HypercubeDate: 10/13/2001 at 21:02:46 From: Bruce Chiarelli Subject: Many dimensions I was reading a book on analytic geometry, and it said that mathematicians are currently (this book was old, so i don't know if that's true) working on a four-dimensional hypercube. I did some research before I came here for help, but all I got was that the fourth dimension was time. Have there been any recent developments in this topic? -Bruce Chiarelli Date: 10/14/2001 at 04:20:08 From: Doctor Jeremiah Subject: Re: Many dimensions Hi Bruce, The fourth dimension in physics is time. The reason why it counts as a fourth dimension is that any event can be nailed down with its location in the universe (three dimensions) and the time it happened (another dimension). However, there could be more than three dimensions of space that we just can't see. If we prove that there are four dimensions, then time will automatically become the fifth. Strangely enough (if I remember right), the unified theory of physics (which makes quantum physics and gravitation into one set of equations) requires at least 22 dimensions of space, plus time on top of that! I have a book on the fourth dimension of geometry where they talk about what a fourth-dimensional cube would look like. Here is one way to approach it: To move from zero dimensions (a point) to one dimension (a line), you double the number of points and create a line for each original point. 1. + 2. + ---> + 3. +-------------+ To move from one dimension (a line) to two dimensions (a face), you double the number of points and lines and create a line for each original point and create a face for each original line. 1. +-------------+ 2. +-------------+ | | V V +-------------+ 3. +-------------+ | | | | | | +-------------+ To move from two dimensions (a face) to three dimensions (a cube), you double the number of points and lines and faces and create a line for each original point and create a face for each original line and create a cube for each original face. So to create higher dimension objects you can use a table like this, where P(-1) means the number of points in the dimension above and L(2) means the number of lines in the second dimension: dimens points lines faces cubes 4Dcubes 0 1 0 0 0 0 1 2P(-1) P(-1) 0 0 0 2 2P(-1) P(-1)+2L(-1) L(-1) 0 0 3 2P(-1) P(-1)+2L(-1) L(-1)+2F(-1) F(-1) 0 4 2P(-1) P(-1)+2L(-1) L(-1)+2F(-1) F(-1)+2C(-1) C(-1) Or an an arbitrary recurrance relation: dimens points lines faces cubes 4Dcubes n 2P(n-1) P(n-1)+2L(n-1) L(n-1)+2F(n-1) F(n-1)+2C(n-1) C(n-1) So if n=3 (3D space) then: dimens points lines faces cubes 4Dcubes 3 2P(2) P(2)+2L(2) L(2)+2F(2) F(2)+2C(2) C(2) Now P92) is the number of points in a 2D square (4), L(2) is the number of lines in a 2D square (4), and F(2) is the number of faces in a 2D square (1), and C(2) is the number of cubes in a 2D square (0), so: dimens points line faces cubes 4Dcubes 3 2(4)=8 4+2(4)=12 4+2(1)=6 1+2(0)=1 1 And for a 4D "cube" you would have: dimens points lines faces cubes 4Dcubes 4 2P(3) P(3)+2L(3) L(3)+2F(3) F(3)+2C(3) C(3) 4 2(8) 8+2(12) 12+2(6) 6+2(1) 1 4 16 8+24 12+12 6+2 1 4 16 32 24 8 1 So according to that reasoning, a 4D "cube" would have 16 points, 32 lines, 24 faces, and 8 cubes. Don't try to draw this at home! You can imagine this in three dimensions (without right angles) as a cube inside a larger cube where each line of the larger cube is connected by a face to its corresponding line of the smaller cube. Of course that's not what it looks like in four dimensions. If you wonder why, think about how you would imagine a 3D cube in a 2D world (a square inside a square where each point on the larger square is connected by a line to the corresponding point on the smaller square...). As for proving that a spatial fourth dimension exists in the real world, I have no idea. But it doesn't matter because we would never know even if it did. Imagine you are a 2D person. on a 2D plane in space. A cube moves down through your plane, but you don't know about "up" and "down." You don't see it as a 3D object; you see a 2D cross- section of it. In the same way, if a 4D cube were to intersect our 3D space, it would look like a 3D object and we would have no idea that it was a 4D object except that it just appeared for no reason and when it moved out of the other side of our 3D space, it would just disappear for no reason. So if something just appears and disappears with no apparent explanation, then it might be a higher-dimensional object intersecting with our 3D universe! Does that answer your question? If not, please write back. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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