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Polyhedra

A set of five laminated posters with background notes, from The Beauty of Mathematics poster collection.

The Polyhedra poster set

The posters illustrate three families of polyhedra and two individual more complex examples:

• The Platonic Solids
• Truncation
• The Kepler-Poinsot Solids
• The Compound of Five Cubes
• The Great Dirhombicosidodecahedron

Each poster describes the polyhedra and also gives brief mathematical or historical information.

See below for more information about the background notes as well as samples from the notes. Larger images of the posters may be seen by going to the Poster Gallery.

• Poster Gallery
• Order
• Other poster sets:
• Curves from Circles
• Tilings
• Curves from Lines
Fractals

“The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.”—Aristotle

The Polyhedra background notes

A descriptive booklet accompanies the posters, explaining the mathematics behind the images in the posters and related topics.

The four pages of background notes give more information about the polyhedra on the five posters and about polyhedra in general. The topics covered are:

• Types of Polyhedra
• Prisms and Antiprisms
• Uniform Polyhedra
• Stellation
• Description and properties of the polyhedra on each poster
• Glossary
• References

The notes help interested students and teachers explore further, and provide ideas for activities connected to the theme of the posters.

Extracts from the background notes

Types of Polyhedra

A polyhedron is a solid bounded by plane polygons, called the faces. Only polyhedra with regular faces are considered here, but regular star-polygons, or polygrams, like that shown alongside, are included as possible faces. The arrangement of faces around a vertex is referred to as the vertex configuration.

A polyhedron is either convex or non-convex. For example, there are nine regular polyhedra, the five convex Platonic solids and the four non-convex Kepler-Poinsot solids, shown on The Beauty of Mathematics posters.

Uniform Polyhedra

A uniform polyhedron has regular faces and identical vertices. Excluding the prisms and antiprisms, there are 75 uniform polyhedra, 18 of them convex and 57 non-convex. Apart from prisms and antiprisms, the convex uniform polyhedra are the five Platonic solids and the thirteen Archimedean (or semi-regular) solids. Sixteen are illustrated on The Beauty of Mathematics poster Truncation; the other two are the tetrahedron, and the truncated tetrahedron shown alongside.

Stellation

Apart from truncation, another method of creating one polyhedron from another is stellation. To stellate a polyhedron, the facial planes are extended in a symmetrical way until they intersect. There are no stellations of the tetrahedron or cube—their planar faces never intersect again however far they are extended. The only stellation of the octahedron is the stella octangula, illustrated on The Beauty of Mathematics poster The Platonic Solids. The diagram alongside shows how the faces of the octahedron are extended.

© A K Jobbings 2004-2008