clifford_a-_pickover_surfing_through_hyperspacebookfi-org

HYPERSPHERES AND TESSERACTS 99
Next time there is a bright blue arch of sky above, gaze at it and recall the
words of Charles Hinton.
Let us return for a moment to Hinton's cubes. Hinton's methods of visualizing
four-space structures in three-space cross sections required hundreds of
small cubes, colored and labeled. Hinton said that he was able to think in four
dimensions as a result of studying his cubes for years. He also noted he taught
the method to his sister-in-law when she was eighteen. Although the girl had
no formal training in mathematics, she soon developed a remarkable grasp of
the 4-D geometry and later made significant discoveries in the field.
Hinton's disciples spent days mediating on the cubes until some thought they
could mentally reassemble these cubes in the fourth dimension—thus achieving
nirvana. Figure 4.7 shows an unraveled hypercube. Although the cubes of this
tesseract seem static, a 4-D person can fold the cubes into a hypercube by lifting
each individual cube off our universe into the fourth dimension. Note that Hinton
used the words "ana" and "kata" in the same way I use the terms "upsilon"
and "delta" to describe motions in the 4-D world as counterparts for terms like
"up" and "down." (I find that upsilon and delta are easier to remember than ana
and kata because of the "up" in upsilon and "d" in delta.)
Unraveling
Take a deep breath, and let your imagination soar. Watch now as a 4-D person
folds a tesseract into a hypercube. What do you see? Not much! All you observe
are the various cubes in Figure 4.7 disappearing, leaving only the center cube
in our universe. The folded hypercube looks just like an ordinary cube in the
same way a cube can appear like an ordinary square to a Flatlander.
What would it be like to be visited by a hypercube? If it came into our universe
"cube-first" (like a cube coming into a planar universe "face first"), we
would just see a cube that disappeared as it finally went through our 3-D world.
Even though you and I are not likely to be able to "see" a hypercube all at once
in the same way that we can see a cube, we can be sure that such an object
would have sixteen vertices. It might even look like just a square when it just
touched our world. However, if the object rotated, the ordinary-looking square
could reveal a starburst of lines (as in Fig. 4.4) corresponding to an object that
really has twenty-four square faces, thirty-two edges, and sixteen vertices. If a 5-D
cube passed through a 4-D universe, it would appear for a while as a hypercube
with thirty-two vertices before it disappeared entirely from the world. 3