Understanding Fourth Dimension Figures

Date: 07/05/98 at 20:06:22
From: Heidi
Subject: fourth dimension equations

I am doing a senior research project on the progression of the fourth 
dimension, gearing toward teaching a high school level geometry 
course. I have located the standard form for equations of a 
hypersphere, but my instructor asked me to include equations for 
several more figures including the tesseract and hypertetrahedron. 

I have searched your archives and have found some very useful 
material. As I mentioned earlier, I am a pre-service teacher and 
cannot wait until I can use this site as a reference for my classroom. 
Thanks for providing such a valuable service for students, parents and 
educators.

Heidi

Date: 07/05/98 at 22:52:21
From: Doctor Tom
Subject: Re: fourth dimension equations

Hello Heidi,

The easiest way to think about 4 (or more) dimensions is to see what 
is going on in 1, 2, and 3 dimensions, and see what the pattern is. At
least that will work for your examples. For example, the tesseract
is just a 4D cube.  Well, what does a 3D, 2D, and 1D "cube" look like?

If we're interested in the points that lie inside the cube, a 1D cube
is just a line segment. If in all cases we're looking at a cube with
length 1, in 1D, the "cube" is all the points such that 0 <= x <= 1.
In 2D, it is a square, so the interior is all the points (x, y)
satisfying 0 <= x <= 1 and at the same time, 0 <= y <= 1. In 3D, it is
the same as the square, except the interior is the set of all points
(x, y, z) satisfying 0 <= x <= 1 and 0 <= y <= 1 and 0 <= z <= 1. To
save space, this is often written as: 0 <= x, y, z <= 1. So in 4D,
the interior of a tesseract is the set of all points (x, y, z, w) such
that 0 <= x, y, z, w <= 1. In other words, points that have all their
4 coordinates between 0 and 1.

The hyper-tetrahedron should be examined in the same way. In 1D, it 
is a line segment. In 2D it is a triangle. In 3D, it is a tetrahedron. 
I think the easiest way to look at these things is in terms of 
barycentric coordinates. For a triangle with the three vertices 
P0 = (x0, y0),  P1 = (x1, y1), and P2 = (x2, y2), all the interior 
points are those that have the form a*P1 + b*P2 + c*P3, where 
a + b + c = 1 and a, b, and c are bigger than zero. In other words, 
all points of the following form:

   (a*x0 + b*x1 + c*x2, a*y0 + b*y1 + c*y2)

For a line going from x0 to x1, it is all the points a*x0 + b*x1 where 
a + b = 1 and a and b are non-negative. You can work out the 3D form 
yourself, but here is the answer in 4-D:

Given 5 points P0 = (x0, y0, z0, w0), P1 = (x1, y1, z1, w1), 
P2 = (x2, y2, z2, w2), and so on for P3 and P4, the interior of a 
hyper-tetrahedron (of dimension 4) is given by all the points 
satisfying: 

   a*P0 + b*P1 + c*P2 + d*P3 + e*P4 

where a + b + c + d + e  = 1 and all of a, b, c, d, and e are 
positive. Or:

   (a*x0+b*x1+c*x2+d*x3+e*x4, a*y0+b*y1+c*y2+d*y3+e*y4, ...)

If this doesn't make much sense, do look up barycentric coordinates.
They're often very useful.

Good luck in your future career.

- Doctor Tom, The Math Forum
Check out our web site! http://mathforum.org/dr.math/