# Uniform Polyhedra

People have always been fascinated by symmetry and symmetric objects. Uniform polyhedra are a series of well-defined, highly symmetric solids. Some of them were discovered in ancient times, but the last addition to the collection dates from 1975.

Uniform Polyhedra are polyhedra with the following properties:

• all faces are regular polygons (which may include star polygons like pentagrams)
• all vertices are equivalent

A few special cases:

• the five regular or Platonic solids (all faces identical convex polygons)
• the thirteen semi-regular or Archimedean solids (all faces convex polygons, but not all identical)
• the four Kepler-Poinsot solids (non-convex, but all faces identical polygons)
• an infinite number of prisms and antiprisms

Excluding the prisms, there are 76 uniform polyhedra. Three of them have tetrahedral symmetry (Td), eighteen have octahedral symmetry (17 Oh, 1 O), the remaining fifty-five have icosahedral symmetry (47 Ih, 8 I). I have made paper models of all of them.

Low-resolution photos of the individual solids can be found by following the links below (loading may be slow, though!).

Descriptions of how to make these models can be found in the book by Wenninger (see below). Unfortunately, the drawings in that book are not completely accurate and will not work for the most complex polyhedra. I wrote a program to draw more accurate intersections, and recently resuscitated it as a Python incarnation: if you are interested, download from here.

There are a number of generalizations of the concept of uniform polyhedra, the most important ones being space-filling tesselations and extension to different numbers of dimensions.

A web-site about uniform polyhedra: http://www.georgehart.com/virtual-polyhedra/uniform-info.html.

Some literature relevant to uniform polyhedra:

Alicia Boole Stott, "On Certain Series of Sections of the Regular Four-dimensional Hypersolids", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam (1e sectie) 1900, 7, 1-24

Alicia Boole Stott, "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam (1e sectie) 1910, 11, 3-21

H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, "Uniform Polyhedra", Phil. Trans. Royal. Soc. London A 1953, 246, 401

Magnus J. Wenninger, "Polyhedron Models", Cambridge University Press, 1971

H.S.M. Coxeter, "Regular Polytopes", 3rd edition, Dover Publications, New York, 1973

J. Skilling, "The Complete Set of Uniform Polyhedra", Phil. Trans. Royal. Soc. London A 1975, 278, 111-135

Roman E. Maeder, "Uniform Polyhedra", The Mathematica Journal 1983, 3, 48-57

Zvi Har'El, "Uniform Solution for Uniform Polyhedra", Geometriae Dedicata 1993, 47, 57-110

Peter R. Cromwell, "Kepler's Work on Polyhedra", The Mathematical Intelligencer 1995, 17, 23-33