## 4D Torus Circus Act

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### 4D Torus Circus Act

Here in 3D a torus laying on its side will not roll. In 4D such a torus will roll anywhere in a 2D plane. It would be possible to make a big torus, lay it on its side, stand on it, and roll it about in that plane with your feet. Getting it into another horizontal plane within the 3D hyperplane would involve those hopping maneuvers one may see in the riding of skateboards.

It would be easier than standing on and rolling a ball, which here in 3D some people and dogs have learned to do. Here's an astounding video of a dog descending a garden stair whilst atop a ball. https://www.youtube.com/shorts/3qX6Sok9_b0

For a 4D circus act, make an even bigger torus and have several people stand on and roll it about footwise. Then earn your bread by walking on your hands, jumping or backflip across the torus, having some people walking forward and others backward, and so forth. That would take real coordination.

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Another difference is that in 3D a torus has a major and a minor radius. The length of these cannot be exchanged, as that would cause the torus to intersect itself. In 4D either radius may be larger than the other. The relatively larger the vertical diameter compared with the horizontal one, the more difficult such walking. In very small horizontal diameters it would be like standing on top of a pole that can roll. Can't have stilts like that though -- the tori are too wide.

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All in all, lately I've been realizing how loose the connection between two perpendicular 2D planes in 4D. I mean really perpendicular, with both significant angles with magnitude of pi/2. When one imposes Cartesian coordinates in 4D the WX plane is free to orient itself however it pleases with respect to the YZ plane, and vice versa. What is surprising to me is that this carries over to the physical world.

I'm beginning to see how Clifford proved that rigid objects could rotate in two planes independently. A rotation is rigid if it does not cause angles and/or lengths between points to change. Another way to say this is that polygons do not change under rotations (except of course for their location). Well, rotations in the WX plane have no effect whatsoever on Y and Z components because the angle between any pair of WX and YZ vectors is always pi/2, no matter what those vectors are doing. As long as the lengths of the vectors doesn't change, the length of the yz-wx vector doesn't change either. No change in angles, no change in lengths, means that the rotation is rigid.

I have the feeling that this is closely related to e1e2+e3e4 of geometric algebra not being a blade, but I can't say how.
Last edited by PatrickPowers on Tue Jul 19, 2022 12:08 am, edited 2 times in total.
PatrickPowers
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### Re: 4D Torus Circus Act

There are several distinct toruses in 4D, which one(s) are you referring to?
quickfur
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### Re: 4D Torus Circus Act

quickfur wrote:There are several distinct toruses in 4D, which one(s) are you referring to?

The sort of torus where w2+x2=C1 and y2+z2=C2.

I woke up this morning and thought, "what I posted was all wrong. The rolling isn't confined to a 2D plane." But no, it is. The idea is that the torus is lying on its side, pinned to a horizontal 3D hyperplane by gravity. So one of the dimensions is constrained to be vertical. Call it Y. Then the torus can roll only in the YZ plane. This means it can roll in any direction in the WX plane. The rider can in effect reset the choice of Cartesian coordinates by hopping the board, temporarily avoiding friction.

At least, so I believe at the moment. It all has to do with the correspondence between Cartesian coordinates and a "real" solid object. The idea is that sometimes ths is arbitrary, sometimes it isn't, and that this is a mixed case.

I'm still not at all sure that I have done this correctly. The quesiont is how much flexibility one has in orienting the Cartesian coordinates on the torus. Can it roll any direction in the WX plane, or only in the Z direction, and how arbitrary is all this relative to the plane upon which it is rolling? I've got to figure this out.
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### Re: 4D Torus Circus Act

Your equations appear to describe a duocylinder, the rotational properties of which has been studied before. This shape consists of two curved surfaces that lie on two orthogonal planes; each curved surface is in the shape of a torus. It has a peculiarity, in that it can only roll along a straight line when place on one of its surfaces, but if you tip it over to land on the other surface, then it will roll along another straight line perpendicular to the first. So the total space it could cover is technically a 2D plane, but it does require an arbitrary number of tippings over in order to actually span a 2D area; otherwise, it will only roll along a line.

Technically, the duocylinder isn't really a torus, it's more like a toroidal cylinder. Or one might argue, that it's a torus with cylindrical properties? I guess it depends on how you look at it.

There are other kinds of toruses possible in 4D. For example, take a 4D sphere, displace it from the origin by some distance R, then rotate it around the origin in a circle. The shape it traces out is a direct analogue of the 3D torus: a donut shape with a hole, but it's actually distinct from the duocylinder! There's also the weird shape you obtain if you start with a 2D sphere displaced from the origin, and roll it around a spherical 2D region. That traces out a different kind of torus (its "donut hole" is of a different dimension than the "donut hole" of the previous torus). I'm not 100% sure whether this is identical to the duocylinder, but my suspicion is that it's yet another different shape.

Higher dimensions are weird.
quickfur
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### Re: 4D Torus Circus Act

Today I'm convinced that the shape I'm looking for doesn't exist. Oh well, no big loss.
PatrickPowers
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### Re: 4D Torus Circus Act

There is such a thing as a "spherinder" (spherical cylinder) in 4D: the extrusion of a 3D sphere into 4D. The lateral manifold that curves around its two spherical lids is curved in 2 dimensions; if I'm visualizing it correctly, that would allow it to roll in 2 dimensions simultaneously and without needing to be tipped over like the duocylinder.

Now take a spherinder and subtract from it another spherinder of smaller radius (but equal length), and you end up with a toroidal object that can roll in 2 dimensions simultaneously. Well OK, it's basically a thickened version of an uncapped hollow spherinder, so geometrically it behaves like a spherinder. But it'd be a topological torus, technically speaking.

What's the difference between a 4D tube and an overweight Dutchman? One is a hollow spherinder, and the other is a spherical Hollander.
quickfur
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### Re: 4D Torus Circus Act

Yes I thought of that. It would be like standing on a cylinder here in 3D, which has never been popular. The advantage of a torus would be that it would be more stable than that. Appears that it is not to be.
PatrickPowers
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### Re: 4D Torus Circus Act

What exactly are you looking for? Something to build a wheel with? Something to stand on and move forwards by stepping backwards?
quickfur
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### Re: 4D Torus Circus Act

quickfur wrote:What exactly are you looking for? Something to build a wheel with? Something to stand on and move forwards by stepping backwards?

A torus lying on its side that you can stand on and move forwards by stepping backwards. I'm not sure whether this is possible with a rigid rotation.
PatrickPowers
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### Re: 4D Torus Circus Act

I've given up on the 4D rolling torus.
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