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<![CDATA[42]]>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!

Since today is Towel Day, 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? The answer is 42.

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. If you do the math, you’ll see the reason this works is that

$\displaystyle{ \frac{1}{3} + \frac{1}{7} + \frac{1}{42} = \frac{1}{2} }$

There are actually 10 solutions of

$\displaystyle{ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{2} }$

with $p \le q \le r,$ and each of them gives a way for three regular polygons to snugly meet at a point. But this particular solution features the biggest number possible!

But why is this so important? Well, it turns out that if you look for natural numbers $a, b, c$ that make

$\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} }$

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, since

$\displaystyle{ \frac{1}{3} + \frac{1}{7} + \frac{1}{42} = \frac{1}{2} }$

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 $g \ge 2$ 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 $\pi/a, \pi/b$ and $\pi/c ,$ and you need

$\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1 }$

for these triangles to fit on the hyperbolic plane.

For example, if you take $a = 2, b = 3, c = 7$ you get this tiling:

A clever trick then lets you curl up the hyperbolic plane and get a Riemann surface with at most

$\displaystyle{ \frac{2(g-1)}{1 - \frac{1}{a} - \frac{1}{b} - \frac{1}{c}} }$

symmetries.

So, to get as many symmetries as possible, you want to make $1 - \frac{1}{a} - \frac{1}{b} - \frac{1}{c}$ as small as possible! And thanks to what I said, the best you can do is

$\displaystyle{ 1 - \frac{1}{2} - \frac{1}{3} - \frac{1}{7} = \frac{1}{42} }$

So, your surface can have at most

$84(g-1)$

symmetries. This is called Hurwitz’s automorphism theorem. The number 84 looks mysterious when you first see it — but it’s 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, called Klein’s quartic curve:

It has

$84 \times 2 = 168$

symmetries. That number also looks mysterious when you first 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.

If you carefully count the triangles in the picture above, you’ll get 56. However, these triangles are equilateral, or at least they would be if we could embed Klein’s quartic curve in 3d space without distorting it. If we drew all the smaller triangles whose interior angles are $\pi/2, \pi/3$ and $\pi/7,$ each equilateral triangle would get subdivided into 6 smaller triangles, and there would be a total of $6 \times 56 = 336$ triangles. But of course

$336 = 8 \times 42$

Half of these smaller triangles would be ‘left-handed’ and half would be ‘right-handed’, and there’d be a symmetry sending a chosen triangle to any other triangle of the same handedness, for a total of

$168 = 4 \times 42$

symmetries (that is, conformal transformations, not counting reflections).

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 surfaces are as symmetrical as possible, reaching the limit set by Hurwitz’s automorphism theorem, the size of their symmetry group is a multiple of 42, thanks to the math I just described.

Puzzles

Puzzle 1. Consider solutions of

$\displaystyle{ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{2} }$

with positive integers $p \le q \le r,$ and show that the largest possible value of $r$ is $42.$

Puzzle 2. Consider solutions of

$\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1}$

with positive integers $a, b, c,$ and show that the largest possible value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is $1 - \frac{1}{42}.$

Acknowledgments and references

For more details, see my page on Klein’s quartic curve, and especially the section on incredibly symmetrical surfaces. Sums of reciprocals of natural numbers are called ‘Egyptian fractions’, and they have deep connections to geometry; for more on this see my article on Archimedean tilings and Egyptian fractions.

The picture of a triangle, heptagon and 42-gon (also known as a tetracontakaidigon) was made by Tyler, and you can see all 17 ways to get 3 regular polygons to meet snugly at a vertex on Wikipedia. Of these, only 11 can occur in a uniform tiling of the plane. The triangle, heptagon and 42-gon do not tile the plane, but you can see some charming attempts to do something with them on Kevin Jardine’s website Imperfect Congruence:

The picture of Klein’s quartic curve was made by Greg Egan, and you should also check out his page on Klein’s quartic curve.

“Good Morning,” said Deep Thought at last.
“Er…good morning, O Deep Thought” said Loonquawl nervously, “do you have…er, that is…”
“An Answer for you?” interrupted Deep Thought majestically. “Yes, I have.”
The two men shivered with expectancy. Their waiting had not been in vain.
“There really is one?” breathed Phouchg.
“There really is one,” confirmed Deep Thought.
“To Everything? To the great Question of Life, the Universe and everything?”
“Yes.”
Both of the men had been trained for this moment, their lives had been a preparation for it, they had been selected at birth as those who would witness the answer, but even so they found themselves gasping and squirming like excited children.
“And you’re ready to give it to us?” urged Loonsuawl.
“I am.”
“Now?”
“Now,” said Deep Thought.
They both licked their dry lips.
“Though I don’t think,” added Deep Thought. “that you’re going to like it.”
“Doesn’t matter!” said Phouchg. “We must know it! Now!”
“Now?” inquired Deep Thought.
“Yes! Now…”
“All right,” said the computer, and settled into silence again. The two men fidgeted. The tension was unbearable.
“You’re really not going to like it,” observed Deep Thought.
“Tell us!”
“All right,” said Deep Thought. “The Answer to the Great Question…”
“Yes…!”
“Of Life, the Universe and Everything…” said Deep Thought.
“Yes…!”
“Is…” said Deep Thought, and paused.
“Yes…!”
“Is…”
“Yes…!!!…?”
“Forty-two,” said Deep Thought, with infinite majesty and calm. – Douglas Adams

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