Reinventing the Wheel: A Conceptual Approach for Mathematics

Mathematics goes far beyond numbers in equations. It is a set of ideas. The philosopher-mathematician Poincare said that:

Math is the science of patterns. It happened whenever there is a pattern, whenever something is repeated in our daily lives on the street during traffic. These are ideas that materialize in all objects ever observed, and perhaps you never realized. Math is life.

It’s common knowledge that mathematics is everywhere, and I want to write about manhole covers, coins, and bicycle wheels. What do these three items have in common? Easy, they’re circles.

A circular shape seems perfect to these uses, but I’ll show you that others could replace them.

We can reinvent the wheel. And it will be math.

You may have noticed that streets are flooded with sewage covers providing access to a maze of underground connections through which one drainage water, power grids and communications, and rats. Have you ever wondered why almost all sewage covers round? A circular shape has one exciting feature: its width is independent of the position in which the shape is found. This width is what one usually calls the diameter of the circumference, or in this case of the lid.

Therein mind that the same isn’t right for other geometrical shapes?

For example, the length of the one side of the equilateral triangle below measures 51 centimeters, but the height is only 44 centimeters. The length of the one side of the square below is 62 centimeters. But if you measure the distance between 2 cross edges, you will get 88 centimeters. On the other hand, the circle is always the same width regardless of position.

However, is the circle the only geometrical shape having this feature?

The British 50 pence coin is one of the best-known shapes with this property. This coin is not circular, but still, it has a constant width regardless of its position. The 50 pence coin is what we call a shape of constant width.

To better understand these shapes let’s do the following. Draw an equilateral triangle, which means a triangle where all signs are equal. Now with the help of a pair of compasses, draw a piece of a circle centered on one vertex and crossing the other two. Then do the same on the different sides. This new shape now appears to have the feature we intended. Being the width of it is independent of its position. The 50 pence coin is obtained in a very similar way starting with seven equal sides polygon. Choosing that particular shape for coins wasn’t done on a whim, though.

The fact that height is independent of its position. It’s very important to prevent coins from jamming when inserted into a vending machine. Furthermore, it’s much easier for mechanics to identify the coin. In a way, the brick to design the 50 pence coin reinvented the wheel.

These shapes of constant width are not just mathematical trivia. And they serve for much more than shaping currency. For example, they’re used in mechanical engineering to transform rotary motion into translatory motion. They can also be used in some contraption.

It is a drill that drills square holes. When one of these shapes wheels inside that square, it’s center also describes a square, which allows making a hole anything but trivial.

And what about the wheels of this eccentric bike created by a Chinese inventor. Mind that although the wheels are not circular, this bike remains steady while riding. One could say that this Chinese inventor, as did the British coin designer, also reinvented the wheel.

But then what’s the connection between the 50 pence coin, those Chinese bike wheels, and manhole covers?

With their circular shape, manhole covers are always the same width regardless of their position, which prevents them from falling down the hole. A round manhole cover could not fall through its circular opening, whereas a square manhole cover might fall in if it were inserted diagonally in the hole. A Reuleaux triangle or other curve of constant width would also serve this purpose, but round covers are much easier to manufacture.

What about square lids? The square lid is not a shape of constant width. It wouldn’t be pretty if this thing fell on them. It’s also particularly important when these lids lay within a vehicle circulation area where they can jump out by accident, or even fall during maintenance work.

In short, a circular manhole cover is a shape of constant width, as is the 50 pence coin and the eccentric will. So what keeps a sewage lid from being the same shape as that Chinese bicycle will. Nothing really… There’s an idea for a local mayor who wants to be original. He’s a madman.

I found tons of books in Argentina!

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