Surface Of a 4D Torus

[PatrickPowers]

I realized that what I thought was the surface of a 4D torus was only 2D. I'm thinking of the product of two circles. The location on each circle can be described by one angle for a total of two, one angle in the wx plane and the other yz. That can't be right, can it? Don't all surfaces have to be 3D? For a torus to have a surface, one of those two circles has to be a disc instead.

I guess I'll call that 2D construct a 2D surface. That's an oxymoron but I think it will do. It's clear enough.

[gonegahgah]

I realized that what I thought was the surface of a 4D torus was only 2D. I'm thinking of the product of two circles. The location on each circle can be described by one angle for a total of two, one angle in the wx plane and the other yz. That can't be right, can it? Don't all surfaces have to be 3D? For a torus to have a surface, one of those two circles has to be a disc instead.
I guess I'll call that 2D construct a 2D surface. That's an oxymoron but I think it will do. It's clear enough.

My head was trying to wrap around the term "product" which I was trying to process as "multiply" which I am struggling to reconcile...
However, if "product" is just a synonym for "result" then isn't a 4D torus just the "product" of a circle and a sphere; with the sphere leaving infinite clones of itself around the circle as the process of producing the 4D torus?

[mr_e_man]

See http://hi.gher.space/wiki/Duocylinder

Sometimes the 2D surface is called the "duocylinder ridge".

[PatrickPowers]

I realized that what I thought was the surface of a 4D torus was only 2D. I'm thinking of the product of two circles. The location on each circle can be described by one angle for a total of two, one angle in the wx plane and the other yz. That can't be right, can it? Don't all surfaces have to be 3D? For a torus to have a surface, one of those two circles has to be a disc instead.
I guess I'll call that 2D construct a 2D surface. That's an oxymoron but I think it will do. It's clear enough.

My head was trying to wrap around the term "product" which I was trying to process as "multiply" which I am struggling to reconcile...
However, if "product" is just a synonym for "result" then isn't a 4D torus just the "product" of a circle and a sphere; with the sphere leaving infinite clones of itself around the circle as the process of producing the 4D torus?

This is a "latitude torus" used in navigation of a 4D planet. The planet has two perpendicular planes of rotation which are roughly analogous to our North and South poles. The Equator is a latitude torus. That's the set of points for which the minimal distance to the "poles" is equal. Let's call these pseudo-poles Circles instead, because instead of points they are great circles on the surface of the planet. So what is the set of points on the surface of the planet equidistant from the two circles?

One Circle is the set [w,x,0,0] where w²+x²=C, the other is [0,0,y,z] where y²+z²=C. So the equator is the set where w₂+x₂ = y₂+z₂. Geometrically take a circle, then for each point in the circle have a perpendicular circle with the same center. You get a 2D "surface" embedded in a 4D space.

One may "construct" the surface of the planet out of latitude tori by ranging over all possible values of c₁ and c₂ that satisfy
c₁² + c₂² = C²
w²+x² = c₁²
y²+z² = c₂²

A 4D object like this would be the set of points such that w²+x² <= c₁² and y²+z² <= c₂². I find the solid version of this shape mindboggling. If placed on a flat surface will it roll freely?

[PatrickPowers]

So what does ChatGPT have to say about this.

A duocylinder, having two independent circular cross-sections (from its definition as
could theoretically roll in two different perpendicular directions at the same time.

That's true, but all solid 4D objects can do that if unconstrained. And it's a bad use of the term "roll". Does one really barrel roll in a direction? Sort of. I think it's basically a mistake.

If constrained to roll on a 3D hyperplane (a 3D slice of 4D space), it might move in a way similar to a sphere rolling in 3D—except it has extra degrees of freedom because of its two circular dimensions.

So it doesn't know either. That makes me feel better about it.

[PatrickPowers]

https://www.qfbox.info/4d/duocylinder

The duocylinder is a peculiar object. Its two torus-shaped boundaries are surfaces that it can roll on, like a wheel. They are mutually perpendicular, so when rolling on one side, the duocylinder can only cover the space of a line. But if you tip it sideways on the other side, it will roll along a perpendicular line. It can always roll no matter which side you stand it on, but in different perpendicular directions depending on which side it is standing on.

So a "latitude torus" is the boundary between the two sides. That's what the duocylinder pivots on when tipping it over. Then we should be able to roll the duo balanced on that ridge, yes? In that case could it roll in any direction? Or could we make a spinning top with a 2D "point" that would move around in some way? Do that by getting both cylinders rotating for top-like stability.

[quickfur]

The duocylinder has two surfaces along which it can roll. Each surface covers the span of a 1D line by rolling, and the lines covered by rolling along each surface is perpendicular. That means the duocylinder can roll no matter which surface it's sitting on, and if you tip it over, it will start rolling in the perpendicular direction.

You can roll the duocylinder by standing it on its "latitude torus" but it will be unstable, like standing a 3D cylinder on one of its circular edges.