Does an N-Cube have Surface Area?

  1. 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.
  2. jcsd
  3. its boundary is not a surface but does have a 3d volume
  4. Would this n-1 dimensional boundary be a hypersurface?
  5. Depend what you mean by hypersurface. Explain.
  6. HallsofIvy

    HallsofIvy 40,563
    Staff Emeritus
    Science Advisor

    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".
  7. Does that mean that a light wave in 4D would have a flux through a volume rather than a surface area?
  8. 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.
  9. Solution of the wave equation is quite different in even dimensions vs. odd dimensions.
  10. Why would that be?
  11. 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)?
  12. 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.
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