I guess it's now time for one (or more) of us to invent square wheels.
Too late! I have these in my files, but still haven't put them on the
WWW. I did put up http://www.amasci.com/exhibits/wheels.html in 1994. A
couple of people have yelled at me since then, but it still didn't cause
me to get off my butt and create a build-it article.
Imagine a cube. Inflate it a bit so it has a slight "pincusion" shape.
Now drill a hole through it, through one pair of diagonal vertices. Stick
two of these pincusioned-cubes on an axle. If the elliptical curve of the
"pincusioned" edges are right, then this device will roll perfectly
smoothly, like a glass ball.
Or, recall that if you view a cube along its diagonal vertices, you see a
hexagon. If you bend the edges of the cube outwards a little, then the
hexagon becomes a circle. Circles can roll!
Or, imagine the volume which is created by the intersection of four
cylinders passing through a central point. Now grind the pointed pyramids
of this volume down, leaving only the "cube" edges. The result looks like
a slightly-swollen cube. The edges are segments of ellipses.
In 1990 we had a Radio Shack RC car with four polished plexiglass "square
wheels" running around the lab. A pair of "square wheels" on an axle
makes a great physics-paperweight. Triangular wheels look even stranger.
I wonder, was this my independant invention? Or is it already well known?
I've never stumbled across any papers on "square wheels" since I started
playing with them in 1994. I wouldn't be suprised if there was some
obscure article about them in a journal somewhere.
Similar things can be done with a tetrahedron (triangular wheels), or
most any polyhedron.
I came up with this while trying to think of some cool science-toy to top
Piet Hein's "Superegg". Square wheels is actually just a ripoff of the
Exploratorium exhibit where swollen triangles are used as roller bearings
(the edges of the triangle are segments of a circle, with center of
curvature placed at the vertex opposite each side). Not a big leap from a
swollen triangle to a rolling cube, eh?
:)