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 Four Kepler-Poinsot Solids

Small Stellated Dodecahedron

Great Dodecahedron

Great Stellated Dodecahedron

Great Icosahedron

The next set of uniform polyhedra, four in number, whose beauty is most striking, is that of the Kepler-Poinsot solids. (Kepler discovered two about 1619, and Poinsot rediscovered these and discovered the two others in 1809.) These solids are all the more interesting because they were unknown to the ancient world. Of course the star polygon, also called the pentagram, is very ancient, possibly as old as the seventh century B.C.³ The Pythagoreans used this five-pointed star as a symbol of their brotherhood. Yet the discovery of solids with star-shaped facial planes belongs to the modern era. It is perhaps correct to say that the ancients missed these b~cause they were concerned only with convex polygons and polyhedra and did not consider the case of intersecting facial planes. These are the properties that enter into the solids to be described in what follows. It is precisely the star-shaped appearance of the Kepler-Poinsot solids that gives them their interest and beauty—a beauty that, again, is greatly enhanced by making them so that each facial plane has its own color.

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 five-pointed 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 Kepler-Poinsot 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 wedge-shaped 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 peg-and-hole 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

The Small Stellated Dodecahedron
To make a model of the small stellated dodecahedron uniform in construction with the models previously described, all you need for a net is an isosceles triangle with base angles 72 degrees and vertex angle 36 degrees. This is triangle number 1 in Figure 20. With a needle, prick through the vertices of this triangle placed as a net on top of six sheets of paper, each of a different color. You will need to repeat this pattern ten times, giving you ten triangles of each of the six colors—sixty triangles in all. Score the lines and trim the tabs as described before. You will find that the more acute angles must also have more acute trimming. This is best done after the tabs have been folded; if it is done before, folding becomes more difficult. (See Figure 21 for suggestions.) Next, five of these triangles are cemented together as in Figure 22. Then the final edges are joine~to form a pentahedral angle—that is, a pentagonal pyramid without a base. You will find it easy to get the correct color arrangements by following the color scheme given below:

Fig. 20

Fig. 21

Fig. 22

(0)   Y   B   O   R   G (1)   W   G   O   R   B (2)   W   Y   R   G   O (3)   W   B   G   Y   R (4)   W   O   Y   B   G (5)   W   R   B   O   Y

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

The Great Dodecahedron
The second stellation of the dodecahedron is known as the great dodecahedron. It may be described as a solid composed of twelve intersecting pentagons. When it is appropriately made in six colors, it readily gives the appearance of a solid star embossed on a pentagon plane; but each such star shares each of its arms with an adjacent star. The net is simply an isosceles triangle, this time with base angles of 36 degrees and a vertex angle of 108 degrees. ‘This is triangle 2 in Figure 20. Ten sheets of each of the six colors are needed. (See Figure 23 for a suggestion about how to trim the parts. The color arrangement is shown in Figure 24.) Again only half the model is shown. A simple method of construction is to cement three triangles in the form of a trihedral dimple, as follows:

Fig. 23

Fig. 23a

Fig. 24

(1)   Y   W   G   (6)   G   O   Y (2)   B   W   Y   (7)   Y   R   B (3)   O   W   B   (8)   B   G   O (4)   R   W   O   (9)   O   Y   R (5)   G   W   R   (10)   R   B   G

Once this half has been constructed, the color pattern itself is evident enough to enable you to continue without further difficulty. The other ten dimples are the usual counterparts. In approaching the last pieces you must take care to have one trihedral dimple left as the last part to be cemented. Let the cement set firmly along one edge of this part, then apply cement to the other two edges and close the triangular hole, with the trihedral dimple serving as the lid. In this way the longer and acute part of the wedge-shaped star arms can easily be pinched together with the fingers.

Great Stellated Dodecahedron

The Great Stellated Dodecahedron
The third and final stellation of the dodecahedron is called the great stellated dodecahedron. This solid makes a lovely Christmas decoration, and is often seen as such in store windows and commercial displays during the holiday season. It can be, and often is, made by adding triangular pyramids to an icosahedron base; but, as mentioned before, this method will be found to lack something of the precision and beauty that can be achieved in a completely hollow model in six colors. You may perhaps be surprised that the icosahedron is used as a base for a stellated form of the dodecahedron. But then there are many surprises in the world of polyhedra, and curiosity about the reasons for them may spur you to further study on your own. (See the Bibliography for related works.)

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

(1)   Y   G   B   (6)   W   G   B (2)   B   Y   0   (7)   W   Y   0 (3)   0   B   R   (8)   W   B   R (4)   R   0   G   (9)   W   0   G (5)   G   R   Y   (10)   W   R   Y

The first five vertices are joined in a ring with the bottom edges forming a pentagon. Then the next five vertices are added to each edge of this pentagon, so that the white edge of (6) is cemented to the yellow of (1), and so forth. The next ten vertices have colors in counterclockwise rotation and are placed diametrically opposite to their counterparts. If you work systematically, as before, you will not find this difficult. Tn fact, the colors will help you if you remember that each triangle is an arm of a five-pointed star and that you want each star to have five arms of the same color. When you have completed these three according to the instructions given here, you will also notice that planes that are parallel to one another are the same color.

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

The Great Icosahedron
But of all the polyhedra so far described perhaps the most beautiful and attractive is the great icosahedron itself, which is the fourth of the Kepler-Poinsot solids. The nets are very simple (see Figure 26). The color arrangement described here gives a final beauty that cannot but appeal to anyone who sees or handles a model. And, fortunately enough, it is not at all difficult to make—.—not as hard as some of the compounds mentioned below. It does require the patience and the time to prick, cut, score, and trim one hundred twenty individual pieces of cardboard for a complete model. But it is well worth the trouble and the effort.

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

(0)   YG   BY   OB   RO   GR (1)   BG   YB   RY   OR   GO (2)   OY   BO   GB   RG   YR (3)   RB   OR   YO   GY   BG (4)   GO   RG   BR   YB   OY (5)   YR   GY   OG   BO   RB

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