Contents Introduction The Five Platonic Solids The Thirteen Archimedean Solids Prisms, Antiprisms, and Other Polyhedra The Four Kepler-Poinsot Solids Other Stellations or Compounds Some Other Uniform Polyhedra Conclusion Notes Bibliography Photographs: Stanley Toogood’s International Film Productions, Nassau, Bahamas

Polyhedron Models for the Classroom by Magnus J. Wenninger

Some Other Uniform Polyhedra

A Faceted Octahedron

Faceted Icosidodecahedra

Faceted Rhombicuboctahedra

Faceted Cuboctahedra

Faceted Rhombicosidodecahedra

dodecadodecahedron

A Truncated Great Dodecahedron

A Faceted Cube

A Quasitruncated Cube

A Quasitruncated Octahedron

Great Icosidodecahedra

Note:- A faceted polyhedron is on that may be derived from a convex polyhedron by the removal of solid pieces. A stellated polyhedron is one that may be derived from a convex polyhedron by the addition of solid pieces. Truncation is also a removal process. Quasitruncation is a combination of removal and addition.

So far this monograph has described the five Platonic solids, the thirteen Archimedeans, the infinite set of prisms and antiprisms, a few miscellaneous irregular polyhedra, the four Kepler-Poinsot solids, and some compounds arising from the stellation process. All of these except the irregulars and the compounds are classified as uniform polyhedra. (A polyhedron is uniform when all its faces are regular polygons [including regular star polygons] and all its vertices are alike.) The question might now well be asked, Are there any more uniform polyhedra? The answer is, Yes, there are more—at least fifty-three others. Coxeter has published a brief history of all these solids and a thorough investigation of the mathematics involved in their classification and discovery. In this book, entitled Uniform Polyhedra, he mentions the fact that Badoureau in 1881 made a systematic study of all thirteen Archimedean solids on the basis of their planes of symmetry and their vertices to discover polygons penetrating the interior parts of each solid. By this method he discovered thirty-seven uniform polyhedra not known before that time.⁵ Brueckner published a classic work on polyhedra in 1900 in which many were beautifully illustrated in photographed models and in drawings.⁶ About 1930 Coxeter himself and J. C. P. Miller discovered twelve other uniform polyhedra by investigating the Schwarz triangles on a spherical surface. Thus the total enumeration of uniform polyhedra given by Coxeter in 1954 comes to seventy-five, but (a most striking fact) he admits that a rigorous proof has still to be given that this enumeration is complete.⁷

As for the construction of models for these polyhedra, this monograph can present only a selection of some of the more simple ones, since some are actually most intricate. Coxeter gives sketch drawings, done by J. C. P. Miller, of all seventy-five and photographs of models in wire, made by M. S. Longuet-Higgins.⁸ As for the most intricate of these, one may well suspect that only the original discoverer has ever had patience, zeal, or perseverance enough to make a model. The drawings alone are amazing enough.

For the purposes of this monograph the following fifteen models are described, as a good representative set of the simpler ones. When you have made them, you will have a good idea of why they are classified as uniform polyhedra. It should not be necessary to give a detailed description of the construction of these models. If you have been successful so far, you will without doubt be able to proceed on your own with a minimum of direction. The drawings shown in Figures 36 to 50 reveal the facial planes and the lines of intersection of planes, as well as the nets required for each model. If you compare each of these with the photograph of its model, you wifi understand the relationship of parts.

One special hint about construction techniques may be in order here. It is this: Where a model has parts that have only edges in conmion, a good way to construct it is by using a tongue-and-slot arrangement. This technique can frequently be used in the models now being described. For example, in Figure 36 you may begin by cementing the hypotenuse of each of the isosceles right triangles to an edge of the equilateral triangle. Then, instead of cementing these isosceles triangles to form a triangular pyramid with all tabs inside, turn one or two sets of tabs out, leaving the other set or sets turned in but not cemented. The tabs turned out may be cemented to form a tongue, which will later be inserted into the slot of another pyramidal part and cemented there. A little experimenting will soon make this technique clear. You will have to use your own judgment about what tabs should serve as tongue or slot. This technique is used in all the figures from 36 through 44.

As for color arrangements, no detailed description should be necessary. You will find the most satisfactory results are obtained if you make facial planes the same color. Try to follow the map-coloring principle. Parallel planes may always be of the same color. Thus in Figure 39 the octagons and squares that are parallel may be of the same color. In Figure 41 the pentagons and decagons that are parallel may be of the same color, since each decagon is on a plane below and parallel to a pentagon. In Figure 42 the pentagon is part of the same facial plane as the decagon, so it is of the same color. In Figure 45 each star is on a plane parallel to the pentagon below it, so a six-color arrangement works out very nicely. Figure 46 is easily recognizable as a truncated form of the great dodecahedron and may thus have the same color pattern. Each star can be of the same color as the decagon below it. In Figure 47 three colors may be used for the octagonal stars, and the same three colors will serve for the squares that lie below them. Then two other colors may be used for the triangles. These color arrangements are well worth the trouble it takes to get them done correctly, because they help so much to bring out the relationship between the intersecting regular and star polygons. Figure 48 is somewhat like Figure 47, but each small square is now coplanar with an octagonal star and thus should be of the same color as that star. In Figure 49 there are eight triangles and their related hexagons, which are actually coplanar. Then there are six octagons, each parallel pair being perpendicular to the other two parallel pairs. Finally the octagonal star is set high up over these octagonal planes. Two colors will serve for the triangle-hexagon set, so they can alternate; three other colors will then fill the needs of the rest of the planes, with parallel planes of the same color. Finally, Figure 50 is best made with a five-color arrangement for the triangles. It will be found that the color arrangement given above for Figure 33 will serve the purpose here. In fact, the construction can be easily done by making the pentahedral dimples each with a pentagon of a sixth color at the bottom. These dimples can then be joined by cementing the star arms to the edges of these dimples. A single net of three star arms joined at their bases will serve here, because all the star planes are of the same color, whereas each triangular plane has its own color. (See the following pages for Figures 36-50.)

Polyhedron Models for the Classroom by Magnus J. Wenninger			Reprinted with permission from Polyhedron Models, copyright 1966 by the National Council of Teachers of Mathematics. All rights reserved.
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		Ragnar Torfason 2006 June 3