Connections

Lets start with the classic question, "Why are manhole covers round?"

The standard answer is that a circular cover can't fall through the opening. There is no way of orienting a round lid so that it fits through a round hole of a slightly smaller (due to a taper or lip) size. This is because the diameter of a circle is the same no matter how it is oriented.

There is a whole family of these constant width shapes called Reuleaux Curves, named after a late nineteenth century engineer, Franz Reuleaux.

The simplest of them, a Reuleaux Triangle can be created by taking an equilateral triangle and joining each adjacent vertex with an arc centred on the opposite vertex. Like a circle, a Reuleaux triangle fits snugly inside a square having sides equal to the curve's width no matter which way the triangle is turned. The rounded triangle can rotate freely inside the square without ever having any room to spare (though the center of the triangle does not stay in one place). This attribute has been exploited by one manufacturer that has created a drilling system that lets one drill square holes or mortises.

A more well known use of the Reuleaux Triangle is in the Wankel Rotary Engine, currently used by Mazda in their RX8. In this clever rotary engine design, the constant width attribute of the Reuleaux Triangle shaped rotor allows it to maintain contact with the cylinder sides while rotating. The cylinder walls are not perfectly straight lines, however, as the rotor is rotated around a circular axis, rather than "squished circle" path shown in the animation.

As mentioned before, there is a while family of constant width curves. They can be constructed around an infinite family of odd-sided polygons. Below are several examples, including ones based on a regular pentagon, regular heptagon, and an irregular pentagon.

Many faceted coins, including the Canadian dollar coin, termed a "Loonie" are actually fabricated as Reuleaux curves. This allows them to roll down fixed width channels without jamming. The large majority of faceted coins have an odd number of sides as a Reuleaux curve can not have an even number of vertices.

Getting back to manholes, according to the Straight Dope, an equilateral triangle would also not fall through a slightly hole. I found one manufacturer that does indeed make triangular manholes and covers! My speculation is that the real reason most manhole covers are round is that they are heavy, and a round cover needs no special attention to orientation when putting it down, making for much less effort than having to align any other shaped cover with its frame!