So Lady Chance is making balls and such out of some design book. It takes shapes (triangles, squares, etc) to make the balls. Most of them use the SAME shape for the entire pattern.
Being a geek I automatically connected this with the dice that are used in role-playing games…4-, 6-. 8-, 10- 12-, and 20-sided are the most popular ones (though others ‘stunt’ dice exist).
And I got to thinking…what other polyhedrons can be made using only the same size and shape squares of material? Soccer balls don’t count…they used pentagons and hexagons.
Any of the geometrically inclined dopers of the world wish to venture an opinion?
I assume you don’t actually mean “squares” here. Otherwise, the question is going to get boring real fast, since the only polyhedron (regular or otherwise) composed entirely of squares is the cube.
It’s possible to make a polyhedron with any even number of sides (well, 6 or greater), simply by gluing two n-sided “pyramids” together at their bases. So there’s that, at least.
It’s a Well-known Mathematical Fact (well-known among people who know these kinds of things anyway, including obviously N9IWP :)) that there are exactly five regular polyhedra. Here’s one (out of many) site that explains what they are and why there can only be 5 of them.
nitpick: should be “the only regular polyhedra”. Consider, for example, a tetrahedally stellated icosahedron: glue a tetrahedron on each triangular face of an icosahedron and you have a 60-faced polyhedron made of equilateral triangles. You’re missing the assumptions of convexity and that interfacial angles are the same.
We’ve been implicitly talking about dividing up the surface of the sphere into pieces that could be made flat; in other words, something that could be used as a die (assuming you flattened out the sides.) If you want to include things like the baseball, where the pieces must be curved, then that’s a more general question: how many ways can you divide the surface of a sphere into n chunks, where all the chunks are identically shaped? Unfortunately, I’m not aware of any well-known answer for that.
What if you make the “pentagons and hexagons” out of triangles? Then it’s all triangles.
Modular Origami is really a pretty entertaining thing. Buckyballs are especially fun to do because it’s easy to break down into subcomponents that don’t require a lot of thought or dexterity. Hint: built the pentagons first.
Clearly there are an infinite number of these. You can split along the lines of longitude on a globe into n crescents.
Similarly, If you use two isosolese triangles pasted together at the base in place of the cresents in a globe, you can produce an infinite number of “balls” with n*2 pieces of the same size and shape.
So there are clearly an infinite number with the same irregular size and shape.
Interestingly, it is impossible to construct a “ball” using an odd number of pieces of the same size and shape (unless you count the degenerate case of one piece).
Nitpick: Some of the vertices of the triangles you use to create the hexagons and/or pentagons won’t be touching the surface of the sphere, so it isn’t exactly a “ball.”
You seem to be mixing a couple of seperate statements.
A geodesic sphere isn’t made out of equilateral (or otherwise all-identical) triangles. All the triangles of a geodesic sphere touch the surface of a sphere in a consistent way, however.
Buckyballs have hexagonal and pentagonal faces, which, again, touch the surface of a sphere in a consistent way.
If you put together five equilateral triangles in place of a pentagon and six equilateral triangles in the place of a hexagon and assembled them like a buckyball, you’d have neither a buckyball nor a geodesic sphere, but you would have a polydehron made entirely out of a single type of regular polygon- which is what the OP asked about. And having made this shape out of paper, most people would probably describe it as a “ball”, even though it does deviate significantly from a perfect sphere.
Random: If someone types “surphase”… is that a typo… or… what is that exactly?
No polyhedra with an odd number of congruent faces, eh? Do you have a reference? I’m not (quite) calling your bluff or anything, but it doesn’t seem that obvious to me at first.
