In 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 10 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 need
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 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
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 10 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 need
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 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
