Does an N-Cube have Surface Area?

dimensionless

Let's say I have a four dimensional cube. Would it have a true surface area? I'm wondering if maybe it would have a surface volume rather than a surface area.

wofsy

its boundary is not a surface but does have a 3d volume

Rasalhague

Would this n-1 dimensional boundary be a hypersurface?

wofsy

Depend what you mean by hypersurface. Explain.

HallsofIvy

In n dimensional geometry, a "hypersurface" is the n-1 dimensional boundary of a bounded n-dimensional region.

As for dimensionless's original question, its really a matter of convention whether you call the 3 measure of the boundary of a 4 dimensional region "area" or "volume". That's why most people just talk about n or n-1 dimensional "measure".

dimensionless

Does that mean that a light wave in 4D would have a flux through a volume rather than a surface area?

wofsy

In general there would be an exact analogue of flux but with light there is a Lorentz metric and I am not sure how that would work.

g_edgar

Solution of the wave equation is quite different in even dimensions vs. odd dimensions.

dimensionless

Why would that be?

Rasalhague

I had in mind an (n - 1)-dimensional "bit" of the given n-dimensional space. HallsofIvy's "the n-1 dimensional boundary of a bounded n-dimensional region" sounds like what I was thinking but more precisely worded that I'd have managed. Wikipedia calls a surface a "two dimensional topological manifold". Would a hypersurface then be an (n - 1)-dimensional topological manifold (and is every manifold at least a topological manifold)?

wofsy

Every manifold is at least topological but may have additional structure such as a differentiable structure.

A submanifold of dimension n-1 is a called a hypersurface. You may be aware that you can have submanifolds of lower dimension as well. For instance in 4 space the Klein bottle can be embedded as 2 dimensional surface.