Reuleaux's triangle... and then some

So what makes a circle so important? Firstly, it is the shape with the highest possible area to volume ratio... in other words, it has the highest area for the least volume. This is one thing that makes circles unique. What I'm going to talk about is something else, that the circle has constant width. That means that there is a set maximum distance between any 2 points, and you can make that distance given any starting point. At first glance, this also seems like something which is unique among shapes since the circle has rotational symmetry. The green lines are other lines which have a total length of less than the red line. The red line represents the maximum width.

The awesome thing is that there are other shapes with constant width, called Reuleaux polygons. These were made famous by Franz Reuleaux, a German engineer. The shape below, the Reuleaux triangle, is the most famous example.

The shape in between the circles is the shape we're looking for. If you pick any spot on the triangle's perimeter, it has a constant width as discussed before.

 

The trick behind the shape is that, as illustrated in the second picture, each corner is actually the centre of a circle, and each side a part of a circle's perimeter. This means that all of the orange lines are the same length, since they are each a radius of one circle. Imagine an orange line sweeping across from one corner to another, and you should be able to visualise it. The pink lines show shorter lengths that you can find. The other side to this is that if you pick any point which isn't a corner, the longest distance across is always between that point and the opposite corner.

This is pretty cool. While there are an infinite number of other shapes which do it (by performing the same trick of rotating a radius of a circle around corners), this is the most famous. The question now is, what's the use of this? Well for starters, manholes need to have constant width so that the lid can't fall down the manhole. Reuleaux triangles can do this. Trust San Francisco to have a Reuleaux triangle manholes :D (above).

The last thing I'll show you is a 3d equivalent of a Reuleaux triangle. A Reuleaux triangle-shaped manhole would roll smoothly, like a circle, due to its constant width. If you get a reuleaux triangle and rotate it around a central axis (as below), you get the shape below. The red lines show the axis of rotation.

If you want to see these shapes in action (and i recommend it highly, it's pretty cool) check out this youtube video. I've had a play with 3d printed versions before and they're mindblowing. Also, check out the other videos this guy's made, they really mess with what you think you know about shapes.

I guess the point of all this stuff is that there's more to geometry than meets the eye. There's certainly more behind these shapes than I could possibly write about in one post (look at the size of this monster already). I expect my house will be full of mathematical curiosities like this by my thirties... certainly more interesting to have on your coffee table than a bunch of fashion magasines and barely touched books about the antarctic.