# Hypercube

#### atomka

Registered Member
Does anyone on the forumn know of the aspect of a hypercube? I would say it is more of a spiritual thing, but it still has to do with science and geometry. If you would like me to try and explain it, reply this thread.

Originally posted by atomka
Does anyone on the forumn know of the aspect of a hypercube? I would say it is more of a spiritual thing, but it still has to do with science and geometry. If you would like me to try and explain it, reply this thread.
What d'you mean by hypercube? A tesseract?

atomka ...

The only reference to a hypercube I can remember is:

The three dimensional representation of a four dimensional cube.

Take care.

By "hypercube" most people mean "tesseract", although a tesseract is only one example of a hypercube (tesseract = 4 dimensional hypercube).

Let's look at it by analogy. A square is a 2-cube (2 dimensional hypercube). It has four sides consisting of straight lines. A cube is a 3-cube. It has six faces consisting of squares. A tesseract is a 4-cube which has eight "faces" consisting of 3-cubes.

Each corner (vertex) of a square has 2 lines. Each vertex of a 3-cube has 3 adjacent faces of the cube. Each vertex of a 4-cube has 4 adjacent cubes.

We can construct a 3-cube by folding a 2-dimensional grid of six squares into the 3rd dimension. Similarly, to construct a 4-cube we need to fold a collection of eight 3-cubes in the 4th dimension.

The problem with tesseracts is that we cannot visualise them in our 3D world. We can imagine various "shadows" though. If you shine a torch onto a 3-cube and look at its 2 dimensional shadow on a wall, the shape of the shadow might be a square, or some type of parallelogram. Similarly, if we could shine a "3D torch" onto a 4-cube and look at its "shadow" in our 3-space, we would see 2 joined and possibly distorted 3-cubes.

Each line in a square is at 90 degrees to the adjacent lines. Each square in a 3-cube is at 90 degrees to adjacent squares. Each 3-cube in a tesseract is at 90 degrees to adjacent 3-cubes. Remember that four 3-cubes meet at each vertex of the tesseract. All of them are mutually at right angles to each other. How is this possible? Answer: It is only possible in 4 dimensions (or more).

5-cubes, 6-cubes and even n-cubes are possible.

James R ...

Re.
Similarly, if we could shine a "3D torch" onto a 4-cube and look at its "shadow" in our 3-space, we would see 2 joined and possibly distorted 3-cubes.
Would one not see six .... etc. if it were a 'wireframe' 4-cube?

And, wouldn't it have to be a 4D torch?

A hypercube is just a representation of a cube in four dimensions (Or at least it's creator explained it as that) of course there is a very uncanny resemblance to some artist pictures of paradoxical preportions. (in fact you can find out about those images by checking out M.C.Escher (1898-1972) http://www.artchive.com/artchive/E/escher.html )

From my understanding though, I didn't really class it as four dimensional to begin with until it's orientation was tilted (which added time to the image) I know that a hypercube can be fractalised, and is a shape you can't build in three dimensions.

Yes, Chagur. Make that a 4D torch. The wire frame thing is harder to explain. You really need a picture. Arguably you'd see all 8 cubes, but some would be distorted. There are some sites on the web which have good animations of 4-cubes projected into 3 dimensions and represented on a 2-D screen.