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Counting Squares and Cubes

Similar observations emerge if students investigate the numbers of vertices, edges, and faces of cubes and hypercubes in various dimensions. Just as there is a hierarchy of subsimplices within each simplex, there is an analogous sequence of squares and cubes within each n-dimensional cube. A 3-cube has 8 vertices, 12 edges, and 6 squares, as can be verified by an actual count. A square, or 2-cube, has 4 vertices, 4 edges, and 1 square. A 1-cube is a segment with 2 vertices and 1 edge, and a 0-cube is point with 1 vertex. This data can form the beginning of another table:

DIMENSION: 0-cubes
(points)
Vertices
1-cubes
(lines)
Edges
2-cubes
(squares)
Faces
3-cubes
(cubes)
Cubes
4-cubes
(hypercubes)
4-Cubes

Point: 1 0000
Line: 2 1000
Square: 4 4100
Cube: 812610
Hypercube:16 ???1

When we try to fill in the missing numbers for a hypercube, the process becomes a bit more difficult. We know how to generate a hypercube — move an ordinary cube in a direction perpendicular to itself. As the cube moves, the 8 vertices trace out 8 parallel edges. This yields 12 edges on the original cube, 12 on the displaced cube, and 8 new edges traced by the movement for a total of 32 edges on the hypercube (Figure 43).

Figure 43. Framework for hypercube: two cubes with joined edges yield 16 vertices and 32 edges.

Figure 44. Shading helps identify two horizontal groups of four parallel squares in the hypercube. There are six such groups in all, three associated with the original cube and its displaced copy and three aassociated with the edges that join the two cubes.

Counting squares presents more of a problem, but a version of the same method can be used to solve it. First observe that there are 6 squares on the original cube and 6 on the displaced one. To these 12 we must add the squares traced out by the edges of the moving cube. It helps to group edges and squares in parallel bundles. The edges in the hypercube come in four groups of 8 parallel edges. Similarly the squares can be classified in four groups of 4 parallel squares, one such square through each vertex. Two horizontal groups are rather easy to see (Figure 44); another group of four vertical faces become clearer when we remove some of the extraneous lines (Figure 45).

Figure 45. A group of four vertical squares in ahypercube determined by the horizontal displacement of the original cube. These squares are easier to see when background lines are removed, as in the lower figure.

Student teams can easily identify the remaining three groups of four squares. It is easier to do this when the four squares do not overlap and relatively more difficult when the overlap is large. The entire set consists of 24 squares.

Grouping edges or faces is particularly effective when an object possesses a great deal of symmetry, as does the hypercube. We can study the relation between symmetry and grouping by looking at different dimensions. Symmetries of a cube, a square, or a segment arise by permuting the edges at each vertex in different ways and by moving each vertex to another position. The collection of all symmetries of the cube or hypercube is an important example of a group, an algebraic structure that reflects geometric properties. The symmetry group of a cube is the collection of permutations of its vertices that preserve its structure. The attempt to codify the relation of permutations to symmetries of algebraic and geometric structures provided considerable impetus for the development of modern algebra during the past two centuries. Even now symmetry groups continue to fuel theoretical work in atomic physics.

The crucial observation about the hypercube is that it is so highly symmetric that every point looks like every other point: if we know what happens at one vertex, we know what happens at all vertices. For example, at each of the 16 vertices of the hypercube there are 4 edges, for a total of 64. But this process counts each edge twice, so the actual number of edges is half of 64, or 32.

At each vertex there are a certain number of square faces. How many? As many as there are ways to choose two edges from among the four edges that meet at the vertex. Once we have chosen one edge from among the four, there remain three possibilities for the second; together, these yield 12 pairs. As before, each pair of edges appears twice in this list, once in each order. So these 12 pairs yield 6 different squares at each vertex. All 16 vertices together then yield 96 squares. But each square is counted four times, once for each of its vertices. Hence the true total is 96/4 = 24 squares in a hypercube. This reasoning confirms the direct count of six groups of four squares that we saw in drawings of the hypercube, but it is reached by a method that would work even if applied to a five-dimensional cube. [an error occurred while processing this directive]