clifford_a-_pickover_surfing_through_hyperspacebookfi-org

appendix c
banchoff klein bottle
It is true that we are all at every moment situated simultaneously in all
the spaces which together constitute the universe of spaces; for whenever
there is disclosed to us the existence of a space which had previously
been concealed from us, we know from the very first moment
that this space has not just come into being, but that it had always surrounded
us without our noticing it. Yet, nevertheless, we are not ourselves
able to force open the gate which leads to a space that has so far
been closed to us.
—Karl Heim, Christian Faith and Natural Science
In the last decade, even serious mathematicians have begun to enjoy and present
bizarre mathematical patterns in new ways—ways sometimes dictated as much by
a sense of aesthetics as by the needs of logic. Moreover, computer graphics allow
nonmathematicians to better appreciate the complicated and interesting graphical
behavior of simple formulas.
This appendix provides a recipe for creating a beautiful graphics gallery of
mathematical surfaces. To produce these curves, I place spheres at locations determined
by formulas that are implemented as computer algorithms. Many of you
may find difficulty in drawing shaded spheres; however, quite attractive and informative
figures can be drawn simply by placing colored dots at these same locations.
Alternatively, just put black dots on a white background. As you implement the
following descriptions, change the formulas slightly to see the graphic and artistic
results. Don't let the complicated-looking formulas scare you. They're very easy to
implement in the computer language of your choice by following the computer
recipes and computational hints given in the program outlines.
Unlike the curves you may have seen in geometry books (such as bullet-shaped
paraboloids and saddle surfaces) that are simple functions of x and y, certain surfaces
occupying three dimensions can be expressed by parametric equations of the
form: x = f(u,v), y - g(u,v), z = h(u,v). This means that the position of a point in
the third dimension is determined by three separate formulas. Because^g, and h
can be anything you like, the remarkable panoply of art forms made possible by
plotting these surfaces is quite large. For simplicity, you can plot projections of
these surfaces in the x-y plane simply by plotting (x,y) as you iterate u and v in a
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