Plaiting Polyhedrons

In Plato's dialogue Phaedo, Socrates tells a story in which the earth, viewed from outer space, appears "many-colored like the balls that are made of 12 pieces of leather." Historians take this to mean that the Greeks made balls by stitching together 12 leather pentagons stained with different colors and stuffing the interior to make the surface spherical. Rigid pentagons that are regular and identical would of course make a regular dodecahedron, one of the five Platonic solids.

There are all kinds of methods for constructing the five regular convex solids out of flat pieces of heavy paper or cardboard, and many problems have been proposed about ways of coloring their faces. The idea of weaving or braiding a regular solid from strips of paper seems to have been explored first by an English physician, John Gorham, who published in London in 1888 a now rare book about it: A System for the Construction of Plaited Crystal Models on the Type of the Ordinary Plait. His techniques were improved by A. R. Pargeter and by James Brunton in papers listed in the Bibliography. This year Jean J. Peder-sen, a mathematics teacher at the University of Santa Clara, hit on an ingenious variation of the plaiting technique. It applies not only to the Platonic solids but also to many other polyhedrons, providing models of stunning multicolored symmetry and suggesting fascinating combinatorial theorems and puzzles.

Unlike Mrs. Pedersen's predecessors, who used crooked and asymmetrical basic patterns, she weaves each Platonic solid from n congruent straight strips. Assume that each strip is a different color and that each model has the following properties:

(1) Every edge is crossed at least once by a strip; that is, no edge is an open slot.

(2) Every color has an equal area exposed on the model's surface. (An equal number of faces will be the same color on all Platonic solids except the dodecahedron, which has bico-lored faces when braided by this technique.)

Mrs. Pedersen has proved that if the above two properties are met, the number of necessary and sufficient bands for the tetrahedron, the cube, the octahedron, the icosahedron and the dodecahedron are respectively two, three, four, five and six.

Let us see how this works for the tetrahedron. Although the model can be plaited with one straight band, it will have some open edges. Therefore at least two bands are necessary. As shown in Figure 59, valley-crease each strip along the broken lines. (Scoring the lines with a hard pencil will facilitate clean folding.) Overlap two triangles as shown and fold the underneath strip into a tetrahedron. Wrap the other strip around two faces of this tetrahedron, then tuck the end triangle into the open slot. If you use construction paper of good quality and strips of different colors, the result is a rigid tetrahedron with two adjacent faces (of course, any two of its faces must be adjacent) of one color and two of the other color.

To construct the cube three strips, each a different color, will do the trick [see Figure 60], Valley-fold each along the broken lines. The reader can have the pleasure of weaving the three strips—it is quite easy—into a rigid cube. He will find that there are two essentially different ways to make a model with two faces of each color.

One method makes a cube that has adjacent pairs of faces with like colors. If you think of each band as being glued together where its end squares overlap, this model consists of three closed bands, each pair of which is linked. Imagine that

Figure 59

Figure 59

Plaiting a tetrahedron

Plaited three-strip cube the surface is flexible and that the cube is stuffed, like the leather dodecahedron mentioned by Plato, until it is spherical. The coloring, as Piet Hein has suggested, is a striking three-dimensional analogue of the familiar yin-yang symbol of the Orient. Like the yin-yang, it is asymmetrical (has either left or right handedness). Piet Hein proposes calling the three regions yin, yang and lee, the last two terms honoring C. N. Yang and T. D. Lee, the two Chinese-American physicists who shared a Nobel prize in 1957 for their role in overthrowing the symmetry law of parity.

The other way of plaiting the three strips produces a cube with opposite faces of like color. Again regard the three bands as being joined at their ends. Inspection reveals an unexpected structure. As Mrs. Pedersen has noted, the bands are topolog-ically equivalent to the Borromean rings that are used as a trademark for Ballantine beer. Although the three bands cannot be separated, no pair is interlocked. If any one band is removed, the other two will slide apart.

The octahedron requires four valley-creased strips, each like the strip shown in Figure 61. These cannot be woven to make a model with opposite faces of the same color. (Can you prove it?) A model is possible, however, with like colors on pairs of adjacent faces, the four colors meeting at one corner and the same four, in reverse cyclic order, meeting at the diametrically opposite corner. A good procedure is to start with the two pairs of overlapping strips held together by a paper clip as shown in the illustration. Fold one pair into an octahedron, then weave the other pair around it, with both of the free ends tucked in slots, to achieve the desired color pattern. After the model is completed you can reach into the interior and remove the paper clips.

The octahedron is more difficult to make than the cube, but if the reader will set himself the task, he will find, as with all

How strips are clipped together to weave an octahedron

such models, that there is an aesthetic delight in feeling the solid acquire permanent rigidity when the final tuck is made. Mrs. Pedersen has found that handsome models of this solid and the other four solids can be made by using colored cloth tape glued to construction paper for rigidity.

The icosahedron is woven with five valley-creased strips [see Figure 62]. A charming model can be constructed with each color on two pairs of adjacent faces, the pairs diametrically opposite each other. All five colors go in one direction around one corner and in the opposite direction, in the same order, around the diametrically opposite corner. Each band circles an "equator" of the icosahedron, .its two end triangles closing the band by overlapping. In making the model, when the five overlapping pairs of ends surround a corner, all except the last pair can be pasted together or held with paper clips, which are re-

Figure 62

Figure 62

The five-banded icosahedron

moved later. The last overlapping end then slides into the proper slot. An expert will soon dispense with the use of paste or paper clips for this model.

Only the dodecahedron cannot be plaited with straight-sided strips so that each face is a solid color. Mrs. Pedersen discovered, however, that by using six strips the dodecahedron shown in Figure 63 can be woven. The obtuse angles made by the valley folds [broken lines] with the strip's sides are each 108 degrees, the interior angle of the regular pentagon. The broken lines must equal the shorter line segments on the sides, making each section of the strip a truncated pentagon.

Figure 63

Figure 63

The six-banded dodecahedron

To construct the dodecahedron, the most difficult of the Platonic solids, Mrs. Pedersen suggests starting with three pairs of strips, each overlapped and glued together to make the curved, bracelet-like structure shown in Figure 64. Using two bracelets, overlap and glue together the pairs of ends to make a pair of braided closed bands. Slip one bracelet inside the other so that each circles a different equator of the dodecahedron. The third bracelet then is woven around a third equator, and its four free ends are tucked into slots on opposite sides of a pair of adjacent pentagonal faces. The technique is similar to the one used for making the cube with opposite faces of like color. Once the construction is mastered it is possible to use only paper clips to keep each bracelet together. The paper clips can be removed after the model is completed.

Note that every face of the finished dodecahedron has two colors. The same two colors are on the diametrically opposite face but are reversed in their arrangement. All diametrically opposite corners are mirror images in the order of the three or four colors that surround them. The model in the illustration,

a dodecahedron

on which like colors are indicated by the same shade, appears asymmetrical, but when the actual model is turned in one's hands, its curious symmetry becomes apparent. The eight corners that are surrounded by exactly three colors mark the ver-texes of an inscribed cube. The four corners surrounded by three triangles mark the vertexes of an inscribed tetrahedron.

It is difficult to explain the exact procedure for plaiting the last two models, so that I shall leave their construction as additional exercises for the patient and intrigued reader. It may help to construct each solid first by conventional means, then weave the required strips around it. I can only promise to report later if and where Mrs. Pedersen publishes instructions for the Platonic solids as well as for more elaborate and less regular polyhedrons that can also be formed by weaving congruent strips.

Mrs. Pedersen has devised a technique, involving the use of gummed tape or adding-machine tape, for folding the strips for all five models without drawing any fold lines. This technique, along with instructions for making what she calls a golden dodecahedron (each face has a pentagonal hole surrounded by five triangles of different colors), are given in her Fibonacci Quarterly article listed in the bibliography.

For years I was puzzled by the fact that Plato, repeating the earlier views of Pythagoras and his followers, identified the universe with the dodecahedron rather than the icosahedron, which I took for granted to be more nearly spherical. I found the answer recently in Volume I of Howard Eves's entertaining work In Mathematical Circles. Contrary to almost everyone's intuition, it is the dodecahedron that is most like a sphere. If the two solids are inscribed in the same unit sphere (a sphere with a radius of 1), the 20-faced icosahedron has a volume of 2.536 +, whereas the 12-faced dodecahedron has a larger volume of 2.785 + . Their surface areas are in the same ratio as their volumes: 9.574+ for the icosahedron, 10.514+ for the dodecahedron. The ancient Greeks had good reason to use the dodecahedron for their leather spheres.

If a cube and an octahedron are inscribed in a unit sphere, the cube has the greater volume and greater surface, and again their surface areas are in the same ratio as their volumes. The octahedron's volume and area are respectively 1.333+ and 6.928 + ; the cube's volume and area, 1.539+ and 8. An interesting mechanical question, difficult to formulate precisely and perhaps even more difficult to answer, is which solid of each pair—cube or octahedron, icosahedron or dodecahedron— rolls more easily when used as a gaming device?

If a cube and an octahedron are inscribed in the same sphere, which solid surrounds the larger inscribed sphere? The surprising answer, as Eves explains, is that the two inner spheres are the same. This is also true of the inscribed spheres of a dodecahedron and an icosahedron that are inscribed in the same outer sphere.

Here are three polyhedron problems:

(1) What is the simplest nonconvex polyhedron that, like the cube, has a surface of n faces, each a unit square?

(2) If each face of a regular tetrahedron is a different color, how many different tetrahedrons can you make by using the same four colors? Rotations, of course, are not counted as different. Can you devise a simple formula that applies to all the Platonic solids, giving the number of different colorings possible when each of the n faces has a different color and the same n colors are used?

(3) If three colors are applied to a cube, each color going on two faces as in Mrs. Pedersen's plaited model, how many different colorings are possible? Again, as customary, rotations are not considered different. How many such cubes can be woven with Mrs. Pedersen's three bands, assuming there are no loose end squares that are not tucked in?

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