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

The Five Platonic Solids

Terahedron

Octahedron

Hexahedron, or Cube

Dodecahedron

Icosahedron

The most ancient polyhedra are the set of five known as the Platonic solids. They derive their name from the great Greek philosopher Plato, who discovered them independently about 400 B.C., though they were probably known before Plato. The ancient Egyptians knew four of them: the tetrahedron, octahedron, and cube are found in their architectural design, and Egyptian icosahedral dice are to be found in an exhibit in the British Museum. According to Heath, the Etruscans were acquainted with the dodecahedron before 500 B.C.¹ All five were studied by the early Pythagoreans before the time of Plato and Euclid. It is in the Elements of Euclid, however, that we find the most extensive treatment of the geometry of these five solids.

Today models of these solids, usually in plastic, are featured in the catalogues of scientific and educational supply houses. But models in heavy paper are so easily made and so useful as a project for students that it is well worth the effort to make a set. The nets, or patterns, for making these models are given in many geometry textbooks. It will be found that the models are even more attractive when they are made with facial polygons of various colors. (Suggested color patterns are set out in Figures 1, 2, 3, 4, and 5.)

For easiest construction and sharpest edges, the material to use is heavy paper with a somewhat hard finish—the type used for file cards. It can be bought in larger sheets, in colors, under the name “colored tag.” Pastels are very suitable, and they are a good alternative to deeper colors. In the method of construction suggested here you need only one triangle, square, or pentagon as a net. Three, four, or five sheets of colored paper may be stapled together and the net placed on top of them. Then, using a sharp needle and pricking through all the sheets at one time, make a hole in the paper at each vertex of the net, which is held as a guide or template. In this way exact copies of each part are quickly obtained. Next give the paper an initial trimming with scissors, with all the sheets still held firmly together by the staples. You must be careful to provide about a quarter-inch margin all around to be used for flaps or tabs to cement the parts together. After this it will be best to treat each part individually. Experience will soon teach you that the accuracy of your completed model is directly proportional to the care you have lavished on each individual part. With a sharp point, such as that of a geometry compass, you must now score the paper, using a straightedge or set square as a guide to connect the needle holes with lines. (Pencil lines are not needed, since the process of scoring sufficiently outlines the shape of each part.) More accurate trimming is next to be done. (For suggest-ions on how to do this, see Figures 6, 7, and 8.) The scored lines then make folding of the tabs a simple and accurate operation.

Fig. 6

Fig. 7

Fig. 8

A good household cement provides the best adhesive, since it is strong and quick-drying. A few wooden clothespins of the coiledspring variety that have been turned inside out make excellent clamps. When these are used, the cementing can proceed rapidly and the clamps can be moved from one part to the next in almost a matter of minutes as each part is successively cut and trimmed. The last part will not give you too much trouble if you first cement one edge and let it set firmly, then proceed to put cement on the other edges and close down the last polygon as you would close the lid on a box. A needle or compass point makes a good instrument to maneuver the last edges into accurate position. Deft fingers and a little practice will do the rest.

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