clifford_a-_pickover_surfing_through_hyperspacebookfi-org

CHALLENGING QUESTIONS FOR FURTHER THOUGHT 209
Figure H.5
Optical aid for "seeing" higher universes. (Invention by William Beaty.)
apart as shown in Fig. H.5. You can try other geometries as well, such as a 12" X
7" X 2" arrangement.) The mirrors are taped together at their edges to form a triangular
tube with a reflective inner surface. You look into the larger triangular end
as you gaze at your hand to see the flesh blobs. (If you were to look into the smaller
end at your friend, Beaty says you would see a "spherical glob-monster covered
with eyes.") This is a wonderful, fun classroom project, and William Beaty gives
practical tips for illumination and safety.
Magic Tesseract
Mathematician John Robert Hendricks has constructed a 4-D tesseract with magic
properties. Just as with traditional magic squares whose rows, columns, and diagonals
sum to the same number, this 4-D analog has the same kinds of properties in
four-space. Figure H.6 represents the projection of the 4-D cube onto the 2-D
plane of the paper. Each cubical "face" of the tesseract has six 2-D faces consisting
of 3 X 3 magic squares. (The cubes are warped in this projection in the same way
that the faces of a cube are warped when draw on a 2-D paper. ) To understand the
magic tesseract, look at the "1" in the upper left corner. The top forward-most
edge contains 1, 80, and 42, which sums to 123. The vertical columns, such as 1,
54, and 68 sum to 123. Each oblique line of three numbers—such as 1, 72, and