Reuleaux Polygons

Reuleaux polygons are shapes that have constant width just like a circle. In a circle, all diameters are of equal length. That is, all segments drawn through the circle's center are of constant length. This is why wheels have been used in transportation, machinery, energy and countless other uses for about 10,000 years. However, the earliest discovery of the wheel dates back to 3500 B.C. in Mesopotamia.

It wasn't until the 1800s that Franz Reuleaux, a German engineer, created a triangle with a curve at each side that had constant width and soon discovered that there are an infinite number of such shapes. 

I posed the question to the children, "Why are manhole covers all circular?" After the children manipulated the box I created to simulate a manhole cover, they realized that a square cover would be able to fall through. This is not very safe for the workers down below. Circles are used because they have constant width. Upon discovering that Reuleaux triangles and pentagons do not fall through either they were able to see that they also have constant width.

Then I asked them whether tires could be made out of Reuleaux shapes. They thought the ride would be quite bumpy. I created a track that the triangles and pentagons could roll underneath, and to their surprise, the ride was rather smooth and maintained parallel distance to the ground. I even showed them a picture of a bicycle invented in China. 

Reuleaux triangles are also used in certain car engines for pistons and as drill bits to create square holes; the children were particularly curious about the square hole and were amazed to watch it form into a square (unfortunately, the square is not perfect; it is slightly rounded at each vertex).

They spend the rest of class constructing equilateral triangles and Reuleaux figures with a compass and straight edge.

The attached pdf will allow them to explore more during the week.