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Phoenix666 [soylentnews.org] writes:

What follows is an essay adapted from a talk, delivered in 2010 to teenagers and parents in my hometown of Cupertino, California. The talk concerned the surfaces of things, like bodies and planets, and abstract surfaces, like the Möbius strip and the torus. The goal was to learn about the world by studying surfaces mathematically, to learn about mathematics by studying the way we study surfaces, and, ideally, to learn about ourselves by studying how we do mathematics.
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  Say you begin with a ball of clay. Its surface is a sphere. You toss it on a potter’s wheel, flattening the base a bit. As the wheel spins, you press on the sides, then depress the center, then squeeze the walls you have formed a bowl. Since each change was so mild, you must never have changed the fundamental shape of the object.1 We must say that the surface of a ball is the same shape as the surface of a bowl.

The branch of mathematics concerned with the study of shape and space is called topology (from ancient Greek topos, meaning place). An oft-repeated and never funny joke is that a topologist cannot tell a coffee mug from a donut. This is not true. But it is true that mathematicians abstract the sensory qualities from a thing (the heft of the mug, the sweetness of the donut). And if both mug and donut were made of a perfectly malleable substance, you might expand the base of the mug to fill up its volume, then shrink the resulting thick cylinder down until it's the same width as the handle, leaving a torus. It’s not that topologists think it desirable that a donut be a coffee mug. It’s that, having promised to allow small deformations and being honorable folk, they are obligated to admit that the mug and donut have the same shape. 2

An accessible article [hypocritereader.com] for those with a casual interest in math.

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