The Five Platonic Solids The Thirteen Archimedean Solids Prisms, Antiprisms, and Other Polyhedra The Four KeplerPoinsot 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 Four KeplerPoinsot Solids
Small Stellated Dodecahedron 
Great Dodecahedron 
Great Stellated Dodecahedron 
Great Icosahedron 
It was Cauchy who pointed out that these four polyhedra are actually stellated forms of the dodecahedron and icosahedron—the first three, of the former; and the last one, of the latter. This process of stellation is in itself a most interesting one. It is most readily understood by investigating the dodecahedron. First it may be noted that the fivepointed star, or pentagram, arises first by producing the sides of a pentagon or by drawing all the diagonals (see Figure 18 and Figure19). If both procedures are combined in one drawing, the result will give three sets of triangles, which provide the parts required for three of the KeplerPoinsot solids: the small stellated dodecahedron, the great dodecahedron, and the g~ steliated dodecahedroiij see Figure 20). The simplicity of this figure, providing such simple parts to be used as nets for the making of these solids, is something greatly to be cherished in these polyhedron models.
Fig. 18 
Fig. 19 
Fig. 20 
Before any description of the construction of these solids is set down, a further description of the stellation process will be helpful. If the facial planes of the dodecahedron, for example, are produced, those planes that intersect will generate certain cells: first a set of twelve pentagonal pyramids; then a set of thirty disphenoids, or wedgeshaped pieces; and finally a set of twenty triangular dipyramids. Each of these sets of cells constitutes the external parts of successive stellations of the dodecahedron. Tn print this may sound complicated, but in models it is very evident.
As for constructing the models, you can do it without making the cells mentioned above. But if you are very ambitious, it would be an excellent project to make all the cells. You can easily discover the nets for these cells by yourself. Then you can show how the stellated forms arise by adding these cells to the basic solid in each case. In fact, with a more solid material, such as wood, plastic, or even plaster, these cells could be made with a pegandhole arrangement to keep the parts together. If a good strong paper with a smooth finish is used, the parts may also be made to adhere with a thin rubber cement.
Small Stellated Dodecahedron 
Fig. 20 
Fig. 21 
Fig. 22 

As the models now come to be more intricate, with both convex and concave parts, a further hint about construction may be useful. The first parts are usually easy to handle while you can work on the interior where the tabs are being joined, using clamps as before. (As you proceed you will notice that the interior actually begins to look just as beautiful as the exterior, but of course this will all be hidden in the completed model. It just happens to be a fact that adds to your interest and enthusiasm for making models and thus lightens the monotony of the work of repeating so many parts.) It might seem that considerable skill or patience is needed to get the last vertex or part cemented in its proper place. But the secret here is to cement only one flap or tab first, let it set firmly, and then close the final opening, as described before. The double tabs make it easy to get the parts to adhere without clamps, since the model has sufficient form by this time to exert its own pressure.
By this method all the solids now being described can be constructed so that they are completely hollow inside. A basic solid could of course be used—for example, beginning with a dodecahedron, you could cement the vertex parts, twelve pentagonal pyramids, one onto each face, and thus obtain a small stellated dodecahedron. But it will be found that the final product will betray its construction when closely examined. In some models you may gain better rigidity by this method; in fact, the construction of the small stellated dodecahedron as described here is one in which the model is not technically rigid. But if the cementing is carefully done along the full length of each edge and a final drop of cement is added to both the acute and the obtuse or concave vertices in the completed model, you will find the result satisfactory.
Again, you may wish to~ economize on the number of separate parts used in a model. You might make one net, for example, in which the five triangles are all of one piece, giving a vertex part that is all of one color. You will find, however, that you have sacrificed beauty for economy. Once more, the rule here is that you get results proportionate to the efforts you put forth. These general comments apply equally to the models now to be discussed.
Great Dodecahedron 
Fig. 23 
Fig. 23a 
Fig. 24 

Great Stellated Dodecahedron 
The net for the great stellated dodecahedron is once more the isosceles triangle with 72 degrees for two angles and 36 degrees for the third. It is triangle 3 in Figure 20. Ten sheets of each of the six colors will suffice. In this model, however, only three triangles are used to form a trihedral vertex, and twenty such vertices are needed for the complete model. One such vertex is shown in Figure 25. The color arrangement for ten vertices is as follows:
Fig. 25 

Nothing has been said about the size of these polyhedra. Perhaps experiment will suggest appropriate measures, depending on how and where you wish to use the models. Different types of paper or cardboard also may be used: shining gold, silver, green, or red would give breathtaking results in a model hung by thread and illuminated with various lights!
A complete set of four dodecahedra—the original dodecahedron and its three stellations—all using the same six colors, and all built to display their exact relationship in size to one another, makes a most attractive classroom display. And when you have made these and examined them at close range, you will better understand the principle
Great Icosahedron 
First staple together five sheets of colored paper, one sheet of each of the five colors. Prick out the patterns to get copies of the net, score, and trim as usual. The dotted line of Figure 26 must be scored on the reverse side, since the small isosceles triangle is folded up. The tabs, as usual, are folded down. You are now ready to cement the parts together. Follow the paired arrangement of colors given below, cementing the pairs first:
Fig. 26 
Fig. 27 

Then five pairs make the fanlike form shown in Figure 27. When the remaining edges are brought together, a vertex is completed. The fold should be down between each member of the pairs given above, and up between the pairs. The smaller isosceles triangles should then be cemented in their respective places to form a pentagonal dimple, from which the vertex rises. Twelve vertices are required; as before, six are counterparts of the other six. You will automatically find the counterparts by reading the color table in reverse, provided you continue the same systematic handling of parts for all vertices. The vertices are joined as shown in Figure 33, where the colors indicated are the colors of respective dimples.
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 