By Iakov Isaevich Khurgin
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Extra info for Did You Say Mathematics?
Example text
7 Now let us return to the triangle. In Fig. off a similar triangle, that is, one having angles. Aside from the properties of all they have the additional property that they lar. Now what does that mean? 42 9 we clip the same triangles, are simi- Fig. 8 Fig. 9 Fig. 10 Fig. 11 Fig. 12 Let us take a sheet of rubber and draw these similar triangles on it (Fig. 10). If the sheet of rubber is stretched lengthwise, the triangles will change but will remain similar (Fig. 11). ) Thus, similarity is a property that is preserved under uniform stretching in some direction.
But a doughnut is nothing but a sphere with a handle (Fig. 19). It looks like a simple weight-lifting device. 48 Fig. 17 Fig. 18 Let us return to transformations in the plane. Draw a cat (Fig. 20) and then another one with a straight line through it (Fig. 21). Now if we compress the entire figure uniformly onto the line, we get a fat cat. Note that in the process all points of the figure, except those of the straight line, are displaced. The latter points stay fixed. Now (Fig. take an arbitrary point 0 inside the figure and turn the cat about this point.
No matter what kind of tension we apply without tearing the sheet, the hole remains. We now have a good deal of observational material. What is there left in common to all these transformations of the rubber sheet? 46 MATHEMATICS AND ART Like art, mathematics takes note of phenomena in real life, combines analogous events, processes and facts, and then generalizes. People's Artist of the USSR, Obraztsov—the celebrated actor, painter and puppet-show man—shows us dogs and cats, and lions and rabbits and with their aid typifies (generalizes) certain humorous, pathetic or unpleasant qualities of human beings.