# Does an N-Cube have Surface Area?

• dimensionless
In summary: 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.In summary, hypersurfaces are submanifolds of dimension n-1. A Klein bottle can be embedded as a 2-dimensional hypersurface in 4-space.
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.

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

Would this n-1 dimensional boundary be a hypersurface?

Rasalhague said:
Would this n-1 dimensional boundary be a hypersurface?

Depend what you mean by hypersurface. Explain.

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".

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

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

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

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.

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

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

Why would that be?

wofsy said:
Depend what you mean by hypersurface. Explain.

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)?

Rasalhague said:
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)?

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.

## What is an N-Cube?

An N-Cube is a geometric shape with N number of dimensions, where N is a positive integer. For example, a 3-Cube is a cube with three dimensions, while a 4-Cube is a shape with four dimensions.

## Does an N-Cube have surface area?

Yes, an N-Cube does have surface area. However, the concept of surface area in higher dimensions is different from the traditional definition in three dimensions. In an N-Cube, the surface area is defined as the total area of all the faces of the shape.

## How do you calculate the surface area of an N-Cube?

The surface area of an N-Cube can be calculated by first finding the length of each edge of the shape, and then using the formula 2^(N-1) * s^2, where s is the length of one edge. This formula can be applied to any N-Cube, regardless of the number of dimensions.

## Is the surface area of an N-Cube different from its volume?

Yes, the surface area and volume of an N-Cube are two distinct measurements. While the surface area is the total area of all the faces of the shape, the volume is the amount of space inside the shape. In general, the volume of an N-Cube is larger than its surface area.

## How is an N-Cube different from a regular cube?

An N-Cube differs from a regular cube in terms of the number of dimensions. A regular cube has three dimensions, while an N-Cube can have any number of dimensions. Additionally, the surface area and volume calculations for an N-Cube are different from those of a regular cube.

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