KaleidoTile is just about the coolest polyhedron exploration program out there. It used to have a hilarious voice component, but no more.

JavaGami is a standalone Java program that lets you create polyhedra, shear/truncate/cap them, put colors and textures on them, and then outputs a flat pattern that can be printed out and put together.

Hypersolids (Regular 3- and 4-polytopes) by Richard Koch (also author of the excellent TeXShop)

The Fourth Dimension is an iPhone/iPad app that lets you rotate a hypercube through 3-space and through 4-space. Excellent.

Hipercubo (free version) is an iPhone game with a colored hypercube that you have to rotate to match given colors.

Tesseract is an iPhone/iPad puzzle/game in which you must unscramble a coloring of the edges of a hypercube. Pretty cool.

Move & Turn (MacOS only) is a polyhedron explorer and a game and... just go look at it.

SeeThrough Studios has a
hilarious embedded YouTube video with a teaser for their game version of
*Flatland*. The game is freely downloadable and
runs on Windows and MacOS. If you are a *Flatland* fan, this is worth
checking out.

Magic Cube 4D is a Java 4D version of Rubik's Cube

Polymake does computations on and visualizations of polytopes. (It requires visualization software such as JavaView or jReality).

Sage is a multipurpose open source system for mathematical software. You can run it over the web or install your own copy. It's constantly under development; I wrote a tutorial for polytopes in Sage. Beyond that, the documentation pages I find most helpful for polytopes are constructor, library, base, and plot. Sage includes lots of other packages (such as lrs and cdd listed below); and some other standalone software can be called from within Sage, which makes it very flexible.

Groups&Graphs (MacOS only) does convex hulls and duality for polyhedra, even though it's primarily software for graph theory and group theory of graphs.

Qhull is a collection of efficient algorithms for computing convex hulls and the like.(I used it to do some computations for my dissertation, in fact.)

lrs does vertex enumeration and convex hulls.

Komei Fukuda's cdd also
(I think) does vertex enumeration and convex hulls.

ZerOne does vertex enumeration for 0-1 polytopes. Yup.

PORTA is a collection of polytope analysis algorithms.

*The Annotated Flatland: A Romance of Many Dimensions* by Ian
Stewart

*A Conversation with Rudy
Rucker*, by Tatiana Shubin (password-protected)

*The Recognition of the
Fourth Dimension*, by C. H. Hinton (password-protected)

*Polyhedra*, by Peter Cromwell

*Beyond the Third Dimension* by Thomas Banchoff

Here's an anti-recommendation: *The
Fourth Dimension Simply Explained*, an edited collection of essays
submitted in a 1910 competition. It's interesting and boring all at the same
time.

**Texts**

*An Introduction to Convex Polytopes*, by Arne Brondsted

*Convex Polytopes*, by Branko Grünbaum

*Lectures on Polytopes*, by Günter Ziegler

*A Course in Convexity*, by Alexander Barvinok

*Computing the Continuous
Discretely*, by Matthias Beck and Sinai Robins

Feeling Your Way Around in Higher Dimensions is a very useful article---it actually tells you how to do an affine transformation and how to find a vertex of a polytope.

Polygons to print out, cut out, and paste into polyhedra: small triangles, big triangles, squares, hexagons

Assembly Required addresses the question of when a polytope is uniquely determined by its skeleton.

Komei Fukuda's really cool FAQ about Polyhedral Computation

Five Open Problems Regarding Convex Polytopes (posted May 2008)

A listing of the past year's arXiv papers in category 52B (Polytopes and Polyhedra), reverse chronological order

How to reasonably compute the genus of Infinite Regular Polyhedra (Coxeter-Petrie polyhedra)

Technical information on and pictures of the Szilassi polyhedron, including a net.

Five Space-Filling Polyhedra are described (and nets are given) in this article reproduced from Math Gazette.

Henry Segerman shows hyperbolic honeycombs

Short films by Gian Marco Todesco: 120-Cell, 120-Cell 2,, polychora movie.

Nesting of platonic solids by Dugan Hammock

Flexible polyhedron; Multistable polyhedron (the net is incorrect, beware); Shaky polyhedron on Mathworld (I haven't tried making this one)

Here's a movie about how to unfold and refold a cube.

George Hart's Encyclopedia of Polyhedra

David Eppstein's Geometry Junkyard page on Polyhedra and Polytopes

The On-line Encyclopedia of Integer Sequences allows you to put in your own sequence for testing.

Hypercube site with text and images that help with visualization

Speculations on the 4th dimension

Nova's Elegant Universe mathematics primer on a few dimensions

Visualizing the Hypersphere another primer

Viewing Four-dimensional Objects In Three Dimensions yet another primer

Bogglers, a
*Discover Magazine* article with puzzles by Scott Kim

Uniform Polytopes in Four Dimensions webpage on 3D and 4D solids (on the Wayback Machine) with tons of mathematics

An Introduction to the Vocabulary of Dimension suggests some excellent investigations

Descriptive text about Platonic Solids in All Dimensions from John Baez

David Fontaine's details about Regular 4-polytopes

Math Expands part of the Math Awareness Month 2000 website on dimensionality

Essays on Dimension another part of the Math Awareness Month 2000 website on dimensionality

NOVA: The Elegant
Universe Here you can watch the Nova episode on string theory. There are
three hours of video here; the parts you want are *Hour 2, Chapter 7
(Multiple Dimensions)*, and *Hour 3, Chapter 4 (Parallel
Universes)*, and *Hour 3, Chapter 6 (Riddle of the Big Bang)*.

Some Notes on the Fourth Dimension: from a course by Davide Cervone. Lots of excellent movies and animations.

Reviews and Bibliographic Data at the MathFiction Higher/Lower Dimensions page

**Books**

**Short Stories**

*The Plattner
Story* by H.G. Wells

*The Mystery of Element
117*, by Milton K. Smith (password-protected)

*The Indian Rope Trick Explained,*
by Rudy Rucker (password-protected)

*Message Found in a Copy of
Flatland*, by Rudy Rucker (password-protected)

*Left or Right?* by Martin Gardner
(password-protected)

*The Church of the Fourth
Dimension*, by Martin Gardner (password-protected)

*Technical Error*, by Arthur
C. Clarke (password-protected)

*Tangents*, by Greg Bear
(password-protected)

*The Next Dimension: A Mathematical
Play in Five Dialogs*, by Vladimir Karapetoff (password-protected)

*The Monster from
Nowhere*, by Nelson Bond (password-protected)

*The Captured
Cross-Section*, by Miles J. Breuer (password-protected)

*The Appendix and the
Spectacles*, by Miles J. Breuer (password-protected)

*The Fifth-Dimension Catapult* by Murray Leinster... This
novella/long-short-story appears in *Science Fiction of the 30's*, ed.
by Damon Knight and contains interesting ideas of how we could see/access a 4th
and 5th spacial dimension.

4-D Polytope Viewer java applet via which you can rotate 15 different objects

Regular 4d Polytope Foldouts (requires Java) are freely rotatable and like unfolding a paper cube, except... Irregular 4d polytope foldouts, too.

Lots of 4+D Objects mindblowing java applets; beware that not all of them work.

M. Newbold's java applets for lots and lots of 4D things... many need red/cyan 3D glasses. (sm can't remember where hers are.)

Bob Allanson's polyhedron-modification applet

*last updated June 2016*