The story of everting spheres in 3-space by regular homotopies is the case history of a nontrivial visualization problem of remarkable complexity and compelling beauty. The task is to show the motion of a spherical surface through itself in space so that, without tearing or creasing, the surface is turned inside out. That so many different graphical methods were applied to the same problem, when it is more in the nature of mathematics to display the versatility of one method by applying it to a variety of different problems, makes it a paradigm for descriptive topology. It is premature to formulate explicit ground rules for comparing the many solutions to this graphical problem. Instead let me tell a very short story about golden rectangles instead. Although the existence of golden rectangles has no direct bearing on eversions, the universal currency of this topic among mathematicians makes for a good example of the “graphical criticism” I have in mind.
KeywordsKlein Bottle Descriptive Topology Dehn Twist Picture Plane Pinch Point
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