# Other Online Resources on 4D

## Discussion forums:

• The Higher Dimensions Forum, and its associated wiki. Originally inherited from Garrett Jones' now-defunct tetraspace page, it has grown over the years into an excellent source of information on the subject of higher-dimensional spaces and speculation related to universes with more than 3 spatial dimensions.

## 4D polytopes:

• Dr. Richard Klitzing's polytopes and incidence matrices. Contains a wealth of technical details about higher-dimensional polytopes (not limited to 4D) and related geometric concepts, including exhaustive information on a large number of 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.

Last updated 08 Jul 2018.