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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.

• 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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# 【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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