# How many equal-sized polygons can be used to cover a soccer ball?

It seems that mostly either 26 or 32 equal-size polygons are used to cover a spherical soccer ball. What other configurations are possible?

• Equal size, not equal shape? Typically it's a truncated icosahedron (with 32 faces, but some of them with 5 and some of them with 6 edges) – Tim Couwelier Oct 7 '14 at 12:17
• Do the polygons have to be regular? I think there are a lot of solutions if you allow right triangles. – Kevin Oct 7 '14 at 13:41
• Soccer balls have polygonal bits of material stitched together with a spherical bladder inside them. I'm assuming that's what you mean when you want polygons to "cover" a sphere? Anyway, see this: en.wikipedia.org/wiki/Geodesic_dome – TheRubberDuck Oct 9 '14 at 13:51

## 3 Answers

There are an infinite number of ways to cover a soccer ball, if you're allowed to use "biangles", polygons with two sides and two vertices. Draw a line from the ball's north pole to its south pole. Repeat this process any number of times. Now you have a "beach ball" pattern, with however many slices you desire, all of equal size.

If you don't allow biangles, but you do allow non-regular polygons, take the above pattern and draw a line around the equator; now you have any number of 90-90-?? triangles.

• well this is a very clever solution! – Martin Frank Oct 8 '14 at 5:31

Most of the possible solutions would be found in the collection of Platonic or Archimedean solids.

The platonic fit the description of same size faces: http://www.korthalsaltes.com/cuadros.php?type=p

In that case the options are 4 6 8 12 or 20. (note: some of them are highly unenjoyable to play with).

The Archimedean solids might offer a better fit: http://www.korthalsaltes.com/cuadros.php?type=a (includes the standard truncated icosahedron which is commonly used)

However, none of them have solutions with equal size polygons, they each consist of a combination of at least two 'regular' polygons.

Once again, with most of these I wouldn't enjoy playing around.

I think the Catalan solids would qualify as well. The faces are all equal polygon, although the polygons themselves don't have equal sides. Biggest would probably be the Pentagonal Hexecontahedron http://en.wikipedia.org/wiki/Pentagonal_hexecontahedron

• Really? I'd have gone with the Disdyakis Triacontahedron. It has twice as many sides. That's probably the most you can get if you insist on each side being equal (other than Kevin's infinite solution or variations thereof). If you allow for some small variation, a geodesic sphere can go much higher - think Epcot Center's Spaceship Earth, which looks more uniform across its surface than the polar-infinite slivers. though the triangles vary in shape slightly. – Darrel Hoffman Feb 7 '16 at 19:15