### Chris Schommer-Pries

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If you try to get several regular polygons to meet snugly at a point in the plane, what's the most sides any of the polygons can have?

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. The reason this works is that

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

There are 17 solutions of

(1/2 - 1/p) + (1/2 - 1/q) + (1/2 - 1/r) = 1

with p ≤ q ≤ r, but this one features the biggest number of all!

But why is this so important? Well, you can take

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

and rewrite it like this:

1/2 + 1/3 + 1/7 + 1/42 = 1

So, 1/2 + 1/3 + 1/7 comes very close to 1. And in fact, if you look for natural numbers a, b, c that make

1/a + 1/b + 1/c

as close to 1 as possible, while still less than 1, the very best you can do is 1/2 + 1/3 + 1/7. It comes within

And why is

1/a + 1/b + 1/c < 1

for these triangles to fit on the hyperbolic plane. A clever trick then lets you get a Riemann surface with at most

2(g-1)/(1 - 1/a - 1/b - 1/c)

symmetries. So, you want to make 1 - 1/a - 1/b - 1/c be as small as possible! And thanks to what I said, the best you can do is

1 - 1/2 - 1/3 - 1/7 = 1/42

So, your surface can have at most

84(g-1)

symmetries. This is called

In particular, the famous mathematician Felix Klein studied the most symmetrical doughnut with 3 holes. It's a really amazing thing! It has

84 × 2 = 168

symmetries. That number looks really mysterious when you see it. Of course it's the number of hours in a week, but the

But why is this stuff the answer to

For more details, see:

http://math.ucr.edu/home/baez/klein.html

and for a webpage version of this post with a few more pictures, see:

http://math.ucr.edu/home/baez/42.html

*The Hitchhiker's Guide to the Galaxy*by Douglas Adams, the number**42**is the "Answer to the Ultimate Question of Life, the Universe, and Everything". But he didn't say what the question was! Let me reveal that now.If you try to get several regular polygons to meet snugly at a point in the plane, what's the most sides any of the polygons can have?

**42**.The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. The reason this works is that

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

There are 17 solutions of

(1/2 - 1/p) + (1/2 - 1/q) + (1/2 - 1/r) = 1

with p ≤ q ≤ r, but this one features the biggest number of all!

But why is this so important? Well, you can take

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

and rewrite it like this:

1/2 + 1/3 + 1/7 + 1/42 = 1

So, 1/2 + 1/3 + 1/7 comes very close to 1. And in fact, if you look for natural numbers a, b, c that make

1/a + 1/b + 1/c

as close to 1 as possible, while still less than 1, the very best you can do is 1/2 + 1/3 + 1/7. It comes within

**1/42**of equalling 1.And why is

*this*important? Well, suppose you're trying to make a doughnut with at least two holes that has the maximum number of symmetries. More precisely, suppose you're trying to make a Riemann surface with genus g ≥ 2 that has the maximum number of symmetries. Then you need to find a highly symmetrical tiling of the hyperbolic plane by triangles whose interior angles are π/a, π/b and π/c, and you need1/a + 1/b + 1/c < 1

for these triangles to fit on the hyperbolic plane. A clever trick then lets you get a Riemann surface with at most

2(g-1)/(1 - 1/a - 1/b - 1/c)

symmetries. So, you want to make 1 - 1/a - 1/b - 1/c be as small as possible! And thanks to what I said, the best you can do is

1 - 1/2 - 1/3 - 1/7 = 1/42

So, your surface can have at most

84(g-1)

symmetries. This is called

**Hurwitz's automorphism theorem**. The number 84 looks really mysterious when you first see it - but it's really there because it's twice 42.In particular, the famous mathematician Felix Klein studied the most symmetrical doughnut with 3 holes. It's a really amazing thing! It has

84 × 2 = 168

symmetries. That number looks really mysterious when you see it. Of course it's the number of hours in a week, but the

*real*reason it's there is because it's four times 42.But why is this stuff the answer to

*the ultimate question of life, the universe, and everything?*I'm not sure, but I have a crazy theory. Maybe all matter and forces are made of tiny little strings! As they move around, they trace out Riemann surfaces in spacetime. And when these surface are as symmetrical as possible, the size of their symmetry group is a multiple of 42, thanks to the math I just described.For more details, see:

http://math.ucr.edu/home/baez/klein.html

and for a webpage version of this post with a few more pictures, see:

http://math.ucr.edu/home/baez/42.html

3

I suppose Adams derived this joke from Dirac's large number hypothesis (http://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis) according to which something like 10^42 is the ration between maximum and minimum structures in the universe, such as the size of the observable universe over the size of the "classical electron radius" (both of which of course debatable and at least one a non-constant! :-) But I think that's where it comes from...)

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