Timeline for The Symmetry of a Soccer Ball
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2010 at 20:43 | comment | added | David Eppstein | Anything that looks combinatorially like a polyhedron can be made with faces that are flat polygons: see Steinitz' theorem, en.wikipedia.org/wiki/Steinitz%27s_theorem | |
| Mar 29, 2010 at 19:32 | comment | added | Douglas Zare | which are squashed versions of regular dodecahedra with pentagonal pyramids attached to two opposite faces. | |
| Mar 29, 2010 at 19:32 | comment | added | Douglas Zare | That's the example I described in the comments on the question. To see that you can make the pentagons flat, start with a nonplanar 12-gon which projects to a regular 12-gon and has alternating vertices in parallel planes. Attach quadrilaterals to each 3 adjacent vertices so that the final vertex is on the line perpendicular to the projected 12-gon through its center. Six of these quadrilaterals meet above at P, and six meet below at Q. Then truncate P and Q. To make this easier to visualize, remember the 10-sided dice from Dungeons and Dragons, | |
| Mar 29, 2010 at 19:14 | comment | added | Anton Geraschenko | An easy not-so-symmetric Fullerene to imagine: fit six pentagons around a regular hexagon, and then fit two of these caps together to get what looks like a squashed dodecahedron. Though now that I think about it, I don't think the pentagons end up flat when you do this. | |
| Mar 29, 2010 at 18:58 | history | answered | David Eppstein | CC BY-SA 2.5 |