For those wondering why my soccer ball got scribbled on (this thread), I was trying to figure out a problem related to polyhedrons. I submit it to the Dopers, (if anyone can figure out what I’m talking about).
What I was trying to figure out is, how many circles can you have around a sphere, such that no more than two cross at any point? By circles I mean great circles, or equatoral lines. Now I know that the answer is infinity, because there is a clear example: you can skew (n) number of circles around a polar axis. However this is somewhat unsatisfying because it has polar or radial symmetry, and what I’m looking for is true spherical symmetry.
Here’s where the polyhedra come in. The above example corresponds to the infinite class of antiprisms. Take each opposite pair of vertices as defining an axis, and plot the equatoral circle midway between them. The same thing works with four of the five Platonic solids. The octohedron gives you three circles, the cube gives you four, the icosohedron six, and the dodecahedron 10 (in a rather lovely pattern of twelve pentagrams; the equivalent is truncating the corners of a soccerball).
What I can’t figure out for certain is whether these are the only patterns possible if you’re limited to no more than two lines crossing at any point. I’ve gone as far as I can without a special program for plotting spherical geometry.
Lumpy: What I was trying to figure out is, how many circles can you have around a sphere, such that no more than two cross at any point? By circles I mean great circles, or equatoral lines.
There can only be one great circle (GC) drawn on a sphere at a time that does not intersect another. E.g., if you draw the equator on the Earth, any other GC must intersect at two points, since half of the 2nd GC must be south of the equator, and half north. So, only two can be drawn “such that no more than two cross at any point”.
*Now I know that the answer is infinity, because there is a clear example: you can skew (n) number of circles around a polar axis. However this is somewhat unsatisfying because it has polar or radial symmetry, and what I’m looking for is true spherical symmetry. *
You can pick any point on a sphere, calculate its opposite point (latitude the same but opposite N/S, longitude would be supplementary angle and opposite E/W), and then have infinite GCs pass through those two points. And there are infinite pairs of these points on a sphere, infintessimally different.
All of your patterns have a fixed number of axes and only rotation through certain angles. “True spherical symmetry” would allow any rotation through any axis that passes through the center of the sphere.
Your soccer ball is not a dodecahedron, which would be 12 pentagon shaped facets. Your soccer ball is a figure of hexagons, and pentagons, which have lengths equal, but not equal surface areas. The regular polyhedra are tetrahedron, cube, octahedron, dodecahedron, and Icosohedron. (twenty equilateral triangles) There are no other regular convex polyhedra. These five are called the Platonic Solids
The circles you can draw around a sphere to coincide with the planar faces of Platonic Solids are not great circles.
to reiterate, the soccer ball is also typically called a “geodesic sphere;” a polyhedron popularized by the architect R. Buckminster Fuller. It is the simplest possible geodesic solid, which, if memory serves, is a solid requiring exactly 12 pentagons and 20 hexagons.
“Buckyballs”, or buckminterfullerenes are an allotrope of carbon (the other two most famous allotropes of carbon are graphite and diamonds) whereby the carbon atoms are located at the vertices of a geodesic solid. A 60 carbon atom buckyball looks exactly like a soccerball.
I’m not contradicting what anyone else said, but the proper name for the shape of a bucky ball or a soccer ball is “truncated icosahedron”.
Having said that, with regards to the OP, I think the answer is infinity, because in general, arbitrarily chosen great circles will not make sets of crossings at the same point. It’s equivalent to the question, how many nonparallel lines can you have in the plane such that no more than two cross at any point? Although I think that bit about the polyhedra is fascinating, I don’t really see what it has to do with your question.