clifford_a-_pickover_surfing_through_hyperspacebookfi-org

102 surfing through hyperspace
with eight corners. Even though we cannot easily visualize the next step in the
process, we can predict that if we were able to move a cube perpendicular to all
its edges, we would generate a 4-D object: a hypercube. It would have sixteen
corners. The trend in the number of corners is a geometric progression (2, 4, 8,
16 . . .), and we can therefore calculate the number of corners in any dimension
by using the formula 2" where n is the number of dimensions.
We can also consider the number of boundaries for objects in different
dimensions. A line segment has two boundary points. A square is bounded by
four line segments. A cube is bounded by six squares. Following this trend, we
would expect a hypercube to be bounded by eight cubes. This sequence follows
an arithmetic progression (2, 4, 6, 8 . . .).
The area of a square of edge length a is a 2 . The volume of a cube of edge
length a is c?. The hypervolume of an w-cube is a".
Past books typically provide wire-frame diagrams for tesseracts produced by
the "trail" of a cube as it moves in a perpendicular direction, similar to the one
in Figure 4.4. Of course, we can't really move in a perpendicular direction, but
we can move the cube diagonally, in the same way a square is moved diagonally
to represent a cube. Now prepare yourself for some wild trails of higher-dimensional
objects rarely, if ever, seen in popular books. To give you an idea of the
beauty and complexity of higher-dimensional objects, I produced Figures 4.12
to 4.17 using a computer program. Modern graphics computers are ideal tools
for visualizing structures in higher dimensions.
Figure 4.12 A 5-D cube produced by moving a hypercube along the fifth dimension.