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

當然，這使它們容易滾動和滑入任何的位置

but there's another more compelling reason

但是還有其他更令人信服的原因

involving a peculiar geometric property of circles and other shapes.

這涉及圓和其他形狀的一種特殊的幾何特性

Imagine a square separating two parallel lines.

想像一個正方形分開兩條平行線

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

當它旋轉時，線先是推動分開，然後復位

But try this with a circle

但是用圓來做嘗試

and the lines stay exactly the same distance apart,

線跟線之間會保持完全相同的距離

the diameter of the circle.

這就是圓的直徑

This makes the circle unlike the square,

這使得圓不同於正方形

a mathematical shape called a curve of constant width.

是一種稱作定寬曲線的數學型態

Another shape with this property is the Reuleaux triangle.

另外一種擁有此性質的形狀是魯洛三角形

To create one, start with an equilateral triangle,

第一步創建一個等邊三角形

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

然後以其中一個頂點為圓心 過其餘兩頂點作圖

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

分別以其餘兩個頂點為圓心 按同樣的方式作出另外的兩個圓

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

它們的重疊區域就為魯洛三角形

Because Reuleaux triangles can rotate between parallel lines

因為魯洛三角形可以在平行線間旋轉

without changing their distance,

且不改變線的間距

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

他們也可以作為輪子，只需要一點創意

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

如果你在旋轉它的同時 使它的中心在一個近圓形的路徑上轉動

its perimeter traces out a square with rounded corners,

它的周界軌跡會是一個圓角正方形

allowing triangular drill bits to carve out square holes.

這使三角形的鑽頭能夠挖出方形的孔

Any polygon with an odd number of sides

任何有奇數條邊的多邊形

can be used to generate a curve of constant width

都可以被用來生成等定寬曲線

using the same method we applied earlier,

使用與我們之前應用的同樣的方法

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

不過，還有其他的定寬曲線 並不是用這種方式生成的

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

例如，如果你使任一定寬曲線繞另一定寬曲線轉動

you'll make a third one.

你將生成第三個定寬曲線

This collection of pointy curves fascinates mathematicians.

這組有尖頭的曲線使數學家著迷

They've given us Barbier's theorem,

他們把這個稱為巴比爾定律

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

任何定寬曲線的周長

not just a circle, equals pi times the diameter.

不僅僅是圓，等於 π *直徑

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

另外一個定理告訴我們:如果你有一堆定寬曲線

with the same width,

寬度相同

they would all have the same perimeter,

他們也會有同樣的周長

but the Reuleaux triangle would have the smallest area.

但是魯洛三角形會有最小的面積

The circle, which is effectively a Reuleaux polygon

圓是一個有效的魯洛正多邊形

with an infinite number of sides, has the largest.

有無數條邊，有最大的面積

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

在三維空間，我們可以生成定寬面

like the Reuleaux tetrahedron,

比如魯洛四面體

formed by taking a tetrahedron,

把一個四面體

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

分別從每個頂點擴展一個觸及相對頂點的球面

and throwing everything away except the region where they overlap.

去除重疊部位以外的區域

Surfaces of constant width

定寬面

maintain a constant distance between two parallel planes.

使兩平面間保持恆定的距離

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

所以你可以在地上扔一堆魯洛四面體

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

把它們當成彈珠一樣平滑地滑過它們

Now back to manhole covers.

現在回到井蓋

A square manhole cover's short edge

方形井蓋的短邊

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

會與洞孔較寬的部分對其，掉進去

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

但定寬曲線的井蓋不會從任何方向掉進去

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

它們通常是圓型的，但是留意身邊

and you just might come across a Reuleaux triangle manhole.

你可能會無意中發現一個魯洛三角形的檢修孔