Reuleaux polygons and tetrahedra
Certain 2D shapes are such that their height is the same regardless of their orientation on the plane. The obvious example is the circle, meaning that your bicycle runs smoothly down the road.
But there are an infinite number of other shapes that share this quality. For the UK contingent amongst you, the 50p and 20p pieces are examples.
You see, they’re not quite heptagons. They’re actually Reuleaux heptagons. That means that where they might have straight edges, the edges are instead slightly curved. And the point on the coin that is opposite that curved side lies at the centre of the circle of which the curve is an arc.
To prove it, take a ledge (e.g. the bottom lip of a laptop screen, or the point at which a radiator guard meets the wall). Place two 50p pieces upright on the ledge. And grab a ruler. Place the ruler across the top of the two coins and drag the coins left or right. You’ll find that the ruler remains the same distance from the ledge at all times.
Simply delightful. And yet more so when you discover that there are Reuleaux triangles!
Now you couldn’t make a bicycle using these shapes, as the centre moves around as the coin rolls. So while the highest point of the wheel would remain a fixed distance from the ground, the point of the axle would wobble go up and down relative to the ground, and so the ride would be a little bumpy.
Now, it gets better. As well as there being an infinite number of Reuleaux planes, there are also an infinite number of Reuleaux solids, which share similar properties. So you can balance a board on a bunch of them, roll it around, and the board will remain a fixed distance from the floor.
You can buy a bag of them here. *reaches for credit card*
(Oh. I also love the guy’s enthusiasm for the subject.)