clifford_a-_pickover_surfing_through_hyperspacebookfi-org

186 appendix c
Figure C.I
Banchoff Klein bottle. (Computer rendition by author.)
computer program. Alternatively, here's a handy formula for viewing the curves at
any angle:
where (x,y,z) are the coordinates of the point on the curve prior to projection and
(9, 4>) are the viewing angles in spherical coordinates.
The Banchoff Klein bottle 1 (Fig. C.I and C.2) is based on the Mobius band, a
surface with only one edge. The Mobius band is an example of a nonorientable
space, which means that it is not possible to distinguish an object on the surface
from its reflected image in a mirror. This Klein bottle contains Mobius bands and
can be built in 4-D space. Powerful graphics computers allow us to design unusual
objects such as these and then investigate them by projecting them in a 2-D image.
Some physicists and astronomers have postulated that the large-scale structure of