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 Thirteen Archimedean Solids

Suggested colour patterns for the Archimedean Solids
1. Truncated tetrahedron:
	4 hexagons 0 R B Y 4 triangles G R B Y
2. Cuboctahedron:
	6 squares Y B R Y B R 8 triangles G
3. Truncated cube:
	6 octagons G B R G B R 8 triangles Y
4. Truncated octahedron:
	6 squares G 8 hexagons R Y O B R Y O B
5. Rhombicuboctahedron:
	6 squares Y 12 squares R 8 triangles B
6. Great rhombicuboctahedron:
	6 octagons R Y B R Y B 12 squares G 8 hexagons 0
7. Snub cube:
	6 squares Y R B Y R B 8 triangles 0 8 triangles B 8 triangles R 8 triangles G
8. Snub dodecahedron:
	12 pentagons 0 15 triangles B 15 triangles R 15 triangles Y 15 triangles G 20 triangles 0
9. Icosidodecahedron:
	3 pentagons R 3 pentagons B 3 pentagons 0 3 pentagons G 20 triangles Y
10. Truncated dodecahedron:
	3 decagons Y 3 decagons R 3 decagons G 3 decagons B 20 triangles 0
11. Truncated icosahedron:
	8 hexagons R 6 hexagons G 6 hexagons Y 12 pentagons O
12. Rhombicosidodecahedron:
	12 pentagons R 30 squares B 20 triangles Y
13. Great rhombicosidodecahedron:
	12 decagons Y 20 hexagons R 30 squares B

Colour Code R = rose, Y = yellow, B = blue, O = orange, G = green, W = white

Once a set of the five Platonic solids has been made, the next project will certainly be to make a set of the thirteen Archimedean solids. These too have an ancient history. Plato is said to have known at least one of them, the cuboctahedron. Archimedes wrote about the entire set, though his book on them is lost. Kepler is the first of the moderns to have treated these solids in a systematic way. He was also the first to observe that two infinite sets of polyhedra, the set of prisms and the set of antiprisms, have something in common with the thirteen Archimedeans, namely, membership in the set known as the semiregular polyhedra.² (A semiregular polyhedron is one that admits a variety of polygons as faces, provided that they are all regular and that all the vertices are the same.)

As in the case of the Platonic solids, so too in that of the Archimedeans the beauty of the set is greatly enhanced by suitable color arrangements for the faces/ Since it is evident that many different color arrangements are possible, you may find it interesting to work out a suitable arrangement for yourself. The general principle is to work for some kind of symmetry and to avoid having adjacent faces with the same color. This may remind you of the map-coloring problem. The fact is that a polyhedron surface is a map, and as such is studied in the branch of mathematics known as topology. In making these models, however, you need not enter into any deep mathematical analysis to get what you want. Your own good sense will suggest suitable procedures. (See page 8.)

The actual technique of construction is the same here as that given above: namely, only one polygon—a triangle, square, pentagon, hexagon, octagon, or decagon—will serve as a net. However, it is important to note that in any one model all the edges must be of the same length. If you want to make a set having all edges equal, you will find the volUmes growing rather large with some models in the set. Of course a large model takes up more display space, so you must gauge your models with that fact in mind. On the other hand, you may want to vary the edge length from model to model and thus obtain polyhedra of more or less uniform volume or actually of uniform height. Here experiment is in order, and a student can have an excellent demonstration appealing to his own experience of the geometrical theorems on the relation of similar figures or solids: linear dimensions are directly proportional to each other; areas are proportional to squares on linear dimensions; volumes, to cubes on linear dimensions.

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