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  • Why are most manhole covers round?

  • Sure, it makes them easy to roll and slide into place in any alignment

  • but there's another more compelling reason

  • involving a peculiar geometric property of circles and other shapes.

  • Imagine a square separating two parallel lines.

  • As it rotates, the lines first push apart, then come back together.

  • But try this with a circle

  • and the lines stay exactly the same distance apart,

  • the diameter of the circle.

  • This makes the circle unlike the square,

  • a mathematical shape called a curve of constant width.

  • Another shape with this property is the Reuleaux triangle.

  • To create one, start with an equilateral triangle,

  • then make one of the vertices the center of a circle that touches the other two.

  • Draw two more circles in the same way, centered on the other two vertices,

  • and there it is, in the space where they all overlap.

  • Because Reuleaux triangles can rotate between parallel lines

  • without changing their distance,

  • they can work as wheels, provided a little creative engineering.

  • And if you rotate one while rolling its midpoint in a nearly circular path,

  • its perimeter traces out a square with rounded corners,

  • allowing triangular drill bits to carve out square holes.

  • Any polygon with an odd number of sides

  • can be used to generate a curve of constant width

  • using the same method we applied earlier,

  • though there are many others that aren't made in this way.

  • For example, if you roll any curve of constant width around another,

  • you'll make a third one.

  • This collection of pointy curves fascinates mathematicians.

  • They've given us Barbier's theorem,

  • which says that the perimeter of any curve of constant width,

  • not just a circle, equals pi times the diameter.

  • Another theorem tells us that if you had a bunch of curves of constant width

  • with the same width,

  • they would all have the same perimeter,

  • but the Reuleaux triangle would have the smallest area.

  • The circle, which is effectively a Reuleaux polygon

  • with an infinite number of sides, has the largest.

  • In three dimensions, we can make surfaces of constant width,

  • like the Reuleaux tetrahedron,

  • formed by taking a tetrahedron,

  • expanding a sphere from each vertex until it touches the opposite vertices,

  • and throwing everything away except the region where they overlap.

  • Surfaces of constant width

  • maintain a constant distance between two parallel planes.

  • So you could throw a bunch of Reuleaux tetrahedra on the floor,

  • and slide a board across them as smoothly as if they were marbles.

  • Now back to manhole covers.

  • A square manhole cover's short edge

  • could line up with the wider part of the hole and fall right in.

  • But a curve of constant width won't fall in any orientation.

  • Usually they're circular, but keep your eyes open,

  • and you just might come across a Reuleaux triangle manhole.

Why are most manhole covers round?

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B1 US TED-Ed width constant manhole circle curve

【TED-Ed】Why are manhole covers round? - Marc Chamberland

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    稲葉白兎 posted on 2015/04/20
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