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topology
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| Topology explores the properties of geometric shapes that have been deformed but not torn. If A1 and B1 are folded so that the arrows point in the same direction, a collar A2 and a twisted band B2 result. A2 has an inner and outer surface and two edges, but B2 has only one of each – precisely because of its twist. B2 is called a Möbius band and, unlike A2, a path beginning at S and traced round the band will, after one revolution, end on the other "side" of the band at T. A Möbius band stretched as in C1 is distorted further into one half of a Klein bottle (C2), which again has only one edge and one surface. Compare with a normal bottle (C3).
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The word comes from the Greek topos for "place," and was introduced into English by Solomon Lefschetz in the late 1920s.
A topologist has been described as someone who doesn't know the difference between a doughnut and a coffee cup. Substitute "care about" for "know" and this becomes more accurate.
Imagine a doughnut-shape, or torus, made of soft clay. A potter can easily shape this into a cup with a handle without removing or creating any new holes. Both shapes, in topology, are said to be genus 1 – objects with a single hole. A sphere, by contrast, is genus 0 (no holes), while an eyeglass frame, with the lenses removed, is genus 2.
For more on topologically intriguing structures, see Möbius band and Klein bottle.
Related entry
• algebraic topology
Related categories
• TOPOLOGY
• SPACE AND TIME
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