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I am attempting to design aliens that live in a universe with a 4+1 space-time--i.e., a universe with four orthogonal spatial dimensions, rather than three, in addition to a fifth time dimension. In this universe, gravity follows an inverse cube law rather than an inverse square law, but it still results in large blobs of solid matter being pulled into (hyper)spheroidal planet shapes. 4D planets, however, have 3D hyperplanar surfaces, not 2D planar surfaces like planets in our universe. While 4D space comes with an array of other interesting features as well (like having an extra Platonic solid, knots made out of sheets instead of strings, and two independent components of angular momentum), it is the 3D nature of planetary surfaces that is most relevant here, as that dictates the geometry of limbs designed to contact and move over them.

Now, to state the obvious as background: humans have two legs, and two arms, derived from creatures that have 4 legs. In 3D space, standing on two legs is unstable, but it is (obviously) feasible to maintain balance on the free axis with active neurological control. There are, however, no creatures which get around on only one leg (modulo the occasional bird that can lock its joints to stand on one leg, nothing actually locomotes with one).

4 legs is a good number for an ancestral creature in 3-space, because in 3-space you need 3 points of contact to maintain stability, and having 4 legs allows you to move one at a time while keeping three points of contact on the ground. Also, having an even number of legs permits bilateral symmetry. Also due to symmetry considerations, we don't see any 3-legged, one-armed animals; having three legs would result in greater stability than two, but Earthling tetrapods that re-purpose some of their limbs as not-legs invariably do so in pairs.

Moving up into 4-space, you need four points of contact on a hyperplanar ground for stability; standing on three legs is as unstable as standing on two legs in 3-space, and standing on two legs in 4-space is as unstable as balancing on one leg in 3-space. Thus, it would seem unlikely for a 4D sophont to have only two legs (and definitely not just one!).

However, while we can reasonably exclude reasonably exclude one-legged and two-legged/two-armed body plans for the aforementioned stability reasons, it is not so obvious what limb arrangements would be reasonable, particularly since adding an extra dimension gives you more directions for limbs to stick out from a single body section (e.g., a shoulder girdle or hip girdle analog).

So: Does it make the most sense to attach only two bilaterally symmetric limbs to each body segment/shoulder girdle/hip girdle? Or, are there higher-order symmetries possible for limb arrangements?

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – L.Dutch - Reinstate Monica Jun 20 at 3:03
  • $\begingroup$ I question whether additional limbs are required, while the 2 point contact system is merely metastable rather than stable, it allows for easy movement and correction. When things change in the +1 dimension (time) we respond accordingly. If there are no additional static forces in the 4th dimension, there's no need for additional static support and the dynamic stability already available should suffice. $\endgroup$ – Separatrix Jun 20 at 10:04
  • $\begingroup$ "In this universe, gravity follows an inverse cube law rather than an inverse square law, but it still results in large blobs of solid matter being pulled into (hyper)spheroidal planet shapes." If you want hard science, I found this page which mentions that all orbits are unstable given 4 spatial dimensions...maybe you could assume that 4D life was seeded by some hyper-advanced civilization that originated in our 3D space, and which artificially maintained the planet's orbit. $\endgroup$ – Hypnosifl Jun 21 at 0:07
  • $\begingroup$ @Hypnosifl- oh, I know. At least in the current design for the universe, the planet is not in orbit, so it doesn't matter that there are no stable ones. It's more like the 5D universe from Greg Egan's novel Diaspora. Planets depend on internal heat and being in a dense stellar cluster for half-way consistent lighting, not on a single bound sun. $\endgroup$ – Logan R. Kearsley Jun 21 at 0:24
  • $\begingroup$ Doesn't Egan describe such a body in a later chapter of Diaspora? I'm thinking of the statue that spans dimensions. $\endgroup$ – rek Jun 21 at 11:56
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To me, one body plan is by far the most likely: 6 limbs in a triangular prism.

To see why, think of the earliest land-dwelling, limbed animals to evolve on Earth. I'm imagining something like a wide-legged lizard. It's low to the ground; effectively its entire body is in a 2D plane. And this makes sense, since opposing gravity takes work, and evolutionary pressure hadn't yet forced creatures to do that work. It has a head, and it wouldn't be advantageous for its limbs to be moving around its head all the time, so it keeps its limbs along the orthogonal axis, 2 on each side (one on each side would be below the minimum number required for stability, as you mentioned in the question). Thus, 4 legs total.

This, fundamentally, is why bilateral symmetry developed on Earth, and not trilateral symmetry. Imagine that lizard having 3 legs: how would it place them around its body without being off-balance when its head is in the plane of the Earth? This logic is important, because, as we'll see, it generalizes simply to 4+1-dimensional space.

Imagine an early land-dwelling animal on our hyper-Earth. Its body crawls along the hyperplane because supporting itself against gravity is hard. For stability, it splays its limbs to the sides of the axis containing its head. The minimum number of limbs it could have is 3, because that spans the plane perpendicular to its head along the ground. But, in analogy with how 2 was the technical, unstable minimum for our proto-lizard, we'd really expect 2 sets of limbs along the spine. Thus, in our case, 6 limbs total: 3 hind limbs and 3 front limbs. I want to stress that trilateral symmetry perpendicular to the spine isn't a problem, because the issues with trilateral symmetry on Earth don't apply on hyper-Earth.

So let's fast-forward the evolutionary clock and ask about hyper-people. I think it stands to reason that they would manipulate objects with their 3 forelegs and walk on their 3 hind-legs. To visualize them walking, it might help to form an analogy with the way people walk on Earth. Our 2 legs are always in the positions of 2 vertices of a triangle, and when we step forward, we change which vertex is missing. This is always unstable due to the missing vertex, but we use momentum to keep our balance. Similarly, the 3 legs of the hyper-humans always form 3 of the vertices of a tetrahedron. Don't actually try to picture it; in 3D, the legs will seem to pass through each other, and it's very confusing to think about.

But suffice it to say, this 3-and-3-limb body plan seems the most logical. Of course, extra legs are always doable, like a 4-and-4 version. This is in analogy to animals with more than 4 legs on Earth, like insects. However, I think I've shown that 6 limbs is, in many ways, the default in 4 spacial dimensions.

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  • $\begingroup$ Makes me start to wonder how many hands are required to hold a 4D object. Can you hold a hypersphere with one hyperhand? What shape does a hyperhand have? This question is a rabbit hole... $\endgroup$ – Muuski Jun 20 at 22:40
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    $\begingroup$ I'd imagine hyper-hands are mostly 3d, like our hands are mostly 2d. They're meant for clinging around the 3d outside surfaces of 4d objects, and you'd probably still only need one or two to hold things, depending on the slipperiness of the object. Also, this question is more precisely a rabbit hyper-cylinder, a hole in 4d space whose cross section is a sphere. :P $\endgroup$ – Gilad M Jun 20 at 23:05
  • $\begingroup$ And I thought the question was confusing... $\endgroup$ – Bilbo Baggins Jun 21 at 3:21
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    $\begingroup$ I'm leaning toward it just being a 3D bias. After all, we're used to bilateral symmetry, and there's a good evolutionary reason for that that I explain, but I don't see any evolutionary reason to prefer 4+4 over 3+3. That said, a lot of animals have way more limbs that I would guess is necessary. Some bugs have 20 or 30 limbs and I have no idea why. I would treat 3+3 as a minimum and a starting place for thinking about evolution in 4D. I'd also personally prefer it if I were designing these aliens, since you don't usually get a good reason to use such a strange body plan in fiction. $\endgroup$ – Gilad M Jun 21 at 17:35
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    $\begingroup$ Side note: these creatures shouldn't have a problem identifying each limb. I'd expect that just as we can tell left from right, they can intuitively sense which of their three limbs is which. Another useful analogy might be the fact that we have left and right arms, but not front and back ones. That's not odd to us. Similarly, these aliens have arms in one of the planes perpendicular to their spine, but none of the other ones. Their versions of terms like "left" or "right" will only refer to directions in that plane, but they could have other words that are the analogy of "front" and "back." $\endgroup$ – Gilad M Jun 21 at 17:44
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Having seen it, I am fairly convinced by Gilad M's answer. However, I think the reasoning can be clarified and expanded on.

So, we start out imagining the simplest creature that could crawl out of the sea onto land and stand up. It will "want" to keep its center of mass low, and span the (hyper)plane for stability. If you zoom in on one limb attachment section of a 3D tetrapod (i.e., the shoulder girdle or hip girdle), you have the major axis of the animal (the spine) defining one dimension of the plane, and then two limbs, providing two points of contact with the ground, defining a second axis; the third point of contact with the ground is provided by something located elsewhere along the spinal axis. That third contact point can be expected to be provided by another pair of limbs, rather than a single limb, since having 4 total permits one to move while maintaining 3-point stability--thus, you end up with matched pairs of limbs. Additionally, since the pairs of limbs do not span the plane by themselves, they also serve as a rotational axis about which the spine can pivot, so 3D tetrapods can stand up. In 3D, if you add additional limbs on the same body segment, you loose the ability for the body to pivot around those limbs--see, e.g., radially symmetric creatures like seastars (flat) or sea cucumbers (elongated), or cephalopods. There is thus no way for such a creature to transition from laying with its primary body axis parallel to the ground to standing with its primary axis perpendicular to the ground.

Moving up to 4D, a primitive creature will still "want" to remain low to the ground, and span the hyperplane for stability. In order to get that static stability, we need at least four points of contact with the ground--plus one to permit movement. However, there are far more different options for how limbs might be arranged. For an axially extended creature (something with a linear spine that can lay down or stand up), it is clear that one will need at least two different limb attachment points, at opposite ends of the spine, to support the entire length of the spine; otherwise, a single limb group would have to exert a lot of torque to support a cantilevered body; one could perhaps imagine something like a 5-legged theropod with its single hip girdle in the middle and the body balanced across it, but that is both not a primitive-looking condition (nothing on our 3D world came out of the ocean balanced on a single limb girdle, despite bipedalism later evolving multiple times), and doesn't lend itself to evolving into something with distinct sets of arms and legs--i.e., it's not very good for a "human-analog". In addition, we should expect each limb girdle to have at least two limbs attached to it, just as in 3D--if there were only one, then when it was that limb's turn to move, the cantilevered body would bee unsupported, and you would end up with a 3-legged-dog situation.

If each limb girdle still has only two limbs attached, then three such girdles would be required, resulting in a six-limbed basal animal, which could raise its front limbs in a direct analog of the 3D mantoid / centauroid body plan, resulting a 4-legged, 2-armed creature. Such a creature would be less stable than a 3D centauroid, as it would not be able to span the surface with the three remaining legs while moving one, but more stable than a 3D biped.

However, it is possible for a primitive 4D tetrapod-analog to have an arbitrarily large number of limbs attached to a single limb girdle, much like radially symmetric 3D creatures, without losing the ability to hinge the spine. The reason for this hinges (pun intended) on the fact that rotations (in space of any dimensionality) do not fundamentally occur around axes, but rather in planes. The utility of linear axes to describe 3D rotations is a 3D-specific artifact of the fact that selecting a rotation plane leaves a single unique perpendicular vector left over to span the space. Radially-arranged limbs on a 4D creature, however, can span a single 2D plane with their attachment points, leaving a second plane containing the spine free to permit rotation of the spine around that set of limbs (hence the (geometric, if not evo-devo) plausibility of the aforementioned 5-legged theropod).

Thus, since we can attach three or more limbs to a single girdle, it is not in fact necessary to adopt the centaur-analog body plan with three limb girdles--which is convenient, as a 4D human-analog with only two sets of limbs is certainly more human-ish than a centauroid! In fact, we can achieve the minimum limb set for statically stable locomotion of an axially-extended creature (5 limbs grouped at least in pairs) with one three-limb set and one two-limb set. Such a creature could stand up, hinging its spin from horizontal to vertical around the 3-limb girdle, to free up 2 arms while remaining as dynamically stable on 3 legs as a 3D human is on 2. This is, geometrically at least, a perfectly plausible body plan, and results in an unusually high degree of freedom for the joint between the spine and the arm girdle. If a consistent symmetry type is to be maintained along the full length of the body, however, that would dictate a 3-leg, 3-arm arrangement, with full trilateral symmetry. This gets us to the basic body plan proposed by Gilad M.

So, to directly answer the question: No, it does not make the most sense to remain restricted to pairs of limbs, thus dictating 3 limb sets total; and yes, arbitrary numbers of limbs with high radial symmetry can in principle be supported on each limb girdle of a 4D creature.

Moving into the realm of greater speculation based on non-purely-geometric arguments, however, the 3+3 hexapod arrangement may or may not actually be optimal. After all, there are no trilaterally symmetric organisms in our world, and very few quadrilaterally symmetric ones. The larger degree of attachment available between radial segments that meet along 3D cell boundaries rather than 2D cross-sectional surfaces may significantly change whatever considerations lead Earthly evolution to favor 5+ symmetries, thus making trilaterally symmetric 4D "humans" perfectly plausible, but as this is well outside the realm of simple arguments to be made about hinging structures, how exactly that would impact 4D creature development is not nearly as obvious. Perhaps the optimal 4D human-analog is actually a ten-limbed, pentalaterally symmetric creature!

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    $\begingroup$ Nice analysis. I also considered a 2+3 arrangement, but it seemed less to me like something nature would do. Plus, radial symmetry is, in some sense, "easier" in 4D, since you have more possible planes to have the symmetry in. If a 4D person is triangularly symmetric in one of the planes perpendicular to its spine, it can still have, for instance, a full range of vision in the other plane. As opposed to 3D, where a triangularly-symmetric person would have to have 3 pairs of eyes around its head, and other annoying constraints. $\endgroup$ – Gilad M Jun 21 at 10:15

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