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