Analytic Geometry in Higher Dimensions: Torus Analogs

[SaxonViolence]

Friends,

Please forgive my lack of proper terminology, but:

Take a Torus. There is a nice analytic formula for a Torus.

Now suppose that we rotate the Torus in the 4th Dimension. There seems to me to be at east 3-ways to go about this.

A.} We can rotate the Torus around its own axis—analogous to how a circle can be rotated to form a sphere.

B.} We can rotate the whole Torus at a distance—analogous to how circle is rotated through space to become a Torus.

And,

C.} I Think that it matters—though I may be wrong—what attitude the Torus is in when it is rotated in 4-Space.

I mean, wouldn't a Horizontal Torus with the center hidden from our 3-D perspective, form a different shape if it was rotated in the 4th Dimension that way, than a Torus that was rotated while facing us head-on like a Bullseye? Or is my power of visualization lacking? Are the two cases the same?

Now, suppose that instead of the Torus staying in the same orientation, it could Rotate around its own axis as it generates what was a "Simple" 4-D Toroidal analog. It can rotate at all sorts of frequencies—making a full Rotation while it makes a full circle around its center, or 2-Rotations, or 100-Rotations—not to mention Fractional Numbers of Rotations.

How can one find the equations of these various—well, what do you call them? They aren't Polytopes are they?

IF we decide to go up to 5-Dimensions, the possibilities become even more staggering.

Now, one way to get a Torus analog is to take a Sphere and rotate it in the 4th Dimension similar to how a circle is rotated to create a torus. Is this yet another 4-Dimensional Toroid, or is it the same as one of the cases we've already considered?

BUT, Suppose that instead of a Sphere, we use a Dodecahedron, an  Icosahedron or one of the Stellated Polyhedra instead, and rotate it in 4-D?

Is there an exhaustive, systematic way that I can answer such questions for myself?

 

Thank You!

 

Saxon Violence

[OceanBreeze]

I think the Theorem of Pappus or Guldinus covers this subject and can answer your questions.

[OceanBreeze]

 

"Point being, if you are at one of those 9 dots then from your perspective the other 8 dots appear to take one arrangement. However, if you are at another dot, than the remaining 8 dots take on an entirely different arrange."

 

Yes, and that is a good explanation why the cosmic microwave background radiation cannot act as a universal reference for all points in the Universe. It may be a good reference on the galactic scale, but it will not be the same at the inter-galactic scale. People who claim the CMB is a universal reference, in contradiction of Einstein, are just flat out wrong

The fractal dimensions of cosmic microwave background (CMB) maps and other sources obtained by the “Planck” mission are investigated. It is shown that the choice of source distribution models used in data processing can significantly affect the fractal dimension of CMB.

 

[SaxonViolence]

Thanks everyone.