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Why are most manhole covers round?
为什么大多数井盖是圆的？
Sure, it makes them easy to roll and slide into place in any alignment
当然，这使它们容易滚动和滑入任何对齐的位置；
but there's another more compelling reason involving a peculiar geometric property of circles and other shapes.
但是还有其它的更令人信服的原因，这涉及圆和其他形状的一种特殊的几何特性。
Imagine a square separating two parallel lines.
想象一个正方形分开两条平行线。
As it rotates, the lines first push apart, then come back together.
当它旋转时，线先是推动分开，然后复位。
But try this with a circle and the lines stay exactly the same distance apart, the diameter of the circle.
但是用圆来做尝试，线跟线之间会保持完全相同的距离，这就是圆的直径。
This makes the circle unlike the square, a mathematical shape called a curve of constant width.
这使得圆不同于正方形，是一种称作定宽曲线的数学型态。
Another shape with this property is the Reuleaux triangle.
另外一种拥有此性质的形状是勒洛三角形。
To create one, start with an equilateral triangle,
第一步创建一个等边三角形，
then make one of the vertices the center of a circle that touches the other two.
然后以其中一个顶点为圆心，过其余两顶点作图。
Draw two more circles in the same way, centered on the other two vertices,
分别以其余两个顶点为圆心，按同样的方式作出另外的两个圆，
and there it is, in the space where they all overlap.
它们的重迭区域就为勒洛三角形。
Because Reuleaux triangles can rotate between parallel lines without changing their distance,
因为勒洛三角形可以在平行线间旋转且不改变线的间距，
they can work as wheels, provided a little creative engineering.
他们也可以作为轮子，只需要一点创意。
And if you rotate one while rolling its midpoint in a nearly circular path,
如果你在旋转它的同时，使它的中心在一个近圆形的路径上转动，
its perimeter traces out a square with rounded corners,
它的周界轨迹会是一个圆角正方形，
allowing triangular drill bits to carve out square holes.
这使三角形的钻头能够挖出方形的孔。
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,
任何有奇数条边的多边形都可以被用来生成定宽曲线，使用与我们之前应用的同样的方法，
though there are many others that aren't made in this way.
不过，还有其他的定宽曲线并不是用这种方式生成的。
For example, if you roll any curve of constant width around another, you'll make a third one.
例如，如果你使任一定宽曲线绕另一定宽曲线转动，你将生成第三个定宽曲线。
This collection of pointy curves fascinates mathematicians.
这组有尖头的曲线使数学家着迷。
They've given us Barbier's theorem, which says that the perimeter of any curve of constant width,
他们把这个称为巴比尔定理，即任何定宽曲线的周长，
not just a circle, equals pi times the diameter.
不仅仅是圆，等于π*直径。
Another theorem tells us that if you had a bunch of curves of constant width with the same width,
另外一个定理告诉我们：如果你有一堆定宽曲线，宽度相同，
they would all have the same perimeter, but the Reuleaux triangle would have the smallest area.
他们也会有同样的周长，但是勒洛三角形会有最小的面积。
The circle, which is effectively a Reuleaux polygon with an infinite number of sides, has the largest.
圆是一个有效的勒洛正多边形，有无数条边，有最大的面积。
In three dimensions, we can make surfaces of constant width, like the Reuleaux tetrahedron,
在三维空间，我们可以生成定宽面，比如勒洛四面体，
formed by taking a tetrahedron, expanding a sphere from each vertex until it touches the opposite vertices,
把一个四面体分别从每个顶点扩展一个触及相对顶点的球面，
and throwing everything away except the region where they overlap.
去除重迭部位以外的区域。
Surfaces of constant width maintain a constant distance between two parallel planes.
定宽面使两平面间保持恒定的距离。
So you could throw a bunch of Reuleaux tetrahedra on the floor,
所以你可以在地上扔一堆勒洛四面体，
and slide a board across them as smoothly as if they were marbles.
把它们当成弹珠一样平滑地滑过它们。
Now back to manhole covers.
现在回到井盖。
A square manhole cover's short edge could line up with the wider part of the hole and fall right in.
方形井盖的短边会与洞孔较宽的部分对齐，掉进去。
But a curve of constant width won't fall in any orientation.
但定宽曲线的井盖不会从任何方向掉进去。
Usually they're circular, but keep your eyes open, and you just might come across a Reuleaux triangle manhole.
它们通常是圆型的，但是留意身边，你可能会无意中发现一个勒洛三角形的检修孔。
