4D Documents
4D Visualization. Ever wondered how to visualize 4-dimensional space? It's not as difficult as it may sound at first! Read the 4D Visualization page to find out more.
Note: (June 2005) This document is a complete rewrite of the original document and has very little text in common. (Update Jan 2006: the old document is no longer available.)
4D Geometric Objects. A collection of descriptions of some simple 4D objects and their 3D projections.
4D Stories
Here are some 4D-related short stories I wrote.
The Legend of the Pyramid. A story about the 4D simplex, also known as the pentatope, or the 5-cell.
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. Probably the novel that started it all.
“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). This site lists their 29 categories.
The Fourth Dimension. After the initial introduction to 4D, this page gives a very good explanation of how the 120-cell is constructed. The only online resource I've found that adequately explains the structure of the 120-cell in intuitive terms.
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.
This game uses wireframes, so it can be confusing, but after you get used to it, it's not too bad.
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.
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 thesis on 4D ray-tracing. Steve's approach is interesting, because it lets you change your 3D viewpoint independently of your 4D viewpoint, allowing you to examine the same view of the 4D scene from different angles.
Geometry for N-Dimensional Graphics [PDF] by Andrew J. Hanson. Good description of geometric formulas for an arbitrary number of dimensions.
