Knot Another Punny Title

So, this is more of an update post than anything else. Work and school have been pretty busy. I'm also writing (well, editing at this point) a short story, which is where the title of my post comes from. How's that work?

For reasons that escape me, I'm writing a short story that heavily features a branch of mathematics known as knot theory. Wait, there's a branch of mathematics about knots? Why yes, yes there is.

Math is about numbers, duh, but it's also about geometry. Many of the ancient Greek mathematicians did math purely through geometry, in fact, because they didn't have algebra to represent general forms or calculus to deal with infinitesimals. Anywho, there was a point in the 19th century at which studying geometry morphed into studying surfaces, and this led to ideas such as differential geometry and topology. Differential geometry is the math behind Einstein's general theory of relativity, and topology tells us that doughnuts and coffee mugs are the same thing.

Doughnuts and coffee mugs are the same thing? Apparently, yes, because in topology, objects are homeomorphic if you can transform (stretch, squeeze, rotate, twist) one into the other without creating any new shapes or holes. There's a little gif on wikipedia showing that this isn't quite as crazy as it sounds.

Now, there's a sub-branch of topology known as knot theory, which studies circles embedded in space (or spheres embedded in 4-space, etc.). It turns out that some circles are just circles (called the unknot), whereas others are true knots that cannot be transformed back into circles without cutting the knot. Take a rubber band, for example. It's just a circle.

No matter how many times you twist it around and tie it into knots, it's still just a circle that can (theoretically) be undone.

But if you cut the rubber band...

...tie it into a knot, and then reconnect the severed ends (and pretend it's not being held together with tape), then you've created a true knot that cannot be transformed back into the unknot.

You can take that one new knot and twist it all around into different-looking knots, but knot theory says that, as long as you don't cut the rubber band again, it is still fundamentally the same knot, the same way a coffee mug and a doughnut can be fundamentally the same topological space.

So, how'd I write a short story about that? Well, it turns out knot theory has uses outside of rubber bands. In fact, Kelvin kind of got some of the credit for starting this whole knot theory business when he hypothesized that atoms were just knots (vortices) in the aether. But then Michelson and Morley kind of threw a wrench in that whole thing.

There are some modern applications of knots, however. DNA gets itself tied up into knots, and knot theory can help explain how it undoes those knots. Quantum field theory can also be described in a knot-like way, and there might also be quantum computers based on knots.

My story extrapolates this all well into the realm of science fiction, to the point that it would probably piss off the 3 mathematicians who study knots and read short science fiction. But I think it goes in interesting directions. I tie in (ha, not intentional) notions of Buddhist and Celtic endless knots, the Gordian knot legend, and the knot-based number system used by the Inca, known as Quipu. Fun stuff, I hope. I currently have two beta readers attempting to determine whether or not it's ridiculously boring. We'll see.

(Hm. I guess that was a little more than just an update.)

The Gravity Thief

(I'm reading The Quantum Thief right now, so I figure vague, sciencey-sounding titles are appropriate.)

My good friend (read: successful and well-liked blogger on whose posts I sometimes leave inane comments) Phil Plait recently blogged about the impending non-Armageddon that is 2012 DA14. The brief summary is that a giant hunk of rock half the size of a football field will just barely miss us this Saturday, and astronomers are so excited they're polishing their mirrors in anticipation. For a slightly more nuanced explanation, check out the Bad Astronomer's blog.

What intrigued me about Plait's post is the minor detail that, currently, 2012 DA14 has an orbital period of 366 days, but after its interaction with the Earth its orbital period will shrink to 317 days. This means, obviously, that the asteroid will go around the sun more quickly from now on. But why? Is it speeding up, or just getting closer to the sun? The answer is yes.

Four hundred years ago, after pouring through Tycho Brahe's enormously detailed astronomical records, Johannes Kepler devised three empirical laws of planetary motion. The relevant one here is the third law, which states that the square of a body's orbital period is proportional to the cube of its semi-major axis (radius). So the closer a planet is to the sun, the shorter its period.

Newton later confirmed Kepler's observation with his universal law of gravitation, which states that the force between two objects is proportional to the product of their masses divided by the square of the distance between them. So, then, if an orbiting object's period decreases, its radius does also, and if the radius decreases, its speed increases.

We know from the kinetic energy formula that a faster object has more energy, which means that 2012 DA14 will gain energy from its interaction with the Earth. How much energy? Well, using Kepler's third law, we can see that if the orbital period shrinks by 49 days, the semi-major axis will shrink by about 9%.

With a bit of calculus, we can figure out that Newton's law of universal gravitation predicts that the gravitational potential energy of an object is inversely proportional to the distance between it and another object. So, a 9% reduction in radius is a 10% increase in energy. (Because gravitational potential energy is defined to be negative, this can be thought of as gaining negative potential energy, or losing positive potential energy. The end result, however, is an increase in kinetic energy).

And energy, as we know, is conserved. So if the asteroid is gaining energy from its encounter with the Earth, the Earth must be losing energy. It looks as if the asteroid has a mass of about 130 million kg, which is less than the mass of the Earth by a factor of 46 quadrillion. Thus, a 10% increase in energy for the asteroid is a 2.4x10^-17 % decrease for the Earth. And if we go backward through the math, we see that the Earth's orbital period will increase by about 1 nanosecond. Put another way, after 28 years, an extra 1 second will have elapsed when compared to the previous 28 years (all else being equal).

The all else being equal part, however, is quite a stickler. The duration of the Earth's orbital period and day vary quite significantly due to effects from the moon, the sun, earthquakes, glaciers, and a whole range of other factors. It's likely that more than just the asteroid's speed changes during its near-Earth encounter, and the same goes for the Earth. Rather than our orbit changing, it might alter the length of the day by a tiny fraction.

But there is an underlying principle here: energy conservation. Energy never disappears completely; it just moves from one object to another, changing forms as it does so.

Don't turn that dial!

Okay, so this is a little late, but I want to talk about my physics lab from last Wednesday. I haven't had a lab course since I took Chemistry my junior year of high school, and that was eleven years ago. Working with a lab partner, writing down results, adjusting the apparatus -- these are things I (until recently) thought I was probably done with. Since I'm trying to become a scientist, however, I may have to get used to this routine.

Anyway, the lab was split into two parts. The first part had me and a partner measuring the spring constant of a spring by pulling a force gauge attached to said spring. After recording the force from the gauge and the distance we'd stretched the spring, we could find the slope of a Force vs. Distance graph, and that slope was the spring constant in newtons/meter -- the higher the value, the stronger the spring.

The exercise was decidedly unrelated to electricity and magnetism (unless you want to talk about the fact that chemical bonds are electric in nature). I think the point was just to acquaint us with doing a lab, which I suppose is necessary given that some of the students haven't taken a lab course in eleven years.

After that, however, the professor had each of us individually play with an analog oscilloscope. We weren't attempting to measure anything with the oscilloscope -- just push buttons and turn dials so that we could familiarize ourselves with its function.

For those unfamiliar with an oscilloscope (I had heard of but never seen one before last Wednesday), it's a device that measures an incoming electrical signal and displays a corresponding Volts vs. Seconds graph.

Here's a picture of the model we were using:

Manipulating and calibrating the o-scope (as the prof called it) is a somewhat tricky enterprise. The display is divided into a grid, and you can adjust the number of volts and seconds per length of grid. As you can see, however, there are a wide variety of other knobs and controls that are required to produce a clean image of the incoming signal. And throughout this whole process, I was wondering, is this really necessary?

I mean, perhaps I'm putting the cart before the horse, but this is 2013; shouldn't there be an iPhone app that can do this for me? (Yes.) In fact, isn't that exactly what any piece of audio equipment does when it translates electric signals into sound -- measure the frequency and amplitude of the wave?

Yeah, I'm in an introductory E&M course. Yeah, we're going over the basics. But is learning to fine-tune an o-scope like a WW2 radio operator really going to be useful in our later physics careers?

I can imagine two ways in which it might be. The first possibility is that while of course there will be software that can identify the sinusoidal wave (or whatever) of an electric signal when we're working scientists, someone has to write that software. In that case, having an intuitive feel for what we're attempting to measure might be useful.

The other possibility I can imagine is that a working scientist will have to play with significantly more complex pieces of machinery, and training on a somewhat antiquated oscilloscope is a necessary first step toward that goal. I mean, you can't start with the LHC, right?

I thought of both these counter-arguments to my original complaint, but I don't know how much stock I put in either of them. Unless I'm actually going to use an oscilloscope as a scientist, isn't there something that more closely resembles a piece of modern scientific equipment that we could be using?

According to wiki:

Oscilloscopes are used in the sciences, medicine, engineering, and telecommunications industry. General-purpose instruments are used for maintenance of electronic equipment and laboratory work. Special-purpose oscilloscopes may be used for such purposes as analyzing an automotive ignition system, or to display the waveform of the heartbeat as an electrocardiogram.

Ah, I did not know that EKGs were essentially specialized o-scopes. That's neat! But I'm a little hazy on what "maintenance of electronic equipment and laboratory work" means for general-purpose o-scopes. A little more reading tells me that they can be used to test the changing voltage of a device to make sure it's working within expected parameters. That's obviously useful, but again I feel as if this is something that could be done by a piece of software. Do we really need a human to turn a dial until the display stops spazzing, or can we just use a do...while loop?

Anywho, I think that's enough griping for now. Perhaps I should be a theorist.

The Community College Cafeteria

Where timeless wisdom and crass marketing come together.

(I'll probably post something substantive later today.)

Let's not drag this out.

So, I've taken another shot at the snow-sticking-to-my-windshield problem. After scouring the internet for answers, I decided to ask the good people at physicsforums.com if they had any idea what might be the cause of my observed phenomenon. Their answer? Friction and drag.

In other words, I was thinking too hard about the problem. Drag from the air pushes a snowflake up the windshield. Gravity pulls it down. And friction resists its movement. It's just Newton's laws, of course.

But it's not quite as simple as that, because the math for determining whether or not the snowflake sticks involves approximating some constants based on the properties of the snowflake, my windshield, etc.

The biggest one is this: what is the coefficient of friction between a snowflake and glass? That is, how easy is it to move a snowflake across a windshield? It turns out there's no simple or single answer to this question, because it depends very much on the properties of pretty much everything: the temperature of the windshield, the moisture of the snow, the shape of the snowflake, etc.

There's a fair amount of research out there on the friction between skis and snow, but not much about snowflakes on glass. The best I could find was this paper, but their one example was a little unclear on the total mass of snow present, so it was difficult to extract a figure for μ.

Anywho, eventually I decided I would simply test the hypothesis that friction and drag are the culprits. Once again, the equation for terminal velocity appears, but this time in the form of the drag equation. It is: F = ½ρv²C_DA. We need to know the area of a snowflake and its drag coefficient. Like a good physicist, I'm going to imagine a spherical snowflake, giving it a drag coefficient of .1 and an area of (assuming a 1 cm diameter) 8x10^-5 m. All in all, assuming a speed of 10 mph, this works out to a force of 10^-4 N pushing on the snowflake.

This force is pushing directly on the snowflake, however, and the snowflake is sitting on an inclined surface. So we have to find the component of that force pointing in the direction of the surface, which means we have to multiply that result by the cosine of the angle of my windshield (about 40°), giving us a force of 7.8x10^-5 N.

Because this drag is pushing the snowflake up the windshield, gravity opposes the motion. The force here is much smaller, amounting to 1.9x10^-6 N. Because the drag force is so much greater than the weight, we know that the snowflake will move upwards and be resisted by friction. The question is, how much friction does it take to resist motion? Here we must use Newton's first law, which says that objects experiencing no net force experience no acceleration.

Therefore, 7.8x10^-5 - 1.9x10^-6 - μN = 0, which we can rearrange as μ = (7.8x10^-5 - 1.9x10^-6)/N to find μ. N is the normal force, which is the windshield's equal and opposite reaction to the snowflake's weight. Solving this equation, I get a μ of 40. What does that mean? Well, it means that my answer is probably wrong. But if it were correct, it would mean the friction holding the snowflake down would have to be extremely strong to resist any motion at all.

There are a couple possible solutions here. One is that there is a lot of friction between a snowflake and a plate of glass because an individual snowflake is a jagged crystal. Another possibility is that the critical speed is much lower than 10 mph. This is possible, except that bringing μ down to what I would expect means the critical speed would need to be something like 1 mph, which is not really what I observed. The only other variable with a lot of uncertainty is the drag coefficient, which I'm quite sure is wrong. A snowflake isn't really a smooth sphere. But a more realistic drag coefficient would probably increase the drag force, leading to an even higher value for μ.

All in all, I'm not satisfied with this answer. The result is less unreasonable than my last one, but still not exactly a perfect match for the data. My conclusion is that there's more going on here than just drag and friction. But I think that's about all the energy I have for this problem.

Vector? I hardly knew her!

(I'm going to post a followup to the Snow Crash post either tonight or tomorrow, but I just needed to make this post tonight.)

So tonight was the first night of Multivariable Calculus. Now, I don't really expect to learn anything on the first day of pretty much any class, but tonight was (with a few exceptions) dreadfully boring. The reason being that we spent about half the class time doing an introduction to vectors.

Vectors are important. This I know. Being able to break up quantities into their components is vital, especially in physics. And the connection between geometry and algebra that vectors allow for is also very useful. This I know, too, because I've been introduced to vectors about a dozen times in my life. That might be an exaggeration, but I'm not sure.

I was tempted (not really -- this isn't the type of thing I would ever do) to raise my hand during class and ask if there was anyone in the class who hadn't gotten a vector intro before. And if no one raised their hand, could we please maybe move on to new stuff? I could not show up to class until we started covering something I didn't know ten years ago, but apparently this is a very popular class/professor and I'd be dropped and replaced if I did that. (This, too, isn't really something I would do.)

Perhaps colleges could offer modules -- short classes that introduce/refresh a topic that will be used in other classes. That way those classes wouldn't be bogged down going over stuff people have already learned elsewhere. Vectors could be one. Polar geometry could be another.

I understand that the remedial math classes at my college are kind of like this. They're modular, and students just take the sections they need so that they can move on to whatever's next.

That said, I did get something useful out of tonight's class. The professor also introduced 3-space, which I've gotten the basics of before as well, but she framed it in a way that was quite illuminating for me. I have a lot of trouble with visualizing objects in more than two dimensions, and anything to help that process is good. She discussed 3-space in terms of the room we were in. One bottom corner of the room was the origin. The front wall was the yz-plane, the side wall the xz-plane, and the floor the familiar xy-plane.

Literally being inside the three-dimensional coordinate system she was describing was immensely helpful for me. Planes made sense; surfaces made sense. And I think I can start to look at the world through the lens of a coordinate system, something that should help the process of abstraction that is necessary for creating models of reality.

Anywho, that's all for today. I should probably think of something to post about yesterday's physics class as well.

Snow Crash

Update: Because I'm neurotic and check my math again and again, I discovered that I made a slight error in my calculations. Somewhere along the way I incorrectly converted some units and ended up with an answer that I'm pretty sure is a million times smaller than the actual value. That is, it takes way more energy to melt (or sublimate) a snowflake than I said, such that my car would have to be going something like a thousand times faster in order to effect a phase change.

Which means my prediction was egregriously wrong. I guess it's a good thing I didn't try to get my research published in a peer-reviewed journal. (I'll note, however, that being off by six orders of magnitude is not quite as bad as being off by 120 orders of magnitude.)

So where does that leave us? I have no idea, really. Perhaps the difference between sticking and not sticking is not related to melting or sublimation. Perhaps the air resistance a moving car experiences pushes snowflakes away from the windshield. I'll have to give this a little more thought...

And now for the original entry:

(Sorry for the month-long absence, my numerous and devoted fans. I'll try to post more frequently. However, school starts back up again on Monday. I might have more to write about, but I'll have less time to write about it. We'll see what happens.)

So we had the first real snowfall of the year this week, and I noticed a rather interesting phenomenon while driving home from work the other day. No, it's not that storm intensity appears to be inversely correlated with driver intelligence; that's depressingly typical.

Rather, I noticed that the flakes hitting my windshield reacted differently at different vehicle speeds. That's not too surprising. In the rain, driving faster means more rain hits the windshield per second, and the rain will hit harder, so the end result is you need to speed up the wipers. But this didn't hold true for snow.

When my car was stopped at a light, the snow would accumulate on the windshield and I'd have to hit the wipers before I started driving again. But at a high enough speed, the flakes wouldn't accumulate. They'd strike my windshield, and then be gone. As best as I could determine while driving down a busy road in the middle of a snowstorm and trying not to reduce the average driver intelligence, the critical speed was about 10 mph.

So what's the explanation? I'll admit upfront that my physics class this past semester was a little light on the thermo, so I'm going to be making a lot of assumptions and ignoring (or being ignorant of) a lot of complicated factors. But my theory is that when my car is stationary, the kinetic energy of a falling snowflake is enough to melt the flake from ice to water, but no more. When my car's forward velocity is added on, however, the snowflake has enough kinetic energy to sublimate directly into gas without first passing through the liquid stage, and thus does not stick to my car.

Materials are only able to sublimate when the temperature and pressure are below that material's triple point. The triple point is a specific temperature and pressure at which a material can exist as solid, liquid, or gas. Water's triple point (snow is really just frozen water with air mixed in) is about .01 &degC and a pressure of .006 atm. You'll notice that Earth's atmosphere has an atmospheric pressure of 1 atm (funny, that), which is significantly higher than .006. The answer to this is that the pressure close to a solid is a result of the solid's vapor rather than the atmosphere as a whole. The upshot is that water can easily sublimate at low enough temperatures.

Okay, then, how much kinetic energy does a falling snowflake have? Our old friend terminal velocity makes an appearance again. According to this paper, the terminal velocity of a snowflake in the conditions I was driving through is about 1 m/s. Wikipedia tells me that a typical snowflake consists of 10¹⁹ water molecules, which is 1.66x10^-5 mol of water. The molar mass of water is 18 g/mol, so a typical snowflake comes in at 3x10^-7 kg. Our formula for kinetic energy is &frac12mv², so a falling snowflake packs a whopping 150 nanojoules.

Let's figure out what we can do with 150 nJ. The energy required to melt ice into water is known as the specific heat of fusion. Under ordinary pressures, this occurs at the familiar 0 &degC. At other temperatures, adding energy changes the temperature, but at 0, it causes a phase transition. The explanation for why this occurs at a particular temperature for a particular material is well beyond my understanding, but the basic idea is that the average kinetic energy of an ice molecule is enough to break the intermolecular bonds holding the ice together. Water's specific heat of fusion is 334 kJ/kg, which means that it requires 334 kJ of energy to transform 1 kg of ice into water.

Now, there are a lot of big assumptions here. Some of them are: all of the snowflake's kinetic energy goes into melting it, the air inside a snowflake doesn't change anything, the temperature of my windshield has no effect on the transition, none of the snowflakes interact with each other, etc. Anywho, if you run the numbers, it looks like it takes 100 nJ of energy to melt 300 &#956g of ice. Thus, falling alone is enough to melt the ice. But is it enough to sublimate the ice?

To sublimate, one need only add the specific heat of fusion onto the specific heat of vaporization (the heat required to turn a liquid into a gas). Water's specific heat of vaporization at 0 &degC is 2500 kJ/kg -- way higher than fusion. So to sublimate, you need a total of 2835 kJ/kg (there's some invisible rounding go on here), which is roughly 8.5 times as much energy. We only have 1.5 times the energy required to melt, so we cannot sublimate from falling alone.

I guesstimated that my car going 10 mph was the critical speed. Let's see what that gives us. 10 mph is 4.5 m/s. We can treat my car as stationary and the snowflake as moving at 4.5 m/s toward my car horizontally. This is easy to believe. When you're driving into a snowstorm, even if the snow is falling straight down, it appears to be moving almost directly at you. The total speed of the snowflake relative to the car is calculated from the Pythagorean theorem: it's the hypotenuse of the right triangle formed from the vertical and horizontal components of speed. This amounts to 4.6 m/s.

Our snowflake is now traveling 4.6 times faster than it was when it was just falling, which makes it 21.25 times as energetic. This means it has more than enough energy to sublimate itself. In fact, we can calculate what the critical speed really is (given our assumptions). The snowflake needs to be traveling √8.5 times faster, or 2.9 m/s. The horizontal component of that speed is 2.7 m/s, or about 6 mph.

I think that was a pretty darn good estimate, if you consider the vast number of factors I was hand-waving away. It's possible that there are a whole bunch of factors swinging one way and an equal number swinging the opposite way, making me only accidentally correct, but regardless, it's pretty nice to have your "theory" match up with your "data." Go Team Science!