Alvin Moon led the Math Circle through some lovely explorations of one of the central concepts of advanced mathematics: limits! We can develop intuition about this deep idea with a few geometric thought experiments. This topic was designed to be accessible to young students but still new and beautiful to more advanced Math Circlers.
See Alvin’s presentation and worksheet attached here for details.
In a previous talk, we showed how to predict the mind-bending phenomenon of time-dilation at near-light speeds using basic geometry and algebra. This time, after a review of the setup from Part 1, we will discuss one of the delightfully impossible consequences of faster-than-light travel, namely what Einstein called “telegraphing into the past”. I’d like to acknowledge wikipedia for the example at the end with easy numbers. A couple things to remember before you read: c is the speed of light, v refers to the speed you’re going, x is the position you’re at, and strange things happen at high speeds! This post was adapted from Davis Math Circle lecture notes from February 28, 2015.
How to Measure Distance and Time at High Velocity
Imagine you are riding a bus going at a nice steady speed down the street. You stand at the back of the bus and toss a ball towards the front of the bus. You see the ball moving at, say, 10mph. However, suppose Bob is standing on the sidewalk and looking at the ball. He will see it moving much faster. If the bus is going 25mph, then the ball will appear to go 25+10=35mph. Well, which is it? Is the ball going 10mph or 35mph? Is it white/gold or black/blue? It depends on your frame of reference.
The frame of reference is a much more profound concept than “a point of view” as people often say. If that ball hits Alice on the bus at 10mph, then she will be fine since the ball isn’t going very fast. If the ball left the bus and hit Bob at 35mph, that could be serious! Especially since Bob would probably be too stunned to realize he’s about to be run over by the bus. Notice that if you had thrown the ball towards the back of the bus, Bob would see it going 25-10=15mph.
Now, instead of tossing a ball, imagine you stand in the middle of the bus and shine a laser towards the front. Don’t point lasers at people, by the way; just imagine. On the bus, the light will travel at a speed of about 299,792,458 meters per second, or about 670,616,629 mph, and the same if you pointed it towards the back of the bus. (I only say “about” because that is the exact speed of light in a vacuum, by definition. I hope for the sake of your imaginary self that you also imagined that there was air on that bus!)
For about 130 years, scientists have very carefully checked that somebody standing on the sidewalk will also see the laser light moving at 299,792,458 meters per second, not 299,792,458 meters per second plus or minus 25mph. Why is light different from balls? I’ll leave that mystery to your physics classes. However, from this simple observation and the Pythagorean theorem, we can derive some of the deepest mathematical formulas you may ever see. (Which is what we did last time. That post will follow when my lecture notes catch up.)
The key concept here is the frame of reference. In Bob’s frame of reference, he is standing still and the bus is going by at 25mph. From your frame of reference, you and the bus are standing still while Bob and the rest of the world goes by in the other direction. Since speeds are being measured so differently, we might also worry that time and distance are being measured differently, too. Outside the bus, Bob uses his tools (a stick and a clock, say) to measure position x and time t, while you use your own tools on the bus to measure x’ and t’.
In the first lecture, we showed that if you’re traveling at speed v, the two systems of coordinates are related by
The first formula is often called time dilation and it implies that Bob will observe your clock running slower than his. The second formula is called Lorentz contraction, named after one of the physicists who proposed these formulas to explain the mysterious Michelson-Morley experiment before Einstein blew everybody away with his new way of thinking about frames of reference, the theory of relativity. Lorentz contraction means that Bob will think your meter stick looks shorter than a meter.
We don’t run into time dilation and Lorentz contraction when riding buses and tossing balls around because that factor is almost exactly 1. Highway speeds are typically under 35 meters per second, and
Speeds where this factor does make a noticeable difference are called relativistic speeds. However, since we are flying through space with nothing but a comically thin layer of gas and a magnetic field between us and a big universe full of constantly exploding nuclear fusion reactors, we do get bombarded by radioactive particles for whom these formulas make quite a significant difference. In the first lecture, we talked in detail about the case of the muon, the electron’s fat cousin.
In today’s Math Circle, we will use the Lorentz transformations to play around with what would happen if we could text each other faster than the speed of light. This sounds somewhat innocent, but every rational thing you hold dear completely falls apart. First, let’s write down one more formula from Einstein:
We’ve all seen the famous formula a million times, so what is this thing? The formula applies to something that is not moving. Which is incredible because it implies that all matter has innate potential energy. Once something gets moving, it has kinetic energy. In the slow world of busses and balls, the kinetic energy from movement, is . At higher velocities, the formula above is much more accurate.
Exercise: For those of you who have seen derivatives or Newton’s binomial theorem, can you show that for v much smaller than c?.
Let’s focus on that term that keeps popping up. Remember that it came from the Pythagorean theorem in our derivation last time. What happens as v gets close to c? Well, v/c gets close to 1, so gets close to 0. If we are dividing by a number close to zero, that is the same as multiplying by a huge number! That means that anything trying to move at speeds closer and closer to the speed of light will need more and more energy. To reach the speed of light or exceed it would take an infinite amount of energy! So, everything around us, no matter how hard we push it, will never go as fast as the speed of light.
Well… except for that one thing. Light! Why does light get to go so fast? Well, there’s a nice work-around in Einstein’s formula: just set the mass to 0, and you can go light-speed without any trouble. Could there be a such work-around for faster than light travel?
If you’ve consumed much science fiction, then you’ve probably run across the word “tachyon” before. Particle names end in “-on” like proton, electron, gluon, etc. What does “tachy-” mean? Fast! By the way, the “ch” is like the “ch” in “Chris”, not “nacho” or “machine”. Tachyons are theoretical fast particles that would move at speeds beyond that of light, a.k.a. superluminal.
But wait! Didn’t we just say it was impossible to make something go from below the speed of light to above it? Yes… any future lawyers or logicians amongst you? What if the object never went slower than the speed of light? What if it came into existence already at higher speeds? Well, then we call it a tachyon. Let’s look at some of their wacky properties.
Since , and . Do you see the weirdness? The energy of a tachyon would be
an imaginary number! But we don’t want that. If it’s going 400,000,000 meters per second, it should have some amount of energy that would be measurable. How can we fix this? Well, you might not think this is much better, but we could assume that it has imaginary mass, . That way, the would cancel, and we’d be back to a real energy. Perhaps even an energy that’s small enough to produce such particles in a particle accelerator!
Ok, tachyons are theoretical particles with imaginary mass that move faster than the speed of light. Got it? In that case, it takes an infinite amount of energy to slow them down to light speed.
All of this so far is exotic, but not necessarily impossible. To the impossible!
Suppose that Alice and Bob have communication devices that use tachyons. This allows them to communicate faster than the speed of light, say at speed . Suppose Alice is at position A and Bob at position B, both in the same reference frame. Alice sends a signal at time and Bob receives it at time . Then the time it took was
since distance over speed is time.
That all seems perfectly ordinary and boring. But now, suppose you are flying by in a space ship going at relativistic speeds. In your reference frame, the time it takes the signal to go from Alice to Bo is
Ok… a lot of algebra happens. So what? Well, if you’re moving fast enough so that , which is possible since , then you would see Bob get the message before Alice sends it!!
Exercise: Get a feel for how much of an effect this is by plugging in different values for a, v, and .
Now let’s figure out what happens from Alice’s perspective if she messages Bob at superluminal speed and then gets a superluminal message back while traveling at relativistic speed. To make the formulas a little simpler, let’s measure time in seconds and distance in light-seconds so that . We also use the standard shorthand of writing (That is the Greek letter gamma, the equivalent of a lower-case g.)
Suppose Alice is moving away from Bob at speed . She sends Bob a tachyonic text with speed , then Bob immediately texts back with the same kind of device when he gets the message.
Alice’s reference frame: She stays at position . Her message travels a distance .
Bob’s reference frame: He remains at position , and his reply will be at since the distance it travels is .
Alice’s reference frame: The reply signal is at
So, if , then
The time it took the message to get to Bob was , so the total time elapsed between when Alice sends the message and when Alice receives the message is
Exercise: Show for yourself that if then
Two-Way Example with Numbers
Let’s try a scenario where Alice sends a message to Bob, then Bob replies. To keep things concrete, we’ll use particular numbers. Let’s measure time in units of days, distance in units of light-days. Once again, this has the nice bonus consequence of in these units.
Suppose Alice and Bob are each in their own spaceship traveling at a relative velocity of . When they pass each other, they synchronize their clocks and set that position to 0.
Alice’s reference frame: Alice stays at while Bob moves along .
Bob’s reference frame: Bob stays at while Alice moves along .
Alice’s reference frame: After 300 days (), Alice sends a message at speed 2.4 to Bob, “I should have played 7 49 53 60 64 4 in the MegaMillions Lottery yesterday!” Then, 150 days later (), the signal is at position light-days. Bob’s position is light-days. So, in Alice’s reference frame, Bob gets the message at .
Bob’s reference frame: We can compute that , so Bob receives the message at due to time dilation. He then immediately replies, “Play 7 49 53 60 64 4 in the MegaMillions when !” In his frame, 135 days later, Alice’s position is now . The position of the signal is , so that is where/when Bob thinks Alice gets the message.
Alice’s reference frame: Here, we also have that , so Alice receives the message at . Almost two months before the lottery at ! Small snag: if she wins the lottery, why would she send the message? But then how did she win the lottery? What does it mean for one event to come before or after another one? I guess it could be worse:
Go ahead and calculate when Alice would receive a reply if she replied to the lotto numbers message. And if they did it again? Can you find a way for Alice to get a message before she ever even meets Bob? Play around with some other numbers. To get nice numbers like I did, use speeds that are multiples of one another, and a that gives a Pythagorean triple in order to get a nice .
Exercise: Pictured below is the Time Code found on the mysterious tattoo that appeared on Fry’s butt which allows for time travel. Can you see where the writers got these particular binary strings?
Many people question the usefulness of abstract math problems in real life. Personally, I question the usefulness of somebody’s definition of usefulness if they are even asking that question! Math is about more than building a better mouse trap; it is about deepening our understanding of everything. Yes, everything! Whether or not a certain piece of mathematics helps us understand phenomena in our life is not up to the math but to our imagination and own willingness to seek out structure and meaning.
Here is a delightful and fun example. The Parable of the Polygons. The game and simulation in this article are based on the work of a Nobel prize-winning economist, Thomas Schelling. Using very simple math and profound insight, we can clarify an issue of great importance in society: segregation. If most people only have a slight bias and actually like diversity, why is there so much segregation in the world? Play the game to see one way in which grand global effects arise from small local changes.
In Math Circle this weekend, we talked about how we can talk about fractals mathematically as sequences of recurring patterns on smaller and smaller scales. The Sierpinski Triangle served as our key example for this discussion. We showed that this object has a perimeter of infinite length but that it contains no area! Such apparent contradictions occur frequently in nature, but we have a developed mathematical understanding of them.
Last Saturday, we had two speakers Eric Brattain (me, our organizer) and Patrick Weed.
I spoke about a topic usually called countability. This is actual a fairly terrible name since it’s about precisely that which can’t be counted in the usual sense, infinite quantities of things. We used the classic example of the Hilbert Hotel to get a feeling for how to map infinite sets into each other.
Imagine a hotel where the rooms are numbered 1,2,3,4,… and for every whole number, there’s a room labeled with that number (maybe in very small font). Now, suppose that when you arrive, the hotel is full! Well, no problem! You can simply request that the hotel manager scoot everybody down one room, 1->2, 2->3, etc. Then, room #1 is available for you! This means that there are not more numbers in the list 1,2,3,4,… than in 2,3,4,5,… which is a little strange.
Since that worked out so well, the next time you visit the Hilbert Hotel, you bring a busload of friends. They are sitting in seats numbered 1,2,3,4,… and for every whole number there is an occupied seat with that number. The hotel manager objects, saying that he can’t possibly have all of the guests march down the hallway forever. Besides, they’d never make enough room for everybody that way! Luckily, being a mathematical enthusiast, you know there’s a simpler way. Simply have each guest double their room number and go to that room. Then, only the even-numbered rooms will be occupied, and you and your party can take the odd-numbered rooms! Hurray! Once again, it’s a bit strange that there are no more whole numbers than there are even numbers. This kind of thinking leads to one of my favorite examples of mathematical ingenuity. We appear to have reached a paradox when we say a subset of numbers has no fewer numbers in it despite leaving some out (like the first one or even all of the odd numbers). What do we do? Declare that this is the definition of an infinite set! Bam!
The next day, you find the hotel manager arguing with somebody who has just brought another infinite busload of visitors. The problem is that the seats in the bus are numbered by fractions! For every fraction a/b, there is an occupied seat. The hotel manager claims that there are simply not enough rooms in the hotel, and it is full besides. The visitor objects that infinity is infinity, so if they accommodated your party, then they must be able to do so for hers. So, we know how to make infinitely many rooms available, but what instructions will allow each fraction-numbered passenger to know what room to go to?
To start, let’s assume they’re positive fractions, a/b, with a,b>0. If we pick a numerator and go through all possible denominators, then we end up with infinitely many… but there are infinitely many numerators to pick from! Infinitely many infinities? Uh oh! We need to be a little more clever. If we restrict our attention to fractions with a+b=n for some n>0, then we only have finitely many. For example, if n=2, then we have just 1/1=1. If n=3, then we get 2/1=2 and 1/2. For n=4, there’s 1/3, 2/2=1, and 3/1=3. There’s just n-1 numbers to consider. Notice that some numbers repeat, like 1/1 and 2/2. To make really good instructions, you’d have to keep track of this if the seats are all in the form of reduced fractions. I’ll leave that as an exercise for you. Basically, for each n, we assign the n-1 passengers with a+b=n to the next available n-1 hotel rooms.
An almost identical system lets you deal with an infinite number of infinite busloads all at once! Check out the picture below from an old NYtimes column by Strogatz.
Now we reach a much deeper conundrum. Yes, it gets deeper than infinity infinities is still infinity. One dark and stormy night, a shuttle arrives where the seats are numbered by decimal numbers, a.k.a. real numbers, like etc. What clever way can we provide these people with hospitality? There is no way! Contemplate this:
No matter how you assign the passengers (imagine the list above as being who goes to room 1,2,3,…), I can always find at least one that is left out despite your claim that you have them all. So… some infinities are bigger than others! Well, no problem. Send the bus down the road to Cantor’s Paradise Resort with rooms labeled by colors. Exercise: Why did that last sentence sort of make sense?
Can you find sets that are bigger than the real numbers? For those of you who’ve held on this long, here’s a little tidbit to knock you off the rails. Is there a set with more elements than 1,2,3,… but less than the decimals? Why or why not? It turns out that the answer to this question is fundamentally undecidable. Math doesn’t always have the answers!
By the way, this wondrous madness came from the mind of Georg Cantor:
If you like this infinity stuff, I highly recommend the graphic novel Logicomix: An Epic Search for Truth.
Speaking of logic, Patrick Weed gave you an entertaining introduction to symbolic logic. Doing serious math or programming without knowing symbolic logic is like deciding to be a writer without knowing basic grammar. No, most writers don’t get excited about grammatical details, but they sure do know them well enough to make jokes about those who don’t.
Here’s a copy of the handout that he provided. Patrick Weed’s Math Circle on Logic
Also, there was some talk of how to map a line onto a plane. By onto, I mean so that every point in the plane is hit by a point from the line. This seems impossible, but if you ponder this illustration of a Hilbert curve, then it will still seem impossible. But it works! :-p
It sounds like everybody enjoyed our day of mathematical biology brought to you by Professor Mogilner and Swati Patel. Applications of mathematics to science are vast, deep, and ever-expanding. Many people are surprised to hear that mathematics can be so successfully applied to biology, probably since these topics haven’t made it very far into the high school curriculum.
Prof. Mogilner’s talk on allometric scaling was based on research published in the 1990s (West_Brown_Enquist_1997) that has been cited thousands of times. Since wikipedia is always a mere click away, you should try to read this. Here is an article that explains the paper a little more clearly than the original: Demystifying the West, Brown & Enquist model of the allometry of metabolism. And, here is an article talking a bit more about the biology of allometry using the example of fiddler crabs with one ginormous claw and one itty-bitty claw. I guess they look like they’re playing the fiddle? Cellist crab might be more apt.
Once you learn about fractals, it’s hard not to see them everywhere in nature.
Swati’s talk about modeling populations provided ideas that get your foot in the door to understanding chaos theory. And who wouldn’t want to know more about chaos??
The only limit to the applications of math is the imagination of the mathematician. We’ll have more professors and graduate students share some examples of applied math with you later this quarter.
I recently read an interesting article supporting what many of us who work in math and mathematics education believe from experience: practice makes you better at math.
However, this somehow manages to be simultaneously completely obvious and nearly impossible to convince people of. Better at algebra than geometry? Practice some more geometry. Trouble with word problems? Practice them as well as your reading in general. The analogy with athletics is completely apt.
As students, we tend to do more of what makes us feel good, which is of course what we are already good at. I have seen many students pride themselves so much on their ability at one thing that they neglect other studies and activities. This will of course lead to the student becoming better and better at that one activity and worse and worse at everything else. Many believe that students showing this one-dimensionality were somehow born different, but I’d venture to say that the vast, vast majority of such students have just engaged in overly narrow practice. Of course, there are a handful of people out there with such enormous capacity at such a young age that it doesn’t seem possible for there to have been sufficient hours in their life for their abilities to be due to practice alone, but I bet the differences between their inherent ability and that of others is much smaller than most people suppose.
What is the point of this rant? You can be good at math! Hurray! The difference between math people and non-math people is that math people do more math, not that they were born with some sort of abstract antenna in their brain that makes them sensitive to mathematical ideas or whatever.
Speaking of getting better at these things, see you at AMC practice on Saturday at 1pm! If you will be joining us for lunch, bringing a few bucks to cover cost of pizza would be much appreciated (but not required).
Email me if you’d like electronic copies of a few old AMC exams for practice.