# 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:

4D Blocks by John McIntosh: an enhanced version of the original 4D maze, this amazing game takes you to an actual 4D world in first-person perspective, and lets you pick up 4D objects and move them around. Comes with a very large set of premade scenes and objects that you can play with, and even sports a train that moves around train tracks in real-time! A very highly recommended way of exploring a truly 4D world.

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.