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As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, and can construct most of the other isohedra, deltahedra, and Johnson solids from memory. The smaller prisms, antiprisms, and trapezohedra are of course trivial. However, I often forget the precise arrangement of faces and vertices for some of the Johnson solids and most of the Catalan solids. Thus, the question that I pose is this:

Where can I find the most complete, robustly indexed, and searchable database of polyhedra?

I would like to use such a database to answer, in short order, questions of the following nature:

Which solid is comprised of exactly eight hexagons and six squares? Which solids are comprised of less than 10 triangles, eight squares, and six hexagons? How many solids can be constructed with exactly 24 edges? What solid with 12 vertices has the most edges (or faces)? etc...

I imagine that such a database does not exist and I am going to be forced to create one, so answers suggesting features for such a database (likely to be web-based) are welcome as well.

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The Wikipedia polyhedra entry has a list of many on-line databases and software packages. But without some kind of restrictions on the types of polyhedra you're interested in a useful database might be overly optimistic, as the numbers grow far too rapidly. – Ryan Budney Jan 14 '10 at 15:56
I am mostly interested in the strictly classified polyhedra. The finite sets (and some simpler subset of the infinite sets) that I named, in particular. There aren't more than a few hundred of those. – Sparr Jan 14 '10 at 16:15

There is a collection of models in Electronic Geometry Models. But they are less models a hobbyist would be likely to want to build and more objects with properties of interest to mathematicians; many of them have more than three dimensions, for instance.

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Try the netlib polyhedra database. It does not seem to be listed in Wikipedia polyhedra page.

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There is George Hart's online Encyclopedia of Polyhedra, which is not a database but is very likely something you will want to consult.

Also not a database, but something you should surely know about if you intend to create one, is the software program called Great Stella. There is a free version and a commercial version (use google). Originally intended to allow to you to create stellations (as for example "The Fifty-Nine Icosahedra" of Coxeter et al) for any polyhedron, and visualize, generate nets, duals etc, this program is now much larger containing information about and the ability to manipulate literally hundreds of polyhedra including Platonic, Archimedean, Kepler-Poinsot, convex prisms and antiprisms, Johnson solids, "near misses (of Johnson solids)", Stewart toroids, compounds, geodesic domes, all uniform polyhedra, zonohedra, Waterman polyhedra, Bruckner polyhedra, rectangular isohedra, duals, nets, facetings, stellations, and much more.

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If you are interested in abstract polyhedra you can consult the atlas of small regular polytopes put together by Michael Hartley, however some of the links seem to be broken right now (I'll follow up with him, so you don't have to).

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The site appears to be working again. – Gordon Williams Mar 31 '10 at 22:16

Coxeter's book Regular Polytopes comes to mind, although this fails to be what you're looking for on several counts:

  1. Much of the book is about higher-dimensional polytopes; it sounds like you're specifically interested in the three-dimensional case.
  2. You don't necessarily want regular polyhedra.
  3. It's a book, and not in any way searchable.

Still, it's not like polyhedra stop existing; if you do try to build such a database, Coxeter's book would be a useful source.

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If you read it sufficiently many times, you end up finding things pretty fast. I've tried :) – Mariano Suárez-Alvarez Jan 14 '10 at 16:46
That's true with any book. – Michael Lugo Jan 14 '10 at 17:04

There are many web sites where the information you want is in essence available.

One place in particular is:

This site has nets of each of the Johnson Solids and it has a table which indicates the number of faces of the different kinds that are present. Also, one can click on each of the solids which takes one to a representation of the solid which can be rotated so that one can see it from "all sides."

Other pages at Mathworld do similar things for other important collections of 3-dimensional solids.

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