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Polyhedral Solids

The study of polyhedra is one of those special areas of Mathematics that allows the amateur and expert to work with an equal delight. Many of the solids displayed here are quite attractive, and, as I hope you will discover, the many relationships between the various solids can be quite surprising and delightful.

What is a Polyhedron, anyway? The term Polyhedron has enjoyed several definitions over time and in various contexts. For our purposes let's simply agree that a Polyhedron is a three dimensional shape bounded by polygons.

The above definition is quite vague, allowing all manner of possibilities. To begin our investigation into Polyhedra, then, we will adopt a set of highly restrictive attributes and see if we can discover what solids have these properties in common. Once this is done we can relax these constraints in various ways to see where they lead.

To begin we will require that the faces be all identical regular polygons, that all the vertices be identical, and that the polyhedron be convex. The set of polyhedra that satisfy these constraints is known as the Platonic Solids. They are by far the most well known of the polyhedra.

Various combinations of the three properties just introduced are shown in following table, the name of the class that they define, and the number of polyhedra that have that combination in common. If you click one of the class names you will be taken to an article about it and containing images of its members.

Identical faces?	Identical vertices?	Convex?	Number	Class
Yes	Yes	Yes	5	Platonic Solids
No	Yes	Yes	13	Archimedean Solids
No	Yes	Yes	infinite	Prisms & Antiprisms
Yes	Yes	No	4	Kepler-Poinsot Polyhedra
Yes	No	Yes	8	Deltahedra
No	Yes	No	53	Nonconvex Uniform Polyhedra
No	Yes	No	infinite	Nonconvex Prisms & Antiprisms
No	No	Yes	92	Johnson Solids

Note: All faces are regular polygons

Some additional topics that you will enjoy are discussed in the following articles:

Stellated Polyhedra
Compound Solids
Dual Polyhedra

Please note that nearly every image is a link to a Virtual Reality (VR) model of that polyhedron. If you have a VRML viewer installed you will be able to "touch" the models with your mouse, spin, zoom and generally play with them! If you do not have a VRML viewer I strongly encourage you to invest the small amount of time to install one; it will be worth it! (See below for links to free VRML viewers.)

To be able to view the 3D animated images you must have an appropriate viewer installed; I use Cosmo Player. To find other VRML plug-ins for your machine and software, check out the large collection at the National Institute of Standards and Technology (NIST) VRML Plugin and Browser Detector.

The construction of models of polyhedrons is very instructive and can be quite satisfying. You may wish to consult my Primer on this arcane hobby and art form to learn the techniques I use for producing high quality paper models.

The images presented in these pages were rendered using the freeware raytracing program Persistence Of Vision. POV is highly recommended for anyone interested in high quality computer generated images.

There is a collection of paper polyhedral models that I constructed on display in the Department of Mathematics at the California State University, Chico. These models are dedicated to the memory of my father, Thomas O. Gettys, and to that of my friend Michael J. Dixon.

I welcome your comments & questions!