Why are manhole covers round? - Marc Chamberland

为什么井盖是圆的？- Marc Chamberland

翻译人员: Cissy Yun 校对人员: Jenny Yang

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

但是还有其它的更令人信服的原因

involving a peculiar geometric property of circles and other shapes.

涉及圆和其他形状的一种特殊的几何属性

想象一个正方形分开两条平行线，

Imagine a square separating two parallel lines.

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

但是用圆来做尝试

and the lines stay exactly the same distance apart,

线间保持完全相同的距离

the diameter of the circle.

--为圆的直径

This makes the circle unlike the square,

这使得圆不同于正方形，

a mathematical shape called a curve of constant width.

是一种称作定宽曲线的数学形态

Another shape with this property is the Reuleaux triangle.

另外一种拥有此性质的形状是鲁洛三角形

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，过其余两个顶点作圆

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.

它们的重叠区域为鲁洛三角

Because Reuleaux triangles can rotate between parallel lines

因为鲁洛三角形可以在平行线间旋转，

without changing their distance,

且不改变线的间距，

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

它们可以作为轮子，只要一点创造性的工程

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

如果你旋转它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,

它的周界轨迹为一个圆角正方形，

allowing triangular drill bits to carve out square holes.

这使三角形的钻头能够剜出方形的孔

Any polygon with an odd number of sides

任何有奇数条边的多边形

can be used to generate a curve of constant width

都可以被用来生成等定宽曲线，

using the same method we applied earlier,

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，还有其他的一些(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,

例如，如果你使任一定宽曲线绕另一定宽曲线转动，

you'll make a third one.

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,

他们给出了巴比尔定理，

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

-- 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

另外一个定理告诉我们：如果你有一群定宽曲线，

with the same width,

宽度相同，

they would all have the same perimeter,

他们会有同样的周长，

but the Reuleaux triangle would have the smallest area.

但是鲁洛三角形会有最小的面积；

The circle, which is effectively a Reuleaux polygon

圆是一个有效的鲁洛正多边形，

with an infinite number of sides, has the largest.

有无数条的边，有最大的面积

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

在三维空间，我们可以生成定宽面，

like the Reuleaux tetrahedron,

比如鲁洛四面体，

formed by taking a tetrahedron,

由把一个四面体，

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

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.

去除重叠部位以外的区域。

Surfaces of constant width

定宽面

maintain a constant distance between two parallel planes.

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鲁洛四面体，

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

就像它们是弹珠一样平滑地滑板通过它们

Now back to manhole covers.

现在回到井盖

A square manhole cover's short edge

方形井盖的短边

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

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(该形状的井盖)不会从任何方向掉进去

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

它们通常是圆形的，但是留意身边，

and you just might come across a Reuleaux triangle manhole.

你可能会无意中发现一个鲁洛三角形的检修孔