4d beings - limb placements?

[Vector_Graphics]

I've currently been using a cube (prism) for the way my 4d beings' limbs are laid out, but recently I've been wondering if I should use an octahedron (antiprism) instead. The only advantage of the cube that I can see is easy analogy.

And no, I will not be accepting squares or disphenoids. I've already made the decision to have more than 2 limbs per set, it makes it easier to balance.

[PatrickPowers]

The human body with two legs is unstable here in 3D. So stability isn't a requirement.

Beings here in 3D have 2,4,6,8, or more legs. An odd number of limbs is unusual but starfish with 5 prove it possible. So evolution can successfully take weird paths. Engineered vehicles are more rational.

[Vector_Graphics]

I mean, we're just based on a quadrupedal model. What would equivalent of "quadrupedal model" be in 4D? 6 or 8 legs?

[gonegahgah]

I mean, we're just based on a quadrupedal model. What would equivalent of "quadrupedal model" be in 4D? 6 or 8 legs?

Hi Vector. I've always favoured, for upright beings, the minimum that would keep them from falling sideways.
This is due to my belief of an evolutionary trend towards sufficient minimalism possibly for the sake of efficiency and simpler co-ordination.
So for my 4D beings they would tend to have three legs (plus three arms from having evolved from six legged animals as a forward and back set of limbs still seems ideal to me).
Though you can certainly have a greater number of legs side-by-side as a circle of legs without (apart from their width) them getting in each others way.
So three, four, five, six, or more bipedal legs would certainly seem plausible but as I said I tend to think evolution favours minimalism.
Interestingly the arms may actually reside above (naturally) but between each of the legs as an improvement on front and back legs having more room to avoid each other.

As 4D allows for more complex brains in a smaller radius I also tend to tend that 4D creatures would be a lot more shorter and, as a ratio, wider than our height to width ratio.
The 2D creatures, if it were even possible (which could also be said for 4D creatures), would have to be taller than us and probably, as a ratio, narrower.

[Vector_Graphics]

4d creatures don't really face the anatomical difficulties of 2d creatures. Also, the "arms are above but between legs" is exactly what I mean by octahedral model.

[gonegahgah]

4d creatures don't really face the anatomical difficulties of 2d creatures. Also, the "arms are above but between legs" is exactly what I mean by octahedral model.

Cool. Good idea.
The only real limitation is how many legs you can fit in their 4D hip as that is the highest radius part of each leg, both for us and probably for them.
Arms take up less radius at the shoulder than the legs at the hip.

[Vector_Graphics]

Nice! I was originally thinking other uniform polytopes for more than 2 rows of limbs, but that was before I thought about it more and realized that torsos are generally more tubes than (hyper)ellipsoids, so stacked antiprisms (in this case octahedra) would do the job.

[gonegahgah]

I like that idea Vector. That is a very interesting shape association in constructing bodies.

[Embi]

Using a cube for 4D limb layout does provide a simple and relatable visual, but an octahedron could add some unique possibilities. It might give your beings a more dynamic and flexible appearance, which could be interesting.

[PatrickPowers]

How about legs in a dodecahedral arrangement? That means maximum volume per leg. Seven times that of a cube arrangement.

[quickfur]

In 4D, there's this very interesting possibility that isn't available in lower dimensions: a 2-legged arrangement that's actually stable. The way this works is that the feet are at 90° angles to each other, such that their projection to 3D forms two opposite edges of a tetrahedron. A tetrahedral arrangement in 4D is inherently stable (analogous to the tripod in 3D). Let's call these two opposite edges E1 and E2. When walking, the creature would lift the foot corresponding to E1, for example, and place it on the other side of E2, thus reflecting the tetrahedron across E2, forming an inverted tetrahedron. Then it lifts foot E2 and places it on the other side of the new position of E1, again forming the original tetrahedron but displaced by twice the distance between E1 and E2. So basically, a 2-legged walking gait.

You can also have the 4-legged equivalent to this, by splitting each foot into 2 legs, and having pairs of legs move together in the same alternating tetrahedral formation. It would be kinda like the "bounding" gait of animals like dogs when they're running, where they move pairs of legs together. Except that in 4D the legs don't have to be forelegs and hindlegs, but located at the same forward distance along the creature's body.

[gonegahgah]

In 4D, there's this very interesting possibility that isn't available in lower dimensions: a 2-legged arrangement that's actually stable. The way this works is that the feet are at 90° angles to each other, such that their projection to 3D forms two opposite edges of a tetrahedron. A tetrahedral arrangement in 4D is inherently stable (analogous to the tripod in 3D). Let's call these two opposite edges E1 and E2. When walking, the creature would lift the foot corresponding to E1, for example, and place it on the other side of E2, thus reflecting the tetrahedron across E2, forming an inverted tetrahedron. Then it lifts foot E2 and places it on the other side of the new position of E1, again forming the original tetrahedron but displaced by twice the distance between E1 and E2. So basically, a 2-legged walking gait.

Hi quickfur, but aren't you then throwing away some of the forward balance that is afforded by our forward extending feet?
For example, if we had one foot pointing forward and the other pointing sideways, sure it would provide even more sideways balance, but wouldn't it make the forward balancing a little more awkward?
Sure our feet usually point outwards a little but it feels like for a 4D person that it would still be a bit like us standing on one leg.
Even if our feet were discs, and the 4Der's feet spheres, it would still be a little awkward to have one less leg, isn't it?

[quickfur]

You're forgetting that this is 4D we're talking about. The feet are still pointing forward. But because of the extra lateral dimension, the legs can be positioned with 2 degrees of freedom while remaining in the hyperplane that faces the direction you're walking in. As I said, this gait is not possible in dimensions below 4D because there isn't enough degrees of freedom to fit everything into. But in 4D it's possible.

[gonegahgah]

You're forgetting that this is 4D we're talking about. The feet are still pointing forward. But because of the extra lateral dimension, the legs can be positioned with 2 degrees of freedom while remaining in the hyperplane that faces the direction you're walking in. As I said, this gait is not possible in dimensions below 4D because there isn't enough degrees of freedom to fit everything into. But in 4D it's possible.

But, feet still basically follow the line principle. A line can only head towards one direction. So they can't end in a forward direction and also end in a around-side direction?

[quickfur]

The feet can point forward and extend sideways in two orthogonal directions. That is, one foot can be oriented 90 degrees relative to the other, yet at the same time both of them can still point forward.

A similar phenomenon occurs with a 4D being's face: you can turn your face around a horizontal plane yet continue to face the same forward direction, at the same time.

[gonegahgah]

The feet can point forward and extend sideways in two orthogonal directions. That is, one foot can be oriented 90 degrees relative to the other, yet at the same time both of them can still point forward.

That's kind of true, after all ducks have splayed feet. However there is only one forward; not two perpendicular forwards.

But, also true is that I don't know of many feet that have evolved to splay directly out sideways.
That probably has to do with concentrating muscles in the forwards direction.
Having feet flapping sideways would probably create some sort of disadvantage I am guessing?

I envision feet that are certainly 4D but that still only face forward.
I envision at least three sets of forward muscles (to our one) to control each foot placement in 4D on the forward ground space.
And inversely at least three sets of rear leg muscles per leg to retract as needed.

We 3D beings certainly have balancing muscles but most of the musculature is concentrated towards making the forward/backwards motion most effective for running when needed.
I do concede that the additional rotating around sideways in 4D is probably of some use when it comes to lining things up.
However, I don't think these need to occur with the same level of maintenance as the forward motion.

[gonegahgah]

This is good though as this discussion gives me a few more thoughts on toe configurations...

It is conceivable that if leg muscles come in sets – either 3 or more – than it is likely that between the sets at ground level will lie big toes; to provide additional, between muscle, foot support.
So, in this case, if there are 3 sets of front, and 3 sets of rear, muscles per leg than it is likely that the 4D person will have 3 big toes per foot. Or 4 big toes for 4 sets, etc.

However if the muscle is continuous, and not in distinct groups around the front of the leg, then every toe is likely to be identical; as their support purposes would all be equal.
It would be interesting to try to speculate which evolution, either muscle sets or continuous muscle, would likely dominate and why?