Model of a Dual of an Archimedean Solid by Richard P. Baker, Baker #547 IX

Description:

This metal model painted white was constructed by Richard P. Baker. A mathematics professor at the University of Iowa, Baker believed that models were essential for instruction in many parts of mathematics and physics. Over one hundred of his models are in the NMAH collections.

A mark in pencil on the side of the model reads: 314940 131 (/) # 547 (/) IX. The model fits in a cardboard box with a label pasted on the lid that reads: No. 547 (/) SPACE DUALS OF (/) ARCHIMEDEAN HALF REGULAR (/) BODIES No. 465, IX, p. 20. A larger label, pasted on the side of the box, reads: Mathematical Models (/) Made by (/) R. P. BAKER (/) No. 547 POLYHEDRON (/) ALL FACES CONGRUENT (/) Dual of Archimedes’ (/) Half-regular body (/) IX (3,3,5,5.).

Mathematicians have known since ancient times that there are only five regular convex polyhedra. The faces such a solid are identical regular polygons and the vertices are all alike (each vertex has the same arrangement of polygons). The Hellenistic mathematician Archimedes showed that there are thirteen other polyhedra that have identical vertices, sides of the same length, and faces that are not all the same regular polygons. These came to be called the semi-regular Archimedean solids. There also are two infinite series of semi-regular polyhedra, the prisms (with a regular polygon on the top, the same regular polyhedron on the bottom, and squares around the sides) and the antiprisms (with a regular polygon with an even number of sides on the top, the same polygon on the bottom, and equilateral triangles around the sides). For examples of these polyhedra, made by Michael Berman, see 1978.1065.006 through 1978.1065.20.

In the mid-nineteenth century, the mathematician Eugène Catalan described another set of polyhedra which have identical faces and form regular polygons at a vertex when that vertex is truncated. However, the faces are not regular polygons and the vertices are not identical. These thirteen polyhedra are called duals of Archimedean solids or Catalan solids.

Baker published a catalog of his models in 1931, and included as #465, numbers I through XV, thirteen Archimedean solids plus two families of prisms and antiprisms. Examples of these do not survive at the Smithsonian. He also made models he called “space duals of Archimedean half-regular bodies,” and might now be described as Catalan solids. These had a general number 547 in Baker’s scheme, and given index numbers I through XV. This is the ninth of them (e.g. IX). Baker’s 1931 catalog includes models assigned numbers as high as 542, suggesting that this model and the other Archimedean duals date from slightly after the catalog.

The thirty faces of the model are equal rhombuses; the polyhedron is a rhombic triacontahedron. Three faces come together at twenty vertices and five come together at twelve vertices. Truncating the vertices would produce twenty equilateral triangles and twelve pentagons.

References:

H. M. Cundy and A. P. Rollet, Mathematical Models, Oxford: The Clarendon Press, 1961.

R. P. Baker, Mathematical Models, Iowa City, Iowa, 1931, p. 20.