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Convex Geometry and Polytopes Resources

Standalone Applications

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

Research Software

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.

Recommended Nonfiction and Texts

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.


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

Articles, webpages, videos, printouts...

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

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.

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.

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

Flatland by Edwin A. Abbott
Sphereland by Dionys Burger
Spaceland by Rudy Rucker
The Boy Who Reversed Himself by William Sleator
The Planiverse: Computer Contact With A Two-Dimensional World by A.K. Dewdney
Flatterland, by Ian Stewart---particularly Chapter 17
Skylark of Valeron by E.E. "Doc" Smith (This is pulp fiction from the 1930's and should be read with that fact in mind; the characters in it make lots of conjectures about a 4th spacial dimension, examine the evidence, and revise their conjectures, and sometimes are wrong.)
Alan Mendelsohn: The Boy from Mars, by Daniel Manus Pinkwater (There's a mystery in this novel that can be explained via use of a fourth dimension.)
Factoring Humanity by Robert J. Sawyer (Lots of descriptions of life in a hypercube.)
The Time Machine by H.G. Wells

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.

Java Web Applications

A Four Dimensional Maze

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

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