Site Map

The above page is from www.keypress.com/space   It is a companion page to the text, which comes with a CD of the above games.

FLATLAND by Edwin Abbott  Click on this link for homework questions related to the book.
For an on line version of the book see here.  Here is a sketch to help you see what a Flatlander sees.

POETRY OF THE UNIVERSE By Robert Osserman  Click on this link for homework questions related to the book.

Speaking of the universe, here's a power of ten examination of it.

THE SHAPE OF SPACE By Jeff Weeks  This is an excellent book I have copied extensively and am using in my classroom with 9th grade Honors Geometry students.  The companion video is very helpful.

Over the last ten years John Sullivan has made a number of computer graphics images of mathematical objects connected with his research in optimal geometry.  His soap bubbles look like the 4-D polytopes we've been studying.  I recommend you check out the very strange images he has from the video Optiverse or watch the video yourself:  full version, and lastly here is his video on a Mobius knot.  Wait! Not lastly how about math+art.

Here is a discussion of Euler's famous formula and for its analogue in 4-D just click the link at the bottom of that page.

The Magnificent Mission by Tim Folger
An audacious spacecraft called MAP is about to answer many of the biggest questions about the universe--how old it is, whether it's expanding or contracting, and if it's really shaped like a doughnut.  The NASA site for MAP is here and contains much information that is of help to students and teachers, with numerous links.  One of which is the:

Links for Polyhedra and the Fourth Dimension

Here are some paintings and other art from Tony Robbins.

Bulatov's Polyhedra Collection of many polyhedral sites

Jonathan Bower's cross sections of polychora. 
-
George Olshevsky's polychora page- George coined the term "polychora". In 2 dimensions you have polygons (many edges), in 3 polyhedra (many faces), and 4 dimensional polytopes are called polychora (many rooms).
This is the world’s only website that tabulates all the convex uniform (i.e., Platonic and Archimedean) polychora (that is, four-dimensional polytopes), and until Norman W. Johnson’s book Uniform Polytopes is published by Cambridge University Press, it remains the only place in the world where you can find this information! !WARNING! You should be fairly well acquainted with the convex uniform polyhedra and their symmetry groups, and somewhat well acquainted with the six convex regular polytopes in four-dimensional space and their symmetry groups, if the following material is to make any sense to you. If you aren’t, perhaps you’d still like to visit my polyhedron models website at Polycell’s Home Page:  
Page 2: What Are Polyhedra? This page displays some more polyhedron models and introduces a working definition of a polyhedron. Use the chart of Greek Numerical Prefixes at the bottom of this page to create formalized names for all kinds of polyhedra.
Page 3: The Regular Polyhedra. This page has a photo of my set of nine regular polyhedron models and describes how to build each one. See an atlas of the regular polyhedra and a table of their various numerical properties (dihedral angles, symmetry groups, circumradii, and so forth). With 10,000+ words and more than a dozen pictures, this page takes a bit of time to download.

Photographs of Polyhedra contructed by Fr. Magnus Wenninger OSB:
Set No. 1, Set No. 2, Set No. 3, Set No. 4, Set No. 5, Set No. 6

-
Mark Newbold's Star Polytope Slicer  This neat applet allows you to slice 8 different polytopes and much more.

Site map to many aspects of 4-D spaces.  Particularly excellent graphics and simple explanations.  Highly recommended.

Puzzles with polyhedra and numbers. In this site one can print copies of
polyhedron puzzles (for non-commercial purposes only) and one can read
several mathematical articles on the subject.

Astronomy Picture of the Day

Have a look. All other links can be found here.

Here is an interesting discussion about the four dimensional analogs of the Platonic solids, which fall in the class of polytopes.  What you will find on the pages linked to this one are a series of computer generated projections of the 6 regular polyhedra that exist in 4-space
1. For a 4-D hypercube sliding through 3-D space.
2. Construction of the 4-D Hyper-Dodecahedron 120 cell.
3. 4-D  hypercube applet, 24-cell applet, cross polytop applet, simplex applet, 
4. 3-D morphing polyhedron
5. Tuncated trickery

A program which can display all possible three and four dimensional regular solids.  You can run it and cycle through each of the faces.  Quite javamazing.

If you're up for the challenge, here is a 4-D Rubik's Cube puzzle. It's actually been solved!

The link below contains homework questions and discussions on 4-D geometry, but mostly on hypercubes:Geometry, Relativity and the Fourth Dimension by Rudolf v. B. Rucker

Experience hyperspace with a stereoscopic 4d game in and around the hypercube.

Very cool site with the polytope foldouts!!!!!!

From cut-the-knot.com, a tesseract with sliders and a game on the tesseract.

Visualizing 4-space and from the same site another good discussion of hyperspheres.

Flexible polyhedra and infinite polyhedra

Four MPEGs of 4-D objects folding up.  Part of a course being taught by Davide P. Cervone
Some of his course notes.  This material is absolutely amazing!

Musings by the amazing Cliff Pickover on the fourth dimension.

The above lesson sends you to the following:  The Topological Zoo is designed a resource for mathematicians and educators. It is a visual dictionary of surfaces and other mathematical objects, consisting primarily of movies, still images and interactive pictures. Its contents can be used to complement classroom presentations, research papers and talks. The coolest thing, to me, is the turning inside out of a sphere, without break or tear.  (Truthfully, I don't really understand it.)

Thomas Banchoff's most recent project, in collaboration with Davide Cervone and Jose Francisco Rodriguez, is a re-creation of the exhibit "Surfaces Beyond the Third Dimension" in Portugal .  Click on Portugal to access many beautiful mathematical constructs.  Additionally, I asked Davide Cervone for real world examples of 4-D.  He was generous enough to reply.

Plato's "The Cave" struck me as anothere example where three dimensional objects were reveals as projections of two dimensional objects.  The shadows on the cave are in 2-D, but the figures dancing in the firelight are 3-D.

One of my favorite books is "Coming of Age in the Milkyway Galaxy" by Timothy Ferris.   It gives a nice history of cosmology. Here is an interview with him.

In trying to "see" Mr. Weeks' 3-torus, I remembered a movie "The Cube", in which some people are trapped in a Rubik's Cube like device. 

Here is an interactive poster for Math Awareness Month, April 2000, maintained at the Mathforum. Clicking on Jeff Week's picture will access may fine geometry programs.

A course intended for inclusion as a short segment in a
late middle school or early high school math course,
designed to convey the beauty and fascination of such
mathematical objects as the sphere, the torus, the Klein
bottle, the Mobius band, the real projective plane and
beyond (3-manifolds), etc. Here are some homework questions I wrote to go along with the website.

Strange Surface: Mathematical models
... Alan Bennett, a retired glass-blower, became interested in Klein bottles ... Glass Triple
Klein bottle by Alan Bennett, 1996. 

To buy a Klein bottle, coffee mug, or hat.   See the world's largest glass Klein bottle being made.

Zimaths Math-e-zine - On blowing glass Mobiusly
... about the professional glass-blower from Bedford, England, who ... this picture of the
Klein Bottle: Alan Bennett did it with glass. 

Escher Web Sketch - Wes Hardaker, Gervais Chapuis; Univ. of Lausanne, Switzerland
A Java applet that allows you to draw repeating patterns. With documentation on how to use the applet, a pattern archive, the authors' favorite patterns, a link to download Escher Web Sketch, the program's source code, and interesting crystallographic.