Solids of Constant Width
Most UK coins are circular, but two coins are seven-sided, the 50 pence coin (shown above) and the smaller 20 pence coin. The reason for their shape is partly to help distinguish the coins from other (circular) coins of a similar size. However the coin is not a regular heptagon, since each of the seven sides is curved not flat. In fact the coin is an equilaterally curved heptagon, or Reuleaux polygon (after Franz Reuleaux, a 19th century German engineer).
The simplest Reuleaux shape is the Reuleaux triangle. This is based on an equilateral triangle, except that each side is a curve, the centre of which is the opposite point of the triangle. The orange shaded figure in the diagram below is a Reuleaux triangle.
A Reuleaux triangle is interesting because it has a constant width - wherever you choose to measure it the diameter will always be the same. So if you constructed a cylinder with the cross-section of a Reuleaux triangle, and placed it under a flat object, you could use the Reuleaux cylinder to slide the object along, and the cylinder would behave in a similar way to a cylinder of circular cross-section - the flat object would slide along on a level plane, and would not go up or down. This is generally referred to as a Reuleaux Rotor.
This feature of constant diameter also applies to other Reuleaux shapes. This is an important feature for the British 50 pence and 20 pence coins, for example, when it comes to using the coins in slot machines or vending machines. These machines would have great difficulty in validating the coins if their width varied. And provided you could find a way of keeping them vertical, you could use the 50 pence piece as a wheel. However what you cannot do is insert an axle. The minute you do this, you would find that your wheel would bump up and down.
Although these shapes may seem a little esoteric, they have a range of real world applications, in addition to being used for coins. They can be used for drilling square holes (or indeed hexagonal holes, and other shapes besides) and they also appear in rotary internal combustion engines.
By rotating a Reuleaux shape about an axis of symmetry, it is possible to create a solid of constant width.These unusual objects can be used for quite a surprising demonstration. Place three of them on a table, in a rough triange. You can then balance a book on top of them. You can then slide the book in any direction you like, and it will move absolutely smoothly, with no up or down movement. Any onlooker will be convinced that the only shapes that do this will be three spheres.
What does change however is that the centre of gravity of each of the solids will move up and down - when the solid is balancing on the tip, for example, the centre of gravity will be higher than when it is resting on the flatter side. One consequence of this is that the solids of constant width will work better fro demonstration purposes if they are made of a light material such as aluminium or plastic. If they are made of a heavy material such as brass, one is conscious of the effort needed when the centre of gravity moves upwards, and this can make it seem as if the rolling action is not smooth.
Since writing this article, we have received a lot of requests for Solids of Constant Width. We didn't find anyone making them, so we have made some ourselves, and we now offer these for sale in the Grand Illusions Toy Shop.
This video was created by Dr Chris Sangwin, whose book 'How Round Is Your Circle?' inspired this article. The book explores how many everyday engineering ideas actually work, and explains the underlying mathematics.