# Hypercube

Discussion in 'Physics & Math' started by atomka, Nov 19, 2001.

1. ### atomkaRegistered Member

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16
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.

3. ### rdeEukaryotic specimenRegistered Senior Member

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278
What d'you mean by hypercube? A tesseract?

5. ### Chagur.Seeker.Registered Senior Member

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atomka ...

The only reference to a hypercube I can remember is:

The three dimensional representation of a four dimensional cube.

Take care.

7. ### James RJust this guy, you know?Staff Member

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31,341
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.

8. ### Chagur.Seeker.Registered Senior Member

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2,235
James R ...

Re.
Would one not see six .... etc. if it were a 'wireframe' 4-cube?

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

9. ### StryderKeeper of "good" ideas.Valued Senior Member

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13,101
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.

10. ### James RJust this guy, you know?Staff Member

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31,341
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.