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Why are most manhole covers round?

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Sure, it makes them easy to roll[br]and slide into place in any alignment

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but there's another more compelling reason

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involving a peculiar geometric property[br]of circles and other shapes.

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Imagine a square [br]separating two parallel lines.

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As it rotates, the lines first push apart,[br]then come back together.

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But try this with a circle

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and the lines stay [br]exactly the same distance apart,

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the diameter of the circle.

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This makes the circle unlike the square,

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a mathematical shape [br]called a curve of constant width.

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Another shape with this property[br]is the Reuleaux triangle.

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To create one, [br]start with an equilateral triangle,

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then make one of the vertices the center[br]of a circle that touches the other two.

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Draw two more circles in the same way,[br]centered on the other two vertices,

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and there it is, in the space [br]where they all overlap.

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Because Reuleaux triangles can rotate[br]between parallel lines

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without changing their distance,

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they can work as wheels,[br]provided a little creative engineering.

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And if you rotate one while rolling[br]its midpoint in a nearly circular path,

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its perimeter traces out a square[br]with rounded corners,

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allowing triangular drill bits[br]to carve out square holes.

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Any polygon with an odd number of sides

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can be used to generate [br]a curve of constant width

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using the same method we applied earlier,

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though there are many others[br]that aren't made in this way.

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For example, if you roll any [br]curve of constant width around another,

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you'll make a third one.

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This collection of pointy curves[br]fascinates mathematicians.

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They've given us Barbier's theorem,

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which says that the perimeter [br]of any curve of constant width,

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not just a circle,[br]equals pi times the diameter.

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Another theorem tells us that if you had[br]a bunch of curves of constant width

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with the same width,

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they would all have the same perimeter,

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but the Reuleaux triangle [br]would have the smallest area.

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The circle, which is effectively [br]a Reuleaux polygon

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with an infinite number of sides,[br]has the largest.

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In three dimensions, we can make [br]surfaces of constant width,

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like the Reuleaux tetrahedron,

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formed by taking a tetrahedron,

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expanding a sphere from each vertex[br]until it touches the opposite vertices,

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and throwing everything away[br]except the region where they overlap.

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Surfaces of constant width

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maintain a constant distance [br]between two parallel planes.

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So you could throw a bunch [br]of Reuleaux tetrahedra on the floor,

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and slide a board across them[br]as smoothly as if they were marbles.

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Now back to manhole covers.

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A square manhole cover's short edge

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could line up with the wider part [br]of the hole and fall right in.

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But a curve of constant width[br]won't fall in any orientation.

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Usually they're circular, [br]but keep your eyes open,

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and you just might come across [br]a Reuleaux triangle manhole.