## Friday, February 10, 2006

### Topologically equivalent

Question: Are males and females topologically equivalent?

If two objects are "topologically equivalent," this essentially means that if they were made out of rubber, they could be stretched into each other without tearing or pasting. The important question, then, is how many holes each item has.

For example, a tennis ball is topologically equivalent to a football, a pencil, an orange, a shoelace, and anything else that doesn't have any holes. These are all topologically equivalent to a sphere, a 0-holed torus, if you will (and please don't).

Question: Why do topologists have trouble eating breakfast?
Answer: Because they can't tell their donut from their coffee cup.

This common joke (common if you surround yourself with mathematicians) is based on the observation that a donut and a coffee cup both have one hole. These are also equivalent to an inner tube, a sock with a hole in the toe, and a severed pierced ear, all of which are topologically equivalent to a 1-holed torus (commonly known as a torus).

Here is my favorite example. Suppose you have a bottle of syrup, the Aunt Jemima kind of syrup where there is syrup in the handle. Then if the bottle is full of syrup and sealed, you have a 1-holed torus (the handle part where you put your fingers through). If you take off and discard the cap and pour out all the syrup (and wash the bottle so that you can recycle it), you now have one hole that starts at the opening where the syrup comes out and connects all the way around the handle and goes through the whole inside of the syrup bottle, and you also have that original hole where you put your finger through in the handle.

That is a sort of complicated example, and I still haven't been able (after almost a year) to mush the plastic around and end up with a pretty 2-holed torus. But one way you can visualize this better is that it's kind of like how many strings can you loop through holes of the torus if the person holding the ends of the string is very far away and each string can only go through one hole?

In the case of the donut, the string goes through the center of the donut (1).
In the coffee cup, it goes through the handle, just like your fingers (1).
In the full syrup bottle, it also goes through the handle (1).
In the empty uncapped syrup bottle, one goes through the inside of the handle, and the other goes around the handle like your fingers (2).

Now that you have some definitions, you can consider the problem of the topological equivalence, or not, of males and females.

Except that there is one problem, or omission, from these definitions and examples. What if instead of just drilling a hole through a ball and making it into a donut, you then drill a hole from the side that goes from the outside of the ball to the already-drilled tunnel? What then? To what is this new surface (since we really only care about the surfaces that remain) topologically equivalent? This is of major importance to the male/female problem, because, for instance, the hole that connects your mouth to your bum also has the holes from your nostrils going into it from the side. So stand by on that while you think about the rest of it.

But wait! We don't care what n-holed torus a male or female is topologically equivalent to; we only care if they are topologically equivalent to each other. So the nostril question, while interesting and important, has no real bearing on the question at hand, because females and males both have nostrils and mouths and the connection is the same. The only thing we have to wonder about is, let us say, the plumbing beneath.

I know from biology classes that there is a pathway, a "hole," if you will, connecting the mouth, the esophagus, the stomach, the intestines, and the bum. I am not so sure about the mechanism for urine. Is there a single pathway? I don't think so. I think the liver collects stuff out of the blood and there is a tube of sorts going from the liver to the orifice for ejecting urine.

This is the same for females and males, but we will consider it anyway: If there's a tube connecting an opening in the skin to the liver, and at the liver end, this tube ends: It doesn't connect, say, to your nose, so it doesn't add an extra hole to the object. Then this is just like if you had an orange and you poked it with a pencil but not all the way through. It would still be topologically equivalent to a sphere, the fact that you poked it would not affect its topology at all. Similarly, if you glued a raisin to the outside of the orange, or if your orange for some reason just had a strange protruding lump on the outside, it would still be topologically equivalent to a sphere. Simple enough.

So the urine-ejecting system does not add an extra hole to the torus. Recalling the raisin discussion above, the fact that males have an extra protrusion in that area also does not change the topology from what it would be if they did not.

Thus, we are ready to consider the most pressing question, which is: Females have an extra hole. Does that mean they are not topologically equivalent to males?

My answer is no, and my reasoning is the same as with the urine system above: That hole connects to the uterus and the ovaries, which don't connect to anything else. So it's like poking into, but not all the way through, an orange.

Do you agree?

Anonymous said...

I do not agree.

Diana said...

Can you give an argument for why you do not agree, or point out the flaw in my reasoning?

Anonymous said...

Hmmmmm. Do acne holes count? Does one acne hole theoretically lead to another resulting in a myriad of extra holes. This is disturbing. And what of hair folicles. Don't women have more hair than men? I am so confused. I'm going to bed.
Goodnight!