Almost a Reuleaux, real Reuleaux and 3D Reuleaux

I saw this Nokia trinket on a colleague's cellphone. It's close to a Reuleaux triangle, but not quite. Darn! A missed opportunit from the folks at Nokia.

A Reuleaux triangle is a shape with constant width. If you roll it, you can set a board on top of it and the board will not bob up and down, it will stay at the same height just as if the roller were circular. You can make a triangle like this easily, just draw three circles that pass through the centers of each other, and the region in the middle is a Reuleaux triangle as shown here:

You can make these with 5, 7, 9 or any odd number of segments by constructing arcs on regular polygons.

George Hart made these solids of uniform width. "Reuleaux acorns" he calls them. They're a 3d generalization of Reuleaux triangles - if you place a flat board on top of them they will roll around smoothly and the board won't wobble up and down. Weird...

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