Hypercylinder

Hyperdimensionality tiem

As we know, a cylinder can be defined in many distinct ways; first, by moving a circle parallel to itself; second, by rotating a rectangle around itself; and third, by folding it so as to connect two opposite edges. The question is, can we generalise it to n dimensions?

Suppose we want to generalise it according to the first construction method. We move the cylinder parallel to the hypothetic 4 axis and we have a 4-cylinder. It has 4 cells, 4 cylinders which form this polytope which we shall call a hypercylinder.

However, we could to generalise it according to the second definition, too. If we fold a rectangular prism in an R^4 space (assuming Euclidean geometry), it follows that the facets formed are the folded prism, and two 3-spheres. However, this definition does not comply with the first.

It seems as though that the definitions agree only for  3-cylinder.

inb4 "LOLOLOLOL THERE IS NOT 4TH DIMENSION UR DUMB"

Your turn.

Kill yourself.  When talking about hyperdimensional objects, there are ONLY two definitions for a three dimensional cylinder, all the others just happen to be ways you can "make" one.

>cylinder
>A three dimensional hypercylinder that can roll along one linear axis. It can be formed by extending a circle into realmspace, or by rotating a square into realmspace.

>hypercylinder
>An n-dimensional cylinder. Hypercylinders are formed by extending or rotating lower-dimensional hypercylinders into higher dimensional space. In realmspace there is only one hypercylinder, the cylinder. In tetraspace, there are two hypercylinders: the cubinder and spherinder.

>cubinder
>A four dimensional hypercylinder that can roll along one axis, but is flat on all the other sides. It can be formed by extending a cylinder into tetraspace linearly.

>spherinder
A four dimensional hypercylinder that can roll around in a planar region. It can be formed by extending a sphere into tetraspace linearly.