance (eyes) and direction (ears). One could go further into possible quadruple-ear-hypotheses (more bilateral symmetries) for triangulating direction in high-D space, but since humans can’t tell direction along our plane of symmetry very well (whether a noise is directly behind you or directly above you) this might not be so important. From this, we might revise our expectations for a decent chance that an organism will have one instance of bilateral symmetry, with the remaining dimensions covered by D-3-spherical symmetry (this also fits with the 3-d world, if you check, because D-3-spherical symmetry drops out).

Going back to our original leg problem helps us as well. We expect D+1 legs on the dominant species if it has D-2-spherical symmetry; bilateral plus D-3-spherical symmetry might instead show D+1+(D+1 MOD2) legs; D-2 bilateral symmetries now constrain the animal to having some fairly symmetric set of legs. Assuming that no single leg is down the center of a line of bilateral symmetry (though it is possible this could happen), we then need the legs to be some number 2^n for the smallest n such that 2^n > D+1. This could be very impractical for an 8-dimensional dog that suddenly needs 16 legs instead of 9 to walk upright; near these cutoff dimensions we might expect limb savings in the form of a human’s heel adaptation to replace a limb, or else several less dimensions of bilateral symmetry. Instead of 6 for our 8-dimensional dog, we could end up with only 4 plus radial symmetry for the other 2 dimensions. This could give us a dog with 4 legs in bilateral dimensions repeated 3 times over the radial plane, for a total of 12, or 2 legs repeated 5 times. At this point, I am no longer sure whether this is a self-consistent picture; please point out if I am eliding over any symmetry constraints that invalidate it. Further, please feel free to add more biology if it at all constrains the symmetries or legs we might expect.

ETA: Brief discussion in the comments of why I have assumed all legs are 1-d, and why we expect few feet greater than 2-d.

Abstract: We have found that for any D-dimensional universe, the most prevalent number of legs will probably be D+1, or some slightly higher correction like D+(D/6) to help the animal move more than one leg at a time. The animal will likely have D-3-spherical plus bilateral symmetry so as to end up with D+1+(D+1 MOD2) legs and stereoscopic sight and direction hearing, or may just have D-2-spherical symmetry. There is also a good chance of D-2 bilateral symmetries, especially at low dimension and when the number of D+1 legs is at or just under some 2^n. The animal will almost certainly have D-2 total symmetric dimensions (in total, p(D-2) symmetry options for the partition function p), but there also should be some quantity of simpler animals with D-1 symmetries and even D symmetries. Less than D-2 symmetries should indicate the higher-dimensional equivalent of sponges, very simplistic organisms.



5 thoughts on “Legs in more than 3 dimensions

  1. This entire analysis makes the implicit assumption that legs are one-dimensional. However, the load-bearing requirements of a leg would favor increasing the size, and therefore the dimension, of the leg. In fact I suspect the leg would have the maximal dimension possible while still providing the flexibility necessary for rapid motion. As long as the legs contact the ground over a higher-than-zero-dimensional space there is no need for further support, so one leg is sufficient.


  2. Ah yes, I should have specified the 1-d assumption.

    However, while increasing dimensions of the legs is helpful for balancing, we can see empirically that it is not often implemented, and theoretically has some flaws. Firstly, one leg is not sufficient; we can see this because 0 3-d animals have 1 leg. Secondly, even large animals like elephants that require heavy load-bearing legs have feet that approximate 0 dimensions, and still require 4 legs. The closest we come to seeing nonzero dimensional feet is in humans, kangaroos, and pangolins, as mentioned in the previous post; these animals tend to have two main 0-d points of contact on each leg, which could be approximated as a 1-d foot. However, these are a clear minority and require special adaptations (kangaroos walk on all fours and only bound on 2 legs, humans have crazy hip structure, pangolins have full body armor because they are so slow). Therosaurs used to be fairly prevalent, and they had close to 2-d feet, but still needed 2 legs. No animals have full legs in multiple dimensions.

    We could theorize that in higher dimensions, we would have more of these multi-dimensional feet, but we would need good evidence for this. In 3-d, many small animals actually have 6 or 8 legs while larger animals tend to have 4 but sometimes 2 (especially if we average in therosaurs). We would then expect only large animals to reduce leg number in favor of foot dimension; in general, animals want to have points of contact with minimal volume wasted, and only for load-bearing reasons must increase leg size. So legs should still be 1-d, but we might expect some 2-d or 3-d feet. Something near to a 3-d foot might develop in especially high dimensions (similar to how the 1-d human foot partially spreads into 2-d), but it is unlikely to save on legs like this in 4- or 5-d. Instead, the minimal volume principle shows that we should expect two 2-d surfaces, or a 2-d surface with an extra 1-d toe, or even just 3 1-d toes sticking out (cf therosaurs). This would be much more efficient for stability without needing to put a 3-d block of material at the end of the leg. For example, in 8 dimensions, it might be reasonable to expect some animals to have 4 legs with an approximation of 3 toes each (or extra toes arranged in some 3-d layout), such that they could have bilateral symmetry and move one leg while still keeping their N+1 = 9 points of contact with the ground. In something like 6 dimensions though, if you had 2 legs with 4-d feet, moving one would make you quite unstable, so you’d probably need to use 4 legs, but now you have extra costly points of contact, and in general it is difficult to find an efficient layout with multi-d feet.

    In general, multi-dimensional legs are unlikely, and multi-dimensional feet can only occasionally make up for their counterparts in limbs.


  3. Thank you for taking the time to respond. I have thought some more about this, and I have more to say, but I might not have the energy to get it in writing. For now, a small question: How would you classify snakes, worms, and caterpillars?


    • Honestly, I’ve been completely glossing over them in this first-order analysis: none of these have the stability problems that I’m talking about in the post because they simply leave their vital organs resting on the ground. But they can be muddled with a 1-legged design, so at some point they should be treated.

      All three seem like good candidates for a 0/1-leg classification. In the language I’ve been using, they’d be 0-leg, since they are essentially the standard tube of vital organs but without specific outgrowths for motility (ignoring caterpillars’ barely outgrown legs and pseudolegs). In this sense, all three are more like amoebas, using weird bodily contortions and points of extra friction to move. To cloud the picture a little, you can step away from the structural classification to a more functional classification based on points of contact (these are essentially collapsed for legged animals, which is why I focus on the legged approximation in the post). But in this POC classification, I would place caterpillars and worms in a 2-POC class, since both move by having one POC fixed and dragging/pushing the other in the forward direction before switching roles. Snakes, on the other hand, would be in some kind of multi-POC class, since they tend to have 2 or more horizontal stretches of themselves on the ground to push themselves forward (a sketchy video: https://www.youtube.com/watch?v=gbxaC2-o0Gs).

      To compare with higher dimensions, I suppose you could ask the question: would we see more or less of these no-legged, tube-of-vitals structures? In the first place, all three of these no-legged designs seem pretty rare on earth, where caterpillars quickly change into something else, earthworms only have that design since they live in a near-solid (not ON a solid, as I’ve been analyzing), and snakes are the exception. In higher dimensions, you have the same kind of problems as in 3-d. Earthworms get pillaged every time they come above ground because they’re so slow (think of robins feasting, esp during rains). Caterpillars require a bunch of pseudolegs to stabilize even their 2-POC movement, meaning higher-dimensional caterpillars would require even more, at which point they should ask themselves why they don’t jump on the leg bandwagon. Higher-dimensional snakes still seem plausible, but since I don’t understand snakes on earth, I can’t well understand snakes on 4-d earth.

      To get back to our question of relative quantity, I think we can ignore earthworms since they are below ground and posit roughly equal but maybe fewer caterpillar designs. This leaves snakes. One could argue that snakes should become more prevalent because they avoid stability issues and have many points of contact for motility. It is also possible that snakes’ snaking 1-d body type makes it hard for predators to focus on the right spot to strike, and this effect would only increase in high-dim spaces, but this is also conjectural. I think the difficulties of moving a body as a snake does would probably not increase in higher dimensions, but I am not sure of that.

      In general, then, I think I would agree with your idea that there might be more snakes in higher dimensions, but I would expect that given the tiny fraction of snakes compared to animals in 3-d, this would not come to represent anything substantial in higher dimensions. I’d also point out that instead of calling them 1-legged, which by my classification would refer to an animal hopping on one leg, we should focus on their many points of contact with the ground and call them a 0-legged, multi-POC animal.


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