clifford_a-_pickover_surfing_through_hyperspacebookfi-org

100 surfing through hyperspace
In this chapter we've discussed how a cube would appear to Flatlanders as it
penetrates their world. As a cube moves corner-first through Flatland, the Flatlanders
first see a point, then a series of growing triangles, then hexagons, then
triangles, and finally a point (Fig. 4.9). Similarly, a variety of shapes would be
produced as a hypercube penetrated our world (Fig. 4.10). In four dimensions,
slicing with a 3-D knife produces an array of strange shapes ranging from distorted
cubes, various prisms, and polyhedra often in unexpected arrangements.
What would the hypercube look like as we rotated it? One way to visualize
this is to begin with the "central projection" in Figure 4.5 and watch the wireframe
change as the hypercube rotates (Fig. 4.11). Starting from the left and
going clockwise, the large exterior cube opens toward the top, flattens out, and
opens inward to form an incomplete pyramid. Meanwhile, the smaller interior
shaded cube opens toward the bottom to form another incomplete pyramid. If
we continue to rotate the hypercube, the unshaded exterior cube will become
the small cube, and the small cube will flatten out and come back to become the
large cube. As Thomas Banchoff notes, each of the eight cubical faces takes its
turn holding all the various positions in this projection. As each of the cubes
flattens out and opens up again during the rotation, it changes orientations. If a
cube contained a right-handed object before the flattening, the object would
become left-handed afterwards.
On the Trail of the Tesseract
In this chapter, we've also used lower-dimensional analogies to help contemplate
the mathematics of higher spaces. Throughout history, mathematicians have
used interdimensional analogies. For example, if mathematicians understood a
theorem in plane geometry, they were often able to find analogous theorems in
solid geometry. (Theorems about circles provide insight into theorems about
spheres and cylinders.) Similarly, solid geometry theorems have suggested new
relationships among plane figures. If history shows that knowledge can be
gained by going to higher dimensions, imagine what we might learn by contemplating
4-D geometries.
The most often-used analogy for contemplating shapes in higher dimensions
involves moving objects perpendicular to themselves. If we move a 0-D point
with no degrees of freedom, we generate a line, a 1-D object with two end
points (Fig. 4.3). A line moved perpendicular to itself along a plane generates a
square with four corners. A square moved perpendicular to itself forms a cube