**Spherical zillij**This beautiful golden pattern was created by someone who goes by the name of Taffgoch. He did it by taking a traditional Islamic tiling pattern made of interlocking hexagons and replacing some of them by

*pentagons*. This lets the original flat pattern 'curl up' and become spherical!

You can see the original flat pattern here:

https://groups.google.com/d/msg/geodesichelp/ayZcCILttBs/bVU6eI5SLUEJIt's of a type called

**zillij**. It's fun to read how Taffgoch transformed it into the round version... and see how he improved it step by step.

**Puzzle:** how many pentagons, and how many hexagons, are in this spherical zillij?

This is similar to a question about

**fullerenes**, which are sheets of graphite - hexagons of carbon - that curl up into spheres because some hexagons are replaced by pentagons. Fullerenes come in different sizes, with different numbers of hexagons. But as long as a fullerene is spherical in its topology, with 3 pentagons or hexagons meeting at each corner, the number of pentagons is fixed!

I'll compute this number now, so if you want to answer the puzzle on your own, maybe you should stop reading. However, this spherical zillij pattern is not exactly the same as a fullerene... so it's not obvious that it has the same number of pentagons.

Here's how it goes. Suppose we have a sphere tiled with P pentagons and H hexagons, with 3 of these polygons meeting at each vertex.

How many edges are there in this tiling? Each pentagon has 5 edges, and each hexagon has 6, but each edge is shared by 2 shapes so the number of edges is

E = (5P + 6H)/2

How many vertices are there? This is where we need to know 3 polygons meet at each vertex. Then by the same reasoning as above, the number of vertices is

V = (5P + 6H)/3

How many faces are there? That's easy:

F = P + H

Now

**Euler's formula**, a fact from topology, says

V - E + F = 2

So, plugging in the equations for V, E, F, we get

(5P + 6H)/3 - (5P + 6H)/2 + (P + H) = 2

or

P + H = 2 + (5P + 6H)/6

or

P = 12

Note that H cancels out, so we learn nothing about how many hexagons there are. But pentagons love the number 12... and ultimately, that's why this shape here has

5 × 12 = 60

rotational symmetries!

**Puzzle:** suppose we have a doughnut with g holes tiled by pentagons and hexagons, 3 meeting at each corner. How many pentagons are there?

#geometry