Polyhedron Model by Martin Berman, Regular Tetrahedron

National Museum of American History

About the work

This painted and varnished paper object is the first in a series of models of convex polyhedra with regular faces constructed by Martin Berman. The faces are four regular triangles.

In 1970 Berman (1938-1984), a physicist at the Pittsburgh Energy Technology Center, constructed a set of models of regular-faced convex polyhedra. A polyhedron is said to be uniform if its faces are regular and its vertices are all alike (so that it has the same arrangement of polygons at each vertex). A polyhedron (uniform or not) is convex if a line segment joining any two of its points lies entirely on or inside it.

A polygon (convex or not) is regular if it is uniform and its faces are all alike. The regular convex polyhedra are the five Platonic solids, which have been known since classical Greece. The ancient Greek mathematician Euclid proved in his Elements of Geometry that there are only five Platonic solids. These are the regular tetrahedron (four sides that are equilateral triangles), the cube (six sides that are squares), the regular octahedron (eight sides that are equilateral triangles), the regular dodecahedron (twelve sides that are regular pentagons), and the regular icosahedron (twenty sides that are equilateral triangles).

The term "semiregular" is used to describe polyhedra that are uniform but not regular. The semiregular convex polyhedra include thirteen solids associated with another ancient mathematician, Archimedes. He lived after Euclid and worked in Syracuse on the Mediterranean island of Sicily. These objects are called Archimedean solids. There also are an infinite number of semiregular prisms. These have like regular polygons on the top and bottom and straight lines joining the vertices of these to form the square sides. A second infinite group of semiregular solids are called antiprisms. These also have like polygons for top and bottom, but twisted so that each vertex of one polygon is joined to two vertices of the other to form an equilateral triangle.

Besides the regular and semiregular solids, there are just ninety-two other convex polyhedra with regular faces. In 1966 the American mathematician Norman W. Johnson, a student of H. S. M. Coxeter at the University of Toronto in Canada, enumerated them. These polyhedra are sometimes called the Johnson solids. In 1969 the Russian Viktor A. Zalgaller offered a computer-based computational proof that Johnson had completed the enumeration of convex polyhedra with regular faces.

Berman made paper models of the Platonic solids, the Archimedean solids, the Johnson solids, a prism with a triangular base, and an antiprism with a square base, for a total of 112 models. He published photographs of these and diagrams for making them in 1971, identifying the models with the same names as those used by Johnson. Berman gave the models to the Smithsonian in 1978.


Accession File

Martin Berman, "Regular-faced Convex Polyhedra," Journal of the Franklin Institute, 291, 1971, pp. 329-352. Includes illustrations of models and of nets for making them.

Norman W. Johnson, "Convex Polyhedra with Regular Faces," Canadian Journal of Mathematics, 18, 1966, p. 169-200;

Viktor A. Zalgaller, “Convex Polyhedra with Regular Faces,” in Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad, vol. 2, English translation: Consultants Bureau, New York, 1969.

Online discussion of Johnson solids on Wikipedia and on Wolfram MathWorld (both accessed November 11, 2015).

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