Poetry and Geometry: inside the donut
Nikolay Oleynikov was a Russian poet-absurdist. He was executed by Stalin’s regime in 1937. Here is my attempt to translate one of his short poems:
O donut, crafted by a baker!
Your true high purpose wasn’t known to your maker!
You seem so simple, hiding secrets under cover:
The convoluted clockwork, the beauty of a flower.
A vulgar man will snap you in his hand.
He’s in a hurry for he cannot stand
Your rings. And, what a shame,
He’s bothered by the hole of mystic fame.
And we are contemplating donuts, their simple grace,
Like architecture of an ancient race,
Attempting to deduce or to recall
What this resembles all in all,
What all those curves are for, and what the circles mean, and all those ugmics?
In vain! The meaning of the donut is escaping us.
Well, it is really hard to study donuts without cutting them! I’ve been playing with Google SketchUp (a free 3D editor with a very intuitive UI, in case you didn’t know), and of course I couldn’t help contemplating some donuts. Take a look at these cut tori:
The one on the left is a punctured torus. It demonstrates one of the mysteries of the donut (or rather of the torus): it has not one hole, but two. Let me explain what I mean. One of the holes is basically the interior of the torus. And the other one is what we call the donut hole. (And when we cut the surface, let’s not call those punctures and cuts “holes”, to avoid confusion). If you look at the punctured torus on the above picture, you can see that the two holes are no different.
The two other things in the above picture are tori with circular cuts. One of them is cut along a “parallel”, and the other one is cut along a “meridian”. These two cuts circle around the two different “holes”, but they both turn the torus into a cylinder.
But if those two holes are not different then how do we know that one of them is the interior? And how come one of those holes can be filled with yummy substance, while the other remains, well, a hole? What’s up with the donut?! This remains a mystery. (By the way, “untrue products” is an anagram of “punctured torus”)
It is well known that a punctured torus can be turned inside out.
In order to do that, you first stretch the puncture around one hole and then shrink it around the other hole. This is clear when you look at the above picture, but it is hard to visualize how it can be done in practice. On the picture below I tried to show one of the methods of turning a real-life (not very elastic) torus inside out:
Typically, this picture doesn’t convince people. “It’s only half way inside-out, and it doesn’t look like a torus at all”.
Well, here is another picture (from Wikipedia) which may be more convincing (observe how the “parallels” turn into “meridians”):
And how about a double-torus (a donut with two holes, a genus-2 surface)? Can it be turned inside out through a puncture? Yes. Here is the easiest way to show that any punctured n-hole-torus can be turned inside-out. Imagine it as a punctured sphere with n handles. Turn the sphere inside out through the puncture. Now the handles are inside. But handles that are inside the sphere are donut holes! So it is still an n-hole-torus. A donut hole is a donut hole, you cannot hide it inside: