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dimensionless
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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.
Rasalhague said:Would this n-1 dimensional boundary be a hypersurface?
wofsy said:its boundary is not a surface but does have a 3d volume
dimensionless said:Does that mean that a light wave in 4D would have a flux through a volume rather than a surface area?
g_edgar said:Solution of the wave equation is quite different in even dimensions vs. odd dimensions.
wofsy said:Depend what you mean by hypersurface. Explain.
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)?
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.
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.
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.
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.
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.