Spiral staircases

[quickfur]

Today I suddenly realized that in 4D, one can build a spiral staircase flush against a wall.

In 3D, the central column of a spiral staircase must be set apart from any walls so that there's space to put the spiralling steps around it, since otherwise the steps would collide with the wall(s).

In 4D, however, thanks to the extra dimension, the central column of a spiral staircase can actually be flush against a wall, and the steps can spiral around it while still remaining flush with the wall. Better yet, since the walls of a 4D room are 3D, there is actually no need for any central supporting column; the steps can be attached directly to the wall (a flat wall!) and spiral upwards at the same time. Furthermore, the resulting staircase will only jut out from the wall as far as the width of each step, and no more, so it could be fitted even between two walls only the width of each step apart. In 3D, the opposite walls of a spiral stairwell must be at least two widths of the steps apart. (Of course, this is compensated in 4D by needing the walls to be sufficiently wide in 2 dimensions.)

After so many years speculating on 4D worlds I didn't think I could find anything more that'd impress me, but I just did.

[gonegahgah]

Nice! I like it.

[PatrickPowers]

Wow! That really is weird.

[anderscolingustafson]

This is something I hadn't really thought about until you mentioned it.

[gonegahgah]

I did realise yesterday that there is probably one problem quickfur?
I was thinking about our 4D people and it is most like they would have a tripod side-by-side stance.
So each step has to at least have a triangular surface and each step needs to be clear of the previous step.

If the steps were like our steps for bipeds then the excellent suggestion of yours would work.
But, with tripeds they can't be like out steps.

I'm guessing 4D creatures would still need do this around a central pole; instead of against a wall?
Shame really though if that is correct. I really did like the idea!

[quickfur]

That's not a problem. The steps can be as wide or as narrow as they need to be; it does not change the structure of the space that lets you define an upwards spiraling path in which the cylindrical envelope of the spiral projects to a plane. The underlying thrust here is that the lateral dimensions needed to construct the spiral fits entirely within a parallel hyperplane to the wall, so the entire structure does not need to protrude beyond the lateral width perpendicular to the wall.

You can construct a spiraling ramp, for example, according to exactly the same principles of construction and it would also work.

[PatrickPowers]

I was thinking about our 4D people and it is most like they would have a tripod side-by-side stance.

I say two legs in 4D. Not stable in 4D, but two legs are not stable here in 3D either and it doesn't stop us.

"Four legs good, two legs better!" -- The sheep in George Orwell's Animal Farm.

[quickfur]

I was thinking about our 4D people and it is most like they would have a tripod side-by-side stance.

I say two legs in 4D. Not stable in 4D, but two legs are not stable here in 3D either and it doesn't stop us.
[...]

The question isn't so much stable or not stable; the question is how (un)stable. Two legs are not stable in 3D, but they are at least stable in the horizontal plane, leaving only a single axis in which the standing creature may fall. Even on a windy day it's not hard to keep your balance against the gust of the wind by pushing against the direction you would fall.

In 4D, however, two legs would mean there are two dimensions along which you may fall, i.e., there's now a 360° circle of directions in which you may fall. That's much harder to control, and would be similar to standing on one leg in 3D. It isn't impossible, but quite hard, and likely requires frequent hopping to restabilize yourself.

Having said that, however, there is a way to be have a semi-stable stance on two legs in 4D: if your feet are perpendicular to each other, and you stand such that they fall on two opposite edges of a tetrahedron on the 3D surface of a 4D planet, say. This configuration is quite stable, and you can walk by lifting one foot (edge of tetrahedron) and putting it down on the opposite side of the other foot (forming a second tetrahedron sharing an edge -- the first foot -- with the first tetrahedron). For maximum stability your feet would have to extend in both directions from your leg, forming an inverted T shape (instead of the L shape we have in 3D).

An equivalent configuration to this is if you have 4 legs at the corners of a tetrahedron, and walk by lifting two legs at a time and planting them on the other side of the other two legs. One could surmise that 4-legged creatures would have such a configuration of legs for upright walking. But I'd expect non-upright creatures to have more limbs than 4; perhaps 6 or 8, in triangular prismic or cubic configuration, say. Or possible in octahedron formation (as triangular antiprism: 3 limbs in the front, 3 limbs in the back in dual orientation).

In dimensions higher than 4D, the number of limbs would probably increase quite quickly in order to maximize stability. Even for upright-walkers, the number of limbs would have to be quite high in order to compensate for the dramatic increase in the number of directions one could fall.

[gonegahgah]

I realised finally yesterday that I could picture having one foot ana, 2nd foot kata, and 3rd foot forward, and they could all equally step left or right.
So I realise now that that is correct.
Translating that back to your example, quickfur, you could have one foot ana, one foot kata, one foot left, and any of those could step forward.
The couldn't all step ana or kata (which I was originally getting mixed up on) as they would have to step over each other.
Likewise stepping towards or away from the left foot is not how we move; we don't step our left leg towards or away from our right foot; instead moving them forward to walk.

So with each foot sideways you can step any of them forward.
As they do so the step could then rotate from its ana-kata-left-right orientation towards one of ana, kata, left or right whilst the wall would be on one of the pair that is not rotated towards.
So if the stairs rotated towards the right; the wall could be on the ana side. Or if the stairs rotated towards the ana; then the wall could be on our left side.
In either case this would begin with forward motion as the direction first being rotated to. We always, as a rule of comfort, have to step forward even it the forward direction is twisting.

@Patrick
One important purpose of 2 legs in 3D is to stop us falling sideways while moving forward. The other purpose is to provide us with a landing leg of course.
Although 2 legs would provide us a landing leg in 4D they would not cover us for sideways movement and we would continually fall sideways as we move.
Alternatively only one leg is needed in 2D as your obviously not going to fall over sideways! 2D people would have to continually hop but without sideways it becomes easier to be stable doing this.

[mr_e_man]

We need to consider not only stability, but also efficiency. If you're hopping on one leg (or running on two legs), there are some moments when you're exerting no force on the ground, and others when you're exerting extra force on the ground. This is tiring. It's more efficient to exert a constant force, which is only possible with at least two legs. This applies just as well in 2D.

I think two legs could be stable enough in any number of dimensions. While walking, if you start falling in one direction, you can move one leg in that direction (only slightly, still moving mostly forward) to push you in the opposite direction. Of course it won't be exactly opposite; you'll start falling in another direction, but it should be slow enough that you can wait until your next step to adjust.

While standing... you just stand. Either you're given good balance, or you're given wide feet, so you don't need to hop to re-stabilize.

True, it is harder to balance with so many directions to fall in; presumably most creatures would have more legs to make it easier. But how hard can it be, given that you're already able to navigate with so many directions to get lost in? A higher-dimensional person necessarily has higher-dimensional abilities.

In 4D, two-legged people are like atoms and solar systems. Supposedly they shouldn't exist, but let's assume they exist anyway.

[PatrickPowers]

Having said that, however, there is a way to be have a semi-stable stance on two legs in 4D: if your feet are perpendicular to each other, and you stand such that they fall on two opposite edges of a tetrahedron on the 3D surface of a 4D planet, say.

Right. The legs take care of left-right, up-down takes care of itself. The feet need to angle into the remaining N-2 dimensions. In 6D feet would need to be proportionally twice as long, in 11D three times as long. It would be necessary to have more muscles.

The feet don't have to be perpendicular though. Our feet stabilize us in the forward-back dimension, but go only forward. Our center of gravity is over the center of the foot. This works in any number of dimensions. So the feet can both angle to the ana. Or kata. It's better because it's narrower and less likely to bump into things. Having feet that go somewhat sideways is a vulnerability that you want to minimize.

[quickfur]

[...]
In 4D, two-legged people are like atoms and solar systems. Supposedly they shouldn't exist, but let's assume they exist anyway. [...]

Well, atoms and solar systems in 4D don't exist if we assume a naïve generalization of 3D physics to 4D via dimensional analogy. That's a pretty big assumption that the laws of physics as we know it, which works so well in 3D, would hold in 4D merely by dimensional analogy. It could well be that a completely foreign set of physical laws in 4D would produce the kind of complexity that gave rise to the 3D universe as we know it.

It's conceivably, for example, that the particle/wave duality in 3D may have an analog not in a mere 4D version of particle/wave duality, but something related to the Hopf fibration which is so prominent in 4D geometry. It could be that instead of electron standing waves around a charged nucleus, 4D atoms are made from isoclinically-gyrating superimposed elementary forms ("gyrons"?) that form a completely new kind of chemistry. In the large scale, there could be a macroscopic weak charged-based force that creates large-scale structures with stable forms that aren't analogous to orbit-based planetary systems and galaxies that we see in 3D.

There's all sorts of possibilities if we would think outside the box a little. (Though it would also be a lot harder to visualize, which is probably why we often just stick to dimensional analogy from 3D instead. )

[mr_e_man]

Yes, that's near my point. If one possible version of higher-dimensional physics or biology requires us to have many legs, another version could allow us to have two legs.