clifford_a-_pickover_surfing_through_hyperspacebookfi-org

HYPERSPHERES AND TESSERACTS
89
Point
Line segment
Square
Cube
Hypercube
Hyperhypercube
Corners Edges Faces Solids Hyper-
1
2
4
8
16
32
0
1
4
12
32
80
0
0
1
6
24
80
0
0
0
1
8
40
volumes
0
0
0
0
1
10
You stare into Sally's eyes with pupils slightly dilated in the dim room
light. "Take a look at the hypercube drawing. Can you see the sixteen
corners? The number of corners (or vertices) doubles each time we
increase the dimension of the object. The hypercube has thirty-two
edges. To get the volumes of each object, all you have to do is multiply
the length of the sides. For example, the volume of a cube is /, 3 where /is
the length of a side. The hypervolume of a hypercube is / 4 . The hyperhypervolume
of a 5-D cube is / 5 , and so on.
"How can we understand that a hypercube has thirty-two edges?"
"The hypercube can be created by displacing a cube in the upsilon or
delta direction and seeing the trail it leaves. Let's sum the edges. The initially
placed cube and the finally placed cube each have twelve edges. The
cube's eight corners each trace out an edge during the motion. This gives
a total of thirty-two edges. The drawing is a nonperspective drawing,
because the various faces don't get smaller the 'further' they are from your
eye" (Fig. 4.4).
You hand Sally a cube of sugar and a pin. "Can you touch any point
inside any of the square faces without the pins going through any other
point on the face?"
"Of course."
"Sally, let's think what that would mean for a hyperman touching the
cubical 'faces' of a tesseract. For one thing, a hyperman can touch any
point inside any cubical face without the pin's passing through any point
in the cube. Points are 'inside' a cube only to you and me. To a hyperman,
every point in each cubical face of a tesseract is directly exposed to
his vision as he turns the tesseract in his hyperhands."
You go to the blackboard and begin to sketch. "There's another way to
draw a hypercube. Notice that if you look at a wire-frame model of a
cube with its face directly in front of you, you will see a square within a