Other Online Resources on 4D
Discussion forums:
Garrett Jones' tetraspace page. An excellent source of information on the subject of 4D space. It also has a 4D discussion forum for discussing 4D-related topics.
Stories:
Edwin A. Abbott's Flatland: A romance of many dimensions. A novel that gives wonderful insight into dimensional analogy.
“And He Built a Crooked House” by Robert A. Heinlein. An entertaining story about a house in the shape of a hypercube.
4D polytopes:
Uniform Polytopes in Four Dimensions, by George Olshevsky. This site has an immense catalogue of uniform 4D polytopes and their various properties.
Uniform Polychora by Jonathan Bowers, who discovered a very large number of uniform 4D polytopes (most of which are non-convex).
The Fourth Dimension. After the initial introduction to 4D, this page gives a very good explanation of how the 120-cell is constructed.
Games & Tools:
An interactive game set in a 4D maze written by John McIntosh. Requires Java. Although this game renders using lines (and so can be rather confusing to look at), it is nevertheless a wonderful opportunity to actually explore a truly 4D world. Once you get used to it, it gives you a very good idea of the various degrees of freedom available to someone living in 4D.
There are a lot of game options available to configure the maze: you can define each dimension individually, control how frequent branches and turns are, etc.. A truly commendable way to explore 4D.
4D building blocks! Written by Henryk Trappmann. A really awesome way of learning 4D, by assembling 4D blocks together. If you can solve all 6 scenarios in the game, you'd have a pretty good grasp on 4D geometry.
Magic Cube 4D: play with a 4D Rubik's Cube! (As if the 3D one isn't hard enough already!) If you can master this game, you will have a very good grasp on 4D rotations.
Magic 120-cell: as if the 4D Rubik's Cube isn't enough, this one is the 4D analogue of the Megaminx puzzle. This puzzle has 120 cells, each of which is a dodecahedron divided into 63 movable pieces, for a whopping total of 2641 pieces with 7560 colored hyper-stickers. If you think you know 4D, prove it by solving this one!
Already mastered 4D? Craving for a challenge? Try Magic Cube 5D. See if you can get into the Hall of Insanity!
Still not satisfied? 5D not enough for you? For a real challenge, try Magic Cube 7D. Guaranteed to blow your puny little mind into 78110 pieces!
Nklein software's multi-dimensional ray-tracer. This one lets you do not just 4D ray-tracing, but ray-tracing in any number of dimensions.
Papers and other technical resources:
Steve Hollasch's M.Sc. thesis on 4D ray-tracing. A very good overview of the issues involved in visualizing 4D space.
Geometry for N-Dimensional Graphics [PDF] by Andrew J. Hanson. Good description of geometric formulas for an arbitrary number of dimensions.
The Polygloss, Wendy Krieger's glossary of higher-dimensional terms designed to avoid the confusion caused by the inconsistent application and generalization of our 3D-centric terminology to higher dimensions. This is the place to go if you want very precise terminology for describing various aspects of higher-dimensional spaces.
