I've got gears on my mind — maybe even

Even the tiniest gears here are indeed turning. This moving picture is by someone called ~zy0rg, and I found it on deviantart.

Let's count the gears on this thing!

If you stare at it, you can see it's based on a regular dodecahedron, a shape with 12 pentagons as faces. The blue gears are the corners of these pentagons. There's a red gear in the middle of each pentagon, and there are 2 yellow gears next to the edge of each pentagon.

Since the regular dodecahedron has 12 pentagons, and there's a red gear in each one, there must be **12 red gears**.

The dodecahedron has 20 corners, since these are the faces of its dual, the icosahedron, which has 20 faces. Or, if you don't know that, you can say: each pentagon has 5 corners, but 3 pentagons meet at each corner, so there are 12 W 5 / 3 = 20 corners. Either way, there are **20 blue gears**.

Finally, the dodecahedron has 30 edges. To see this we can use Euler's formula

V - E + F = 2

so
20 - E + 12 = 2

so
E = 30

Or, we can say each pentagon has 5 edges, but 2 pentagons share each edge, so there are 12 × 5 / 2 = 30 edges. Either way, we get 30 edges, and 2 yellow gears for each edge, so 60 yellow gears.

So, there's a total of 12 + 20 + 60 = 92 gears. It's often not enlightening to total up parts of different kinds like this, and I think it's not enlightening here. 92 is not a number I run into often in my studies of geometry and group theory. Factorizing it shows why: it's 2 × 2 × 23. The number 23 is not a big player in these games.

This animated gif was created by someone named TaffGoch, and you can find other interesting things of theirs on deviantart.

It may have been built by the Phrygians in the 8th-7th centuries BC... or maybe by Hittites fleeing the Phrygians. It seems to have been enlarged much later in the Byzantine era.

But here's the cool part: it's the largest of over 200 underground cities in the Cappadocia region of Turkey... and it's connected by a tunnel to the second largest one!

Why did people build so many underground cities there? I don't know - can you find out? It was relatively easy to do, because the area has a lot of soft volcanic rock. But as any detective show will teach you, there must be motive, not just means and opportunity.

Half of Derinkuyu is open to tourists... have you been there?

That's what emperor Julian of the Roman Empire said about Commodus, the son of Marcus Aurelius, who was emperor from 180 to 192 AD.

He never took after his father the philosopher, but he first showed his true colors after a failed assassination plot in 181. The would-be assassin was tortured and revealed a plot involving his sister.

Later one of the plotters revealed a second plot, and Commodus became paranoid. He sacked all his top commanders and started executing anyone he took a dislike to, including senators... and settled into a three-year binge of debauchery with a string of male lovers and a harem of 300 women: like Saddam Hussein, he had henchmen grab any woman who appealed to him.

As time went on, he only became worse. He developed a taste for voyeurism and had his political favorites have sex with his concubines while he watched. By 188 he had removed all responsible politicians from his inner circles and surrounded himself with a freak show. Often he would amuse himself by cutting off someone's foot or blinding them in one eye. He practiced surgery on live people and let them bleed to death. He liked to sit people at a fancy banquet and serve them food mixed with shit, to see their reactions. I could go on, but it becomes even more disgusting, and I don't want to spoil your day.

As time went on, his delusions of grandeur increased. He renamed the months: August became 'Commodus', and so on. He had always enjoyed killing animals — killing 100 lions with javelins, slicing off the heads of ostriches with special crescent-headed arrows, and so on — and fighting as a gladiator in the Colosseum. But eventually he declared that he was Hercules! He cut off the head of the sun god outside the Colosseum and replaced it with his own portrait, adding a club and a lion to make him look like Hercules. He announced his plan to kill 12,000 gladiators with one hand tied behind his back.

In 193 he was finally killed, in a plot led by his favorite mistress. An attempt to poison him failed, so his gym trainer strangled him.

All this is from an excellent biography of Marcus Aurelius by Frank McLynn. Marcus Aurelius was wise in some ways, but leaving the empire to Commodus was a colossal failure of judgement.

The photo above, by William Storage, shows a sculpture of Commodus at the Getty Museum in Los Angeles. For more of his beautiful photos go here:

- William Storage, Roman Imperial Portraiture.

This beautiful golden pattern was created by 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!

Here is the original flat pattern:

Taffgoch says it's based on a Moroccan tile pattern of the type known as zillij, but I'd say it's an example of girih, or 'strapwork'. It's fun to see how Taffgoch transformed it into the round version... improving 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?

© 2014 John Baez

baez@math.removethis.ucr.andthis.edu