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### Hypercube

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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!

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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