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

Other Stellations or Compounds

Compound of Two Tetrahedra(Kepler's "Stella Octangula")

Compound of Five Octahera

Compound of Five Tetrahedra

Compound of Ten Tetrahedra

Model at top is a stellation of the octahedron. The rest are stellations of the icosahedron

The principle of stellation can be applied to only two other Platonic solids: the octahedron and the icosahedron. If you have studied the principle as it applies to the dodecahedron in the three stellated forms described above, you may also understand very readily why the cube and the tetrahedron cannot be stellated. It is interesting that the octahedron has only one stellation. This is the eight-pointed star—or stella octangula, as Kepler called it—which actually turns out to be a compound of two tetrahedra— It is even more interesting to find among the stellations of the icosahedron other compounds; but more about these later. None of these compounds is classified with the uniform polyhedra, precisely because they are compounds; specifically, they are intersecting polyhedra or interpenetrating polyhedra, not intersecting polygons. Nevertheless these compounds are true stellations, and they do make pleasing models. It is for this reason that a description of them is included in this monograph.

To make a model of the stella octangula, all you need for a net is an equilateral triangle. Since there are eight trihedral vertices, the color arrangement may be as follows:

Fig. 28

(1)   B   Y   O (2)   O   R   Y (3)   B   Y   R (4)   R   O   B

One vertex is shown in Figure 28. The other four vertices are the counterclockwise counterparts of this, and you should find no difficulty in making the proper assembly if you remember that in this arrangement of colors each of the four sides of each tetrahedron is a different color, but each pair of parallel planes of the polyhedron is the same color.

The icosahedron has some very interesting stellations. Including the compounds mentioned above, the total enumeration comes to fifty-nine, if one follows the complete analysis of the problem given by Coxeter.⁴ It is indeed surprising that a compound of five octahedra, a compound of five tetrahedra, and a compound of ten tetrahedra appear among the stellations of the icosahedron. Such a fact would have delighted the mind of Plato.

Fig. 29

These compounds make very attractive models. To understand how the nets are obtained, it is necessary to know something about the stellation pattern for the icosahedron. This is analogous to the dodecahedral pattern of Figure 20. The icosahedral pattern is shown in Figure 29. Actually, the innermost equilateral triangle (numbered 0) is one of the faces of the icosahedron; and the outermost equilateral triangle is one of the facial planes of the great icosahedron, the fourth of the Kepler-Poinsot solids. If each side of this large triangle is divided by two points according to the “golden section”—a linear section, discussed in Euclid’s Elements, that is approximately 1:0.618 —the pattern is quickly and easily drawn. The numbering will show what parts are used for each net. (The nets are given in Figures 30, 32, and 34.) All of these compounds can be made by the methods described above—that is, by using parts with tabs left for cementing the pieces together and constructing the models so that they are completely hollow inside. The color patterns are such that in the case of the compounds of five octahedra and of five tetrahedra each solid is of one color. In the compound of ten tetrahedra each two tetrahedra that share facial planes also share a color.

Compound of Five Octahera

To construct the compound of five octahedra, make thirty copies of the net in Figure 30, six of each of the five colors. First assemble the vertices as though they were small pyramids without their rhombic bases. Then follow the color pattern shown in Figure 31, where each rhomb is a vertex. This shows a ring of five vertices at the center. Between the extending arms of this ring a second set of five vertices is cemented, but their orientation is such that the short slant edge of each pyramidal vertex continues on a line with the grooved edge between vertices of the central ring. You may find this a bit puzzling; but if you remember to keep the basic octahedral shapes in mind, you will see them begin to develop, and the color will then help you proceed correctly. The color pattern of Figure 33 now begins to appear. By comparing the numbering of Figure 31 with Figure 33, you will see this. This hollow model is not completely rigid, but it will be satisfactory nevertheless.

Fig. 30

Fig. 31

Fig. 33

Compound of Five Tetrahedra

For a model of the compound of five tetrahedra all you need is twenty copies of the net in Figure 32, four of each of the five colors. First make trihedral vertices with the bottom edges looking rather jagged. If you begin by making a ring of five vertices cemented together with the edge marked AB of one adhering to the identical edge of the other, you will find the points numbered 2 forming a dimple in the center of the ring. Once you have built this much of the model, the other vertices will easily find their places according to the color scheme, making each tetrahedron entirely of one color. This is perhaps the most difficult model to construct because of all the jagged edges. The points numbered 2 fit into three different and adjacent dimples. The secret here is to worry about only one edge at a time. Always begin with the edge AB. Once it is cemented, let it set firmly, and then give your attention to the other edges. You may find that the last vertex will call for considerable skill, not to mention patience; but it can be done. Here you have a real challenge. This model is rigid, both technically and practically; and it is also aesthetically pleasing. (Folding the parts up instead of down gives the model a reverse twist —a twist that, by the way, will not be noticed by most people.)

Fig. 32

Fig. 33

Compound of Ten Tetrahedra

For a model of the compound of ten tetrahedra, begin by making sixty copies of each of the two parts shown in Figure 34—that is, twelve copies of each of the five colors. Suggestions for trimming the tabs are shown with the nets for the sake of clarity. Note that the left arm of the net in Figure 34 is cut without leaving a tab on its right side, and that the cut is to be made clean into the center point between the arms. The triangle numbered 5 can then be folded down. Once five parts have been cemented together to form a pentagonal dimple, the other triangle numbered 5 can be cemented to the edge lacking a tab by following the color arrangement shown in Figure 35 (basically the same as that shown in Figure 33). The short dotted lines in Figure 35 indicate overlapping parts, but these parts are to be folded down so that the bases of the small triangles can be joined. This is done by folding up the tab of one to adhere to the undersurface of the other, which lacks a tab. This completes one part. Twelve of these parts are needed for a complete model, six of which are counterparts of the other six, as explained before. Some skill and patience is needed for this model also. But it can be done, and it makes a very pleasing polyhedron.

Fig. 33

Fig. 34

Fig. 35

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