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Hypercube


Date: 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/   
    
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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