clifford_a-_pickover_surfing_through_hyperspacebookfi-org

HYPERSPHERES AND TESSERACTS 91
"Sally, ever wonder what a hypersphere would look like projected into
our universe?"
She smiles. "Sure, every day, every waking hour."
You bring out a globe made of glass with all the continents marked.
You shine a light on it and look at the projection on the wall. "First, let's
consider the projection of an ordinary sphere onto a plane." You point at
a spot projected on the wall. "Notice that the two hemispheres will overlap
on one another, and that the distance between our FBI headquarters
and China seems very short. Of course, that's only because we're looking
at a projection. In fact, every point on the projection represents two
opposite points on the original globe. China and America don't actually
overlap because they are on opposite sides of the globe."
Sally studies the projection on the wall. "What we're seeing looks like
two flat discs put together and joined along their outer circumferences."
"Right. Watch as I rotate the globe. The projection makes it appear
that the Earth is rotating both right and left simultaneously. Would a
2-D being go insane trying to picture an object rotating in three
dimensions?"
"Imagine our difficulty in visualizing a 4-D rotating planet!"
You nod. "You can imagine a space-projection of a hypersphere into
our world as two spherical bodies put through each other and joined
along their outer surfaces. It would be like two apples grown together in
the same regions of space and joining at their skins."
You place the globe on an old Oriental carpet covering the hardwood
floor of your office. "Let's return our attention to hypercubes. Another
way to represent a hypercube is to show what it might look like if it was
unfolded." You bring out a paper cube that has been taped together and
remove some pieces of the tape. "By analogy, you can unfold the faces of
a paper cube and make it flat" (Fig. 4.6).
You then bring out a paper model of an unfolded hypercube. "Sally,
we can cut a hypercube and 'flatten' it to the third dimension in the same
way we flattened a cube by unfolding it into the second dimension. In
the case of the hypercube, the 'faces' are really cubes" (Fig. 4.7).
You point at a poster on the wall. "The hypercube has often been used
in art. My favorite is the unfolded hypercube from Salvador Dali's 1954
painting Corpus Hypercubus (Fig. 4.8). By making the cross an unfolded
tesseract, Dali represents the orthodox Christian belief that Christ's death
was a metahistorical event, taking place in a region outside of our space