Shapes Of Constant Width – The Hidden Properties Of The New Maths Scholar’s Pin Badge
Have you received your new Maths Scholar’s Pin Badge yet? These badges are new for 2020 and have been given to this year’s Scholars in recognition of their achievement in becoming a Maths Scholar.
Hopefully your badge has got you wondering – it looks nice, but what exactly is it? Is it just a fun shape or is there some deeper meaning? When a pupil asks about your pin badge you want to be able to explain the mathematics behind it. In fact, you might want to unclip it and start rolling it, as it is a shape of constant width! This is rather surprising, even if you have already read about shapes of constant width. At first glance, the Maths Scholar’s Pin Badge doesn’t look like a shape of constant width, mainly because it has an irregular pentagon at the centre, meaning that it is somewhat disguised.
Reuleaux Triangle
https://en.wikipedia.org/wiki/Reuleaux_triangle#/media/File:ReuleauxTriangle.svg
The most famous shape of constant width is the Reuleaux triangle. Take an equilateral triangle and then round off the straight edges. (Can you see how?) You are left with a shape which has a constant width. This means that if you place it between two parallel lines there will be a constant distance, regardless of which direction you do this from.
https://en.wikipedia.org/wiki/Reuleaux_triangle#/media/File:Reuleaux_supporting_lines.svg
A shape of constant width can roll along between two parallel lines just like a circle could. In fact it calls into question the very definition of roundness, a concept which is explored thoroughly in the book How round is your circle by John Bryant and Chris Sangwin.
Shapes of constant width as wheels
It is only natural to want to use shapes of constant width as wheels. One of the best examples is in this bicycle which was invented in China. This is obviously a novelty, but there are other serious uses of curves of constant width, such as in vending machines where it is important that a coin has a constant diameter.
Beyond the Reuleaux Triangle
There are many other shapes of constant width. In fact, it is possible to construct a curve of constant width based on any triangle. Take a look at this demonstration.
https://demonstrations.wolfram.com/ArbitraryCurvesOfConstantWidth/
If you want your pupils to have a go, then here is further information on how to construct a curve of constant width based on any triangle.
Other polygons
It is actually possible to construct a Reuleaux Polygon which is based on any regular polygon which has an odd number of sides. If you take a regular polygon such as a heptagon, you can construct the shape of the 50p coin – we are all more familiar with shapes of constant width than we think!
It is also possible to construct curves of constant width which are based on any irregular polygon with an odd number of sides. This is where the Maths Scholar’s Pin Badge is derived from. Take a look at the turquoise pentagon at the centre – this is the basis for the outer curve. Imagine a regular Reuleaux pentagon then deform it in your mind – can you now see its relationship to your pin badge?
This is shown clearly in this YouTube video. The whole video is really interesting but if you are short of time start watching from 4 minutes in.
Solids of constant width
If we are being honest, it is quite hard to roll your pin badge along, mainly because it keeps slipping over when placed between two planes, as it isn’t secured by any kind of axle. This is why it is also interesting to show your pupils solids of constant width which are the 3D versions of these amazing shapes. With 3D solids of constant width, it is much easier to see them roll.
Uses in the classroom of shapes of constant width
Shapes of constant width have endless uses in the classroom. They have a wow factor – they go against all common sense and reason, and they have practical uses in the real world. They can be used as part of an investigation and are also great for practicing construction skills and geometrical understanding. Use your Maths Scholar’s Pin Badge as a starting point – and perhaps even go on to purchase some solids of constant width for your classroom?
Shapes of constant width have some really deep mathematical properties and the more you look, the more you will find. Barbier’s Theorem even states that that all curves of constant width have a perimeter which is Pi times the width, regardless of the type of shape. If you measure your pin badge, it has a width of 3cm, meaning that according to Barbier’s Theorem it has a perimeter of 3πcm. Even more similar to a circle than first thought!
Luckily for us there are some free to download teaching resources on the Think Maths Website:
https://www.think-maths.co.uk/downloads/shapes-constant-width
There is also a great Numberphile video which will show your pupils how to stand on solids of constant width (so you don’t have to).
And finally here is a link to a very old book called Mathematical Models which might be useful for your own understanding.
https://archive.org/details/MathematicalModels-/page/n215/mode/2up