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