Most of our argument so far has been in terms of the local structure of a polyhedron or a polytope--that is, the faces in the neighborhood of a point or an edge. In order to gain a better appreciation of the global structure, the shape of the object as a whole, we can use the device of fold-out patterns.

To illustrate this device, we go back to lower dimensions. It would be possible to give the King of Lineland a kit for building a square, just four segments of equal length with instructions telling which endpoints should be attached to each other. The King could begin the assembly by making three of the attachments, but he would be left with just a hinged rod four times the length of the sides of the square. He could not make the fourth connection without having the sides overlap. If it were possible for two sides to occupy the same space on the line, the King could just place one hinged rod with two sides on top of a similar figure and connect their endpoints. But he still could not construct a true square in the line since this "collapsed" quadrilateral has two different kinds of angles, two of them straight angles and two of them zero angles. Of course these are the only kinds of angles available in Lineland. We have to go to the plane to obtain a square, having all edges the same length and all angles equal.

	The unfolded square in Lineland.

	Overlapping construction of a collapsed square.

We encounter similar challenges when we move from the plane into three-space. We can design a prefabricated polyhedral structure in the plane by laying out the polygons and indicating which edges are to be attached to which. A preparatory crew in Flatland could begin the assembly, but they could not complete the project.

A good example is the fold-out pattern for a cube. We place down one of the square faces and attach the four adjacent squares to obtain a cross-shaped pattern. We then indicate which sides of these squares are to be attached to which, giving the fold-out pattern for an open box. There are exactly four free edges not yet attached to anything else, and these are ready for the last square, which we place at the bottom of the cross pattern. An engineer in Flatland could supervise the accurate construction of the individual pieces, but when it came time to assemble the object, it would be necessary to create unstraight dihedral angles, and all but the original square would simply disappear as the other squares folded into three-space. If some faraway light caused the folding cube to cast a shadow, each of the four side squares would be replaced by a shadow rectangle, which would shrink back to a side of the original square.

	The unfolded cube in the plane (left); the unfolded hypercube in three-space (right).

The analogous figure in three-dimensional space is an unfolded hypercube. Our three-space engineers could manufacture the eight cubical faces of the hypercube, and start the construction by arranging a cube surrounded by six cubes. After we see how neighboring square faces match up, there will remain six unattached squares. These are ready to receive the eighth cube, which we place at the bottom of the figure. Unfortunately no one from our space can fully appreciate the process that puts this hypercube together. Analogous to the previous case, if some faraway light in four-dimensional space caused the folding cubes to cast shadows in our space, we could watch as each of the six cubes became a shadowy rectangular box that gradually flattens out to one of the faces of the original cube.

In 1954 Salvador Dalí used the unfolded hypercube as the main symbol in his painting Corpus Hypercubicus, representing a Christ figure suspended in front of this unfolded cross-shape from the fourth dimension. In 1976 Dalí contacted us at Brown University to discuss some of the mathematics in a project he was working on in stereoscopic oil painting, and he very much liked our folding and rotating hypercube model. A copy of this model is on exhibit in the Salvador Dalí Museum in Figueras, Spain.

To construct this model, begin with six square cylinders, each composed of four squares. Attach each of these cylinders to the edges of a cube, and attach a seventh cylinder to the boundary of the bottom piece. The resulting object will flatten into a plane and rotate freely as a sort of "universal joint" when we hold onto two opposite cubes. This model is the primary subject of Robert Heinlein's story, ". . . and He Built a Crooked House," the tale of an architect who builds a home in the shape of an unfolded hypercube. The house suddenly folds up into the fourth dimension, taking with it the shocked owners who have to figure out what happened and how to get back to their own dimension.

	The folding hypercube (left); construction of the folding hypercube (center); this srip of four squares can be folded to make a square prism, with two additional squares for the top and bottom (right).

Folding cubes and hypercubes.

The fold-out patterns for regular polyhedra reveal that their polygonal faces seem to arise in strips. We can think of a cube as a strip of four squares fitting together to form a polyhedral cylinder, capped by the two other squares. A strip of four triangles folds together to form a tetrahedron. Six regular triangles form a polyhedral cylinder having two triangular rims in parallel planes. Fill in two triangles and we have an octahedron. The same construction applied to a strip of 10 regular triangles yields a polyhedron with two pentagonal faces in parallel planes. This object is the pentagonal antiprism we used to construct the regular icosahedron. We may carry this construction out in six different ways, describing the icosahedron as a pattern of woven strips. Finally, a strip of 10 pentagons capped by two more pentagons forms the dodecahedron.

	A strip of four triangles can be folded to make a tetrahedron (left); a strip of six triangles can be folded to make a triangular antiprism, with two additional triangles for the top and bottom of an octahedron (right).

	A strip of 10 triangles can be folded to make a pentagonal antiprism, with 10 additional triangles for the top and bottom caps (left); a strip of 10 pentagons can be folded to make all but two pentagons of the dodecahedron (right).

To carry this idea into the next dimension, rather than start with a strip of polygons with extra polygonal faces attached, we start instead with a stack of polyhedra with extra polyhedra attached. In this way we can obtain the fold-out models of all of the regular polytopes. An example of this construction for the 24-cell is included in the author's paper in the volume Shaping Space. The Canadian geometer H. S. M. Coxeter gives a similar description for the 120-cell and the 600-cell in his book Regular Polytopes.

The search for regular polytopes stretches over more than a century, and discoveries are still being made. Some of the properties of these polytopes were identified fairly early, but it has taken a long time to gain a full appreciation of the beauty of these objects. Only with the interactive computer are we able to turn these objects around in front of us, experiencing the same fascination that one of our remote mathematical ancestors felt when he or she first turned over and over again a die in the shape of an octahedron or a dodecahedron. The fascination will continue on into the future as we begin to explore a myriad of intriguing shapes, all beginning with the search for the regular polytopes.

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			The Self-Dual 24-Cell