This section describes how to make the puzzle pieces for the following 9 models which naturally divide themselves into three types.
The Platonic ^{3} Puzzles: (Do you notice any interesting pattern here? Do you see why we've written the dodecahedron last, although it's usually written before the icosahedron?)
There are some general comments that apply to each of these 9 models. In each case you should first make the required pattern pieces (or strips), by folding the gummed tape. Then glue the strips onto colored paper (when more than one strip is involved glue each strip onto paper of a different color). When gluing the strips onto the colored paper, make certain the paper you plan to use for each strip is large enough. Then place a sponge (or washcloth) in a bowl and add water to the bowl so that the top of the sponge is very moist (squishy). Next moisten one end of the strip by pressing it onto the sponge and then, holding that end, pull the rest of the strip across the sponge (This part of the process is often messy!) ^{4} Make certain the entire strip gets moistened and then place it on the colored paper. Use a hand towel (or rag) to wipe up the excess moisture and to press the tape into contact with the colored paper. Put some books on top of the tape so that it will dry flat. When the tape is dry, cut out the pattern piece, trimming off a small amount of the gummed tape (about 1/16 to 1/8 of an inch) from the edge as you do this. This trimming procedure serves to make the model look neater and, more importantly, it allows for the increased thickness produced by gluing the strip to another piece of paper. Refold the piece (only along the lines you need for your particular model) so that the raised (mountain) folds are on the colored side of the paper. You will then have your puzzle pieces and can proceed to construct your model. We begin the actual puzzles with the two dipyramids to let you get a feel for the way your materials behave. We will be fairly explicit here, to get you started, but we'll be less detailed when we come to the Platonic Puzzles.
Figure 5(a) shows the left-hand end of the 31-triangle strip used to construct the pentagonal dipyramid (with the colored side up). You should mark the first and eighth triangles exactly as shown (note the orientation of each of the letters within their respective triangles). To assemble the model place the first triangle over the eighth triangle so that the circled letters A, B, C are over the uncircled letters A, B, C, respectively. Holding those two triangles together in that position, you will notice that you have the frame of a double pyramid for which there will be five triangles above and five triangles below the horizontal plane of symmetry (the plane containing the vertices 1, 2, 3, 4, 5 in Figure 5(b)). Now hold this configuration up and turn it so that the long strip of triangles falls around this frame. If the creases are folded well, the remaining triangles will fall into place. When you get to the last triangle there will be a crossing of a strip that the last triangle can tuck into, and your model will be complete and stable. If you have trouble because the strip doesn't fall into place there are two frequent explanations. The first (and most likely) reason is that you have not folded the crease lines firmly enough. In that case all you need to do is crease them again with more gusto! The second possible reason is that the strip seems too short to reach around the model and tuck in. This may be remedied by trimming a tiny amount from each edge of the strip. An analogous construction may be made for the triangular dipyramid shown in Figure 5(c). This model can be made from a strip of 19 equilateral triangles. Knowing what the finished model should look like and that you should begin by forming the top three faces with one end of the strip should be sufficient hints. You may discover that you can construct each of these dipyramids with fewer triangles, but we chose the construction that produces the most balanced model. You will note that both of these constructions place precisely three thicknesses of paper on each face, except where the last triangle tucks in (producing four thicknesses). In both cases, you could remedy this small defect by cutting off half of the first and last triangle on the strip. Now let us turn to the Platonic Puzzles.
Figure 6 shows a typical puzzle piece (or strip) next to each solid, and
tells you how many are needed. In each case the puzzle is this: take the
required strips and braid them together to from the required solid in such a
manner that
The tetrahedron, octahedron, and icosahedron all involve strips obtained from the D^{1}U^{1}-folding procedure. All you need to do is prepare the pattern pieces as we described above and try to assemble the models. You may note that, on all of these models, if you take into account the coloring of the surface, they will have lost some of the symmetry you would expect to find on Platonic Solids; that is, not all edges will look the same. For some edges the two adjacent faces will have the same color but, for other edges, the two adjacent faces will have different colors. (We will propose another type of construction that corrects this defect in Section 5). If you manage to get all three of these together without any hints you are really an expert! If you have trouble getting your models together check the hints given in Section 4. The hexahedron (cube) pattern pieces may be made by making exact folds on the tape. All that you need to remember is that if you fold the tape directly back on itself you will produce an angle of precisely p/2, and if you bisect that angle you will know exactly where to fold the tape back on itself to produce a square. Once you have one square on the tape you may then simply fold the tape back and forth, accordion style, on top of this square to produce the required number of squares. There are actually two ways to braid these three pieces together to satisfy the conditions for the puzzle. One of these ways produces a cube with opposite faces the same color and the other way produces a cube with certain pairs of adjacent faces the same color. From the point of view of symmetry the first is more symmetric because, on that model, all edges abut two faces of different colors. If you have trouble assembling this model check the hints given in Section 4. The dodecahedron involves strips obtained from the D^{2}U^{2}-folding procedure. But notice on the final pattern piece you should only fold the pattern piece firmly along the short crease lines (ignoring the long lines) after you have cut out each piece. We should tell you that on this model four sections of each strip will overlap (for stability). It may also be helpful to let you know that the strips go together in pairs and the construction is then similar to that of the more symmetric cube you have constructed above - and, if coloring is taken into account, the completed model loses a lot of the symmetry you expect to see on a dodecahedron. You may now have enough hints, but if you have difficulty consult Section 4. The diagonal cube involves four strips each containing 7 right isosceles triangles as shown in Figure 7(a). The strips for these pieces may be folded by the exact procedure similar to that described above for the cube in the Platonic Puzzles. Just remember that this time you want to emphasize those crease lines that make an angle of p/4 with the edges of the tape. To assemble the cube you begin by laying out the four pieces as shown in Figure 7(b), with the colored side of the paper not showing. You may wish to put a small piece of tape in the exact center to hold the strips in position. Now, thinking of the dotted square surrounding the center as the base of your cube, you begin to braid the strips to make the vertical faces, remembering that each strip should go successively over and under the strips it meets as it goes around the model. When you get to the top face you will find that all the ends will tuck in to produce a very beautiful and highly symmetric cube; indeed, none of the symmetry of the cube has been lost. You will notice that every face has a different arrangement of four colors and that every vertex is surrounded by a different arrangement of three colors. The golden dodecahedron involves strips obtained from the D^{2}U^{2}-folding procedure. But notice on the pattern pieces you should only fold the pattern piece firmly along the long crease lines (ignoring the short lines) after you have cut out each piece. To complete the construction of this model, begin by taking five of the strips and arranging them, with the colors showing, as shown in Figure 8(b), securing them with paper clips at the points marked with arrows. View the center of the configuration as the North Pole. Lift this arrangement and slide the even-numbered ends clockwise over the odd numbered ends to form the five edges coming south from the arctic pentagon. Secure the strips with paper clips at the points indicated by crosses. Now weave in the sixth (equatorial) strip, shown shaded in Figure 8(c), and continue braiding and clipping, where necessary, until the ends of the first five strips are tucked in securely around the South Pole. Above all, keep calm, you can even take a break - the model will wait for you! Just make certain that every strip goes alternately over and under each strip it meets all the way around the model. When the model is complete (with the last ends tucked in) you may remove all the paper clips and the model will remain stable. We notice that this constructed dodecahedron is aesthetically very satisfying - more so than the dodecahedron previously described. This is due to its amazing symmetry - none of the possible symmetry has been lost. Before giving you the hints for the remaining models we cannot resist showing you a lovely use for the cube with three strips, the diagonal cube and the golden dodecahedron. An interesting question^{5} that has been asked by geometers (see [KW]) is "How many disjoint pieces - both finite and infinite (or unbounded) - are formed by the extended face planes^{6} for each Platonic solid?" As it turns out we can use our braided models to answer some of these questions in the case of the unbounded regions.
Let us use the ordinary cube to show how the braided models are useful.
Notice that the edges of the three strips used to create the braided model
lie in 6 planes which interesect each other to form a cube. Figure 9,
suitably interpreted, shows that the extended face planes of a cube partition
space into 27 pieces. There is, of course, the cube itself, which is
bounded. Then come the unbounded regions. There are
What turns out to be true is that the braided models partition the surface of the polyhedron into mutually disjoint sets of 'polygons' where each polygon is covered by 0, 1 or 2 thicknesses of paper. The polygons where there are holes (0-thickness) define unbounded polyhedral regions, the polygons which are narrow slits (1-thickness) define unbounded wedges, and the polygons where the strips actually are crossing each other (2 thicknesses) define unbounded prisms. The shapes of these unbounded regions may vary with the braided model, but these general statements always hold.
If we are to be able to answer our question for the octahedron we need a
braided model with 4 strips so that their edges will define 8 face
planes - and, of course, the braided model should also have the same
symmetry group as the octahedron. Fortunately the diagonal cube satisfies
our conditions (see [Math] concerning the duality of the cube
and octahedron). Figure 10 shows the octahedron with
some of its face
planes extended so that you can see a typical finite region and typical
unbounded regions of each type. The braided models don't help to count the
bounded regions (in this case, however, we can see from the part of
Figure 10
labeled (a) that there is a tetrahedron on each face of the original
octahedron). The rest of the labels in Figure 10
indicate unbounded regions
and Figure 11 reproduces
those regions as they are associated with the
surface of the diagonal cube. Thus, using Figure 11,
we may now count the
unbounded regions. They are (using the labels in Figure 11):
So the golden dodecahedron must also be useful. In fact it is composed of six strips and the planes defined by the edges of those strips intersect inside this model to form a dodecahedron. Thus the surface of the golden dodecahedron can be used to see that there are 122 unbounded regions created by the extended face planes of the dodecahedron. You might like to try to count them yourself using your golden dodecahedron (or see [P] for more details). What about the icosahedron? Figure 11 shows how a model, braided from 10 straight strips, may be made from the D^{2}U^{2}-tape that can be used to count the 362 unbounded regions created by the extended face planes of the icosahedron (see [P] for more details). We have not written down anywhere how to make the model shown in Figure 11, but we're sure the interested reader will be able to figure it out from what we have said and the illustration. You may be asking yourself why we have slighted the tetrahedron. The answer is that the tetrahedron does not have faces lying in opposite parallel planes, so our models are not useful here. However, it is not difficult to imagine extending the face planes of the tetrahedron and seeing that you have one finite region (the tetrahedron) and 14 unbounded regions (4 from vertices, 6 from edges and 4 from faces).
^{3} As any Greek scholar will tell you, the names of the Platonic solids are designed to show that they have 4, 6, 8, 20, 12 faces, respectively. ^{4} Perhaps we should have included some very old clothes as optional (or even essential) materials. ^{5} A question is always interesting to mathematicians if the answer is not obvious but they can see a possible way to answer it. ^{6} The planes in which the faces of a polyhedron lie are called the extended face planes of the polyhedron. |