Beyond the Third Dimension

Date: 5/16/96 at 17:23:49
From: Anonymous
Subject: Beyond the Third Dimension
 
I am searching for information on beyond the third dimension.  
Everywhere I go I get technical information that I don't understand.  
I am in high school and I have to write a five page paper on this.  
I need as much information as well as pictures or diagrams on this 
subject as I can find.  Every source I get has a different explanation 
and I am very confused at this point.  Any help would be incredibly 
appreciated.  

Thanks in advance,
Rob

Date: 5/16/96 at 21:0:40
From: Doctor Jodi
Subject: Re: Beyond the Third Dimension

Hi there! Actually, the subject line (Beyond the third dimension) is 
also the title of a book by Thomas Banchoff.  This is a great book 
designed to introduce the fourth dimension - you might give it a read 
if you have a chance.

Here's something to start with:

From: Bob Hesse
Subject:    Viewing Four-dimensional Objects in Three Dimensions
Date:       Tue, 6 Sep 1994 15:54:12 GMT

Viewing four-dimensional objects in three dimensions:

Given that humans only visualize three dimensions, how is it
possible to visualize four-dimensional, or higher, objects?
This question is the underlying idea of a short novel written
over a hundred years ago by Edwin A. Abbot called FLATLAND.
FLATLAND is a story about two-dimensional creatures--triangles,
squares, circles and other polygons--that live on a plane.
The story contains a section where one of the squares is
visited by a three-dimensional object, a sphere.  The sphere
explains to the square the existence of higher dimensional
objects like itself, and ways in which the square can
understand the form of such objects.  The method the sphere
gives to the square can be generalized so that the form of
four-dimensional objects can be seen in three dimensions.
This method of viewing higher-dimensional objects as well as
others is one way people can understand the shape of higher-
dimensional space.

Before attempting to view four-dimensional objects in
three-dimensional space, let us consider viewing a three
dimensional object in two-dimensional space.  In FLATLAND, the
method in which the sphere showed its form to square was by
raising its body through the Flatland surface.  The square saw
at first a point that quickly grew to a circle, which continued
increasing in size, and then started decreasing in size until
it became a point, and then it disappeared.  So the square
perceived the sphere to be an infinite collection of circles
pieced together...

Before continuing further, it should be mentioned that for
simplicity's sake and for aesthetic purposes, the forms which we
will consider viewing are polytopes, the generalized term for
polyhedra and polygons.

A second way to view three-dimensional polytopes in two dimensions
is by means of a projection. Projection is a popular method for
Cartographers to create maps of the world from a globe.  For
instance the United Nations flag is created by a projection of the
globe about the south pole. One especially useful type of
projection in mathematics is called stereographic projection. 
Stereographic projection takes a sphere and maps it over the entire
plane in the following manner. If one lays a sphere on a plane,
the point of the sphere touching the plane stays fixed while the
point directly opposite it, i.e. "the North Pole" gets sent to
infinity. Any other point on the sphere is sent to the unique
point on the plane found by intersecting the plane with a line made
from the point at the north pole and the point on the sphere...

A third way to view polyhedra in two dimensions is through a method
defined by Barbara Hausmann and Hans-Peter Seidel as "Cut-Throughs"
and "Fold-Downs."  Since polyhedra have as faces regular polygons, one
could cut a polyhedra on the edges and fold it in a way so that all
the faces are lying on the plane...

As you may have already surmised, all of the above methods can be used
to visualize four-dimensional polytopes in three dimensions.  But 
before showing these different ways of viewing polytopes, an 
explanation of how these polytopes are constructed is in order.  As 
regular polyhedra are constructed from regular polygons, so are 
regular 4-dimensional polytopes constructed from regular polyhedra.   
Recall that there are only five regular polyhedra:

1. The tetrahedron, constructed from four equilateral triangles.
2. The cube, constructed from six squares.
3. The octahedron, constructed from eight equilateral triangles.
4. The dodecahedron, constructed from 12 regular pentagons.
5. The icosahedron, constructed from twenty equilateral triangles.

There are only six four-dimensional polytopes.  They are the 
following:

1. The 4-simplex, constructed from five tetrahedra, three tetrahedra
   meeting at an edge.
2. The hypercube, constructed from eight cubes meeting three per edge.
3. The 16-cell, constructed from sixteen tetrahedra, with four 
   tetrahedra meeting at an edge.
4. The 120-cell, constructed from 120 dodecahedra, with three 
   dodecahedra meeting per edge.
5. The monstrous 600-cell, constructed from 600 tetrahedra, with five
   tetrahedra meeting at an edge.


Since the above examples of viewing three-dimensional polytopes in two
dimensions all contain the cube, let us continue viewing in the fourth
dimension by looking at the hypercube.  First, let us look at some
projections of the hypercube.  Figure 5 is a series of diagrams 
created by rotating the hypercube about a plane in four dimensions, or 
a combination of plane rotations.

Figure 6 is a sequence of pictures of a slicing of the hypercube into
three dimensions.  Note that as the hypercube passes through our
three-dimensional space, it is growing and then shrinking from various
polyhedra shapes.  This is analogous to the slicing of the cube in the
plane shown earlier.

Figure 7 is a set of pictures showing the step-by-step approach of the
hypercube being cut-through and folded-down.  Note that in each stage 
of the process a cube pops out of the hypercube, which is analogous to 
a square coming out of a cube as one dissects a cube.


In FLATLAND, the square discovered that the sphere was an infinite
collection of circles.  However the square was unable ever to actually
view the sphere in the same way we three-dimensional beings are able.
Similarly, we can discover what some four-dimensional objects look 
like by viewing aspects of them in three dimensions.  But like the 
square, we are limited in understanding the whole nature of these 
objects.

This article is based on an interview and a seminar given by Barbara
Hausmann at the Geometry Center... [see]

http://www.geom.uiuc.edu/docs/forum/polytope/   
__________

As far as reading about the fourth dimension goes, there are quite a 
few good books around.

BOOKS

- Flatland by Edwin Abbott  
       This classic novel focuses on the life of our 2D hero, 
       A. Square, who is declared mad and imprisoned for discussing 
       his travels into a 1D world and his encounter with a 3D sphere. 
       *GREAT INTRO

- Planiverse
       Talks about lots of sticky details about life in 2D--from 
       zippers and doors to playing volleyball.  It's an awesome story 
       and again should give you ideas about 4D. 

- Sphereland
       
- Shape of Space by Jeff Weeks
       Asks the question, "What is the shape of the universe?" and 
       explores the mathematics of 4D by introducing elementary 
       topology. No college math needed. (also a Geometry Center 
       video)


ON THE WEB

http://www.geom.uiuc.edu/docs/forum/polytope/   
   - an illustrated article by Bob Hess called
     "Viewing four-dimensional objects in three dimensions"

http://mathforum.org/dr.math/problems/dixon24.html   
   - a simple answer to the question "What is the fourth dimension 
     mathematically?"

http://www.uccs.edu/~eswab/hyprspac.htm   
   - 4D polytopes, the 4D equivalent of polygons and polyhedra *GREAT 
     pictures

http://www.graphics.cornell.edu/~gordon/peek/peek.html   
   - description of Peek, an interactive software program to look at 
     projections and cross-sections of higher dimensional objects 

http://mathforum.org/~sarah/HTMLthreads/articletocs/4d.visualization.html   
  - interviews with mathematicians:  visualizing 4D in their work

OTHER

A MATHEMATICS SAMPLER, by W. Berlinghoff and K. Grant

Chapter 8, "Mathematics of Space and Time," includes some
detailed explanations of hypercubes, hyperspheres, 
hypercylinders, etc., how to examine them by looking
at cross sections, and the like.  It's written for a 
first-year college liberal-arts audience, so the mathematics
is not very sophisticated.  However, the reading level
might be a little above the comfort range of some high
school students.  In any event, the material is easily 
adaptable to high-school mathematics.

The book is published by Ardsley House (New York);
3rd Edition, 1992.

The final section of the chapter is a 6-page overview
of "4-Space in Fiction and in Art" which supplies some 
connections between this branch of mathematics and 
some well-known artistic and literary works of the 
past 100 years or so (including Dali's "Corpus 
Hypercubus" cited in Heidi B.'s earlier posting.)

Good luck with your project and let us know how it goes!  
Write back if you need more help!

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