HyperCubes

Date: 3/21/96 at 19:32:8
From: Art Mabbott
Subject: 4th Dimension

To the Geometry Center and the Geometry (Math) Forum:

Do you folks know of any videos that show the hyper-cube in action
that would be appropriate for the high school level?  Thanks. 

After showing The Shape of Space video, my students have more 
questions than I have answers.  It was (is) great.  Thanks for the 
loan, GC. 

Art Mabbott

Date: 3/22/96 at 23:29:57
From: Doctor Jodi
Subject: Re: 4th Dimension

Hi there! Good to hear that Shape of Space is making it into the 
classroom. Olaf Holt and Paul Humke made a short video called 
"Voyager from the Fourth Dimension." If I remember correctly, it 
shows several sequences of hypercubes and nothing more.  This is 
also in the Geometry Center archive.

I also highly recommend Jeff Weeks' book _Shape of Space_ by the 
same title, either for yourself or for interested high school 
students.

On the web, if you'd like to check out some of the shapes used in the fly-
thru's, take a look at the torus and Klein bottle at

http://www.geom.uiuc.edu/zoo   

There are also some Mobius strips (though you can make those 
easily) and rotations of the hypercube (tesseract) and Klein 
Bottle.  (To make a nice torus, try connecting the two ends of a 
slinky.)

Good luck!

-Doctor Jodi,  The Math Forum

Date: 3/23/96 at 1:11:9
From: Doctor Jodi
Subject: Re: 4th Dimension

Hi again Art - I neglected to mention that comparing 3D to 2D 
first is particularly useful. It looks like you've done this with 
Flatland.  I also particularly recommend Planiverse, which talks 
about lots of technical details, from zippers and doors to playing 
volleyball in a 2-D universe.  It's an awesome story and again 
should give you ideas about 4D.

Here's an article by Bob Hesse from the Web.  The correct address 
is given at the bottom.  Enjoy!  

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.  Figure 1 is a series of pictures that
show the sphere as it rises through the plane as the square
saw it:

The flatlander square just as easily could have seen what a
cube looked like by the following rising of the cube through
space in figure 2.

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.  
Figure 3 is an example of a cube which is contained in the sphere,
stereographically projected onto the plane.  Such a picture is 
also called a Schlegel diagram.  

Note that instead of projecting the Cube in the manor shown above, 
the cube could have been rotated so that its faces were not 
parallel and perpendicular to the plane, but rather at different 
angles which would result in a different projection. 

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.  Figure 4 is an 
example of the cube after it has been cut-through and folded-down.

As you may have already surmised, all of the above methods can be 
used to visualize four dimensional polytope 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.  Figures are available 
via anonymous ftp from mathforum.org in the 
/pictures/articles/polytopes directory.  If a Web browser is 
available, the article by Bob Hesse, "Viewing Four-dimensional 
Objects In Three Dimensions," is available at 

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

-Doctor Jodi,  The Math Forum