College Math Teaching

June 7, 2016

Pop-math: getting it wrong but being close enough to give the public a feel for it

Space filling curves: for now, we’ll just work on continuous functions $f: [0,1] \rightarrow [0,1] \times [0,1] \subset R^2$.

A curve is typically defined as a continuous function $f: [0,1] \rightarrow M$ where $M$ is, say, a manifold (a 2’nd countable metric space which has neighborhoods either locally homeomorphic to $R^k$ or $R^{k-1})$. Note: though we often think of smooth or piecewise linear curves, we don’t have to do so. Also, we can allow for self-intersections.

However, if we don’t put restrictions such as these, weird things can happen. It can be shown (and the video suggests a construction, which is correct) that there exists a continuous, ONTO function $f: [0,1] \rightarrow [0,1] \times [0,1]$; such a gadget is called a space filling curve.

It follows from elementary topology that such an $f$ cannot be one to one, because if it were, because the domain is compact, $f$ would have to be a homeomorphism. But the respective spaces are not homeomorphic. For example: the closed interval is disconnected by the removal of any non-end point, whereas the closed square has no such separating point.

Therefore, if $f$ is a space filling curve, the inverse image of a points is actually an infinite number of points; the inverse (as a function) cannot be defined.

And THAT is where this article and video goes off of the rails, though, practically speaking, one can approximate the space filling curve as close as one pleases by an embedded curve (one that IS one to one) and therefore snake the curve through any desired number of points (pixels?).

So, enjoy the video which I got from here (and yes, the text of this post has the aforementioned error)

January 27, 2016

A popular video and covering spaces…

Filed under: media, popular mathematics, topology — Tags: , , , , — collegemathteaching @ 11:16 pm

Think back to how you introduced the sine and cosine functions on the real line. Ok, you didn’t do it quite this way, but what you did, in effect, is to define $sin(u) = Im(e^{iu})$ and $cos(u) = Re(e^{iu})$ and then use “elementary trigonometry” to relate the “angle” $u$ to the arc length subtended on the circle $|z| = 1$. One notes that the map $\rho: R^1 \rightarrow C^1$ defined by $\rho(u) = e^{iu}$ has period $2\pi$

Note: the direction “to the right” on the real line is taken to be “counterclockwise” on the circle (red arrows).

Skip if you haven’t had a topology class
The top line is known as the “universal covering space” for the circle. The reason for the terminology has to do with topology. Depending on how long ago you had your topology course, you might remember that the fundamental group of the real line is trivial and the associated group of deck transformations is infinite cyclic (generated by the map $d(u) = u + 2\pi$ ). One then shows that the fundamental group of the circle is the quotient of the group of deck transformations with the fundamental group of the real line; hence the fundamental group of the circle is infinite cylic.

Resume if you haven’t had a topology class

Notice the following: if one, say, “takes a walk” along the line in the direction of the red arrow, the action of the “covering mapping” is to take the same walk in the counter clockwise direction of the circle. That is, the covering action does the following: a walk on the line in the direction of points $A_1, B_1, C_1, A_2, B_2, C_2....$ corresponds to a walk on the circle $A, B, C, A, B, C....$. That is, walking from $A_1$ to $A_2$ corresponds to a complete lap of the circle.

(that is, on the real line, $A_{n+1} = A_{n} + 2\pi$)

Now note the following: for BOTH the line and the circle, the direction is well defined. “To the right” on the real line” is “counter clockwise” on the circle.

However: on the real line, it makes perfect sense to say that $A_1$ is “before” $B_1$ which is “before” $C_1$ which is “before” $A_2$ and so on; this is merely:

$A_1 < B_1 < C_1 < A_2 < B_2 ...$. This is order is valid no matter where one starts on the line.

However, this “universal ordering” makes no sense on the circle, UNLESS one specifies a start point. True, one moves from $A$ to $B$ to $C$ and back to $A$ again..but if one started at $B$ and started to walk, it would appear that $A$ came AFTER $B$ and not before.

So what?

This quirky animation from CraveFX starts off innocently enough, a janitorial worker mops up a leaky refrigerator and then picks up a coin on the ground. It’s not until you see what causes the refrigerator to leak and why the coin is on the ground that you realize that you’re watching an intricate moving puzzle piece before your eyes. The characters are stuck in an infinite loop caused by another character in their own infinite loop. It’s chaotic and great and hard to keep up with.

The video is below. Now the question: “what action occurred before what other action”? and the answer is “it depends on when you started watching”. The direction of time corresponds to the red arrows in the above diagrams; THAT is well defined. Why? The reason is the Second Law of Thermodynamics; spills do NOT reverse themselves, hence the direction is set in stone, so to speak. But as far as order, it depends ON WHEN THE VIEWER STARTED WATCHING.

Anyway, this video reminded me of covering spaces.