What Are Polyhedra?

Above: Photo of my model of a quasitruncated small stellated dodecahedron: a nonconvex uniform polyhedron with 90 edges whose faces are 12 regular decagrams (green) and 12 regular pentagons (pink), two decagrams and one pentagon meeting at each of 60 vertices (corners or points). The pentagons are quite extensive, being almost the same diameter as the whole polyhedron, which is approximately 37 cm. Its Wythoff symbol is 2 5 | ⁵/₃.

By closed I mean that the set of faces of a polyhedron divides three-dimensional space into two regions: the interior and the exterior of the polyhedron. The interior has a finite volume and is bounded by the faces; the exterior extends to infinity in all directions away from the polyhedron. One cannot join an interior point to an exterior point without passing through a face of the polyhedron (or passing out of three-space!). Convex polyhedra have very well-behaved interiors and exteriors, but star-polyhedra may have exterior regions that are completely surrounded by interior regions.

We owe the word polyhedron to the ancient Greeks. It means “many seats,” the “seats” in this case being the polyhedron’s faces, on one of which it must always sit when placed on a horizontal surface. Traditionally, we name polyhedra by the number of “seats” they have, using the Greek numerical prefix for this number in place of the prefix poly. A polyhedron with four faces is thus called a tetrahedron, the prefix tetra meaning “four.” A list of Greek numerical prefixes appears below on this page.

Most polyhedra are unbelievably complicated, because there are infinitely many large numbers but only finitely many small numbers. So the polyhedra with gazillions of faces far outnumber the polyhedra with, say, 20 faces. For example, the Great Prismosaurus section displayed at the beginning of my four-dimensional uniform polytopes website has only about 700 faces, but each face is diced up by the other faces into innumerable external facelets, making it so complicated that a physical model probably could not be built with existing technology. Fortunately, many interesting polyhedra have a manageable number of faces and can be modeled successfully in paper.

What makes a polyhedron interesting? Constraints and conditions. We might want to know, for example, What are all the different kinds of convex polyhedra that have ten vertices? Or, What are all the different kinds of convex polyhedra that have regular polygons as faces (these are the Johnson polyhedra)? (A polygon is regular if its sides are all of equal length and its interior angles are all the same size. A polyhedron is convex if the line segment connecting any two points of its interior lies entirely inside the polyhedron and doesn’t emerge into the exterior. A star-polyhedron is a nonconvex polyhedron some of whose faces intersect one another.)

The most famous constraints on a polyhedron are that all its faces be congruent regular polygons and that the same arrangement of faces exist around each vertex. Of all the uncountably infinite possible polyhedra, only nine different kinds satisfy these conditions: the regular polyhedra. (Two polyhedra are of the same kind, or similar, if one is just a copy or mirror image of the other, possibly at a different size. In discussing sets of polyhedra, we usually ignore the differences that are due solely to similarity: Once you’ve seen one cube, for example, you’ve seen them all.) Five of the regular polyhedra—specifically, the ones whose faces do not intersect one another—had already been discovered by the time the ancient Greeks turned geometry into the first branch of mathematics. The two with star faces were described by Johannes Kepler early in the 17^th century, and the two with star vertices were described by Louis Poinsot early in the 19^th century. The four regular star-polyhedra are usually called the Kepler-Poinsot polyhedra.

If we relax the constraint on the regular polyhedra to have all the faces necessarily be congruent, but retain the constraints to have the faces be regular polygons and to have the same arrangement of faces at each vertex, the resulting polyhedra are known as uniform polyhedra. The regular polyhedra are all uniform, of course, but the collection also includes the semi-regular polyhedra, whose faces are more than one kind of regular polygon. The 13 convex semi-regular polyhedra were first enumerated by the classical Greek mathematician Archimedes, but the 53 beautiful semi-regular star-polyhedra were completely enumerated only in 1954 by H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller. This is the monograph that introduced the Wythoff symbols for the uniform polyhedra. It was not until 1970 that S. P. Sopov proved that the enumeration was complete. In 1975, J. Skilling added one more figure (the 76^th nonprismatic uniform “polyhedron”) to the roster, but because it has some coincident edges, it is not a uniform polyhedron according to some definitions and was thus excluded from the original enumeration. Many of the polyhedron models depicted at this website are of uniform star-polyhedra.

Above: Photo of my model of a great dodecahedron: a nonconvex regular polyhedron with 30 edges whose faces are 12 regular pentagons, five meeting at each of 12 vertices. This is one of the four Kepler-Poinsot polyhedra. Its diameter is more than 40 cm. I used six different colors for the pentagons: olive, blue, red, orange, cream, and beige, arranged so that wherever two faces intersect they have different colors. (Here the olive is a bit washed out and looks gray, and there’s not much difference between cream and beige. Vagaries of color photography!) Its Wythoff symbol is ⁵/₂ | 2 5. For this illustration, I manually blacked out the background for effect; the model was originally photographed in the same setting as the other models on this website.

Above: Photo of my model of a great dodekicosidodecahedron: a nonconvex uniform polyhedron with 120 edges whose faces are 12 regular decagrams (blue), 12 regular pentagrams (5-pointed stars; white), and 20 equilateral triangles (pink), two decagrams alternating with one triangle and one pentagram at each of 60 vertices. Its diameter is approximately 38 cm; its Wythoff symbol is 3 ⁵/₂ | ⁵/₃.

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Page 3: The Regular Polyhedra. This page has a photo of my set of nine regular polyhedron models and describes how to build each one. See an atlas of the regular polyhedra and a table of their various numerical properties (dihedral angles, symmetry groups, circumradii, and so forth). With 10,000+ words and more than a dozen pictures, this page takes a bit of time to download.

Page 4: Specifications and Prices. Still more photos of models here, along with descriptions of the materials and methods I use in my craft. Find out how much I would charge to custom-build a polyhedron for you.