Math

In 1471, Johannes Müller asked where you should stand so that a vertical bar appears longest.

To be more precise, suppose a vertical bar is hanging so that the top of the bar is a distance a above your eye level and the bottom is a distance b above your eye level. Let x be the horizontal distance to the bar. For what value of x does the bar appear longest?

Note that the apparent length of the bar is determined by the size of the angle between your lines of sight to the top and bottom of the bar.

Please don’t give solutions in the comments. I’ll post my solution tomorrow, and you can give your solutions in the comments to that post if you’d like.

Source

Update: See this post for more historical background.

Fermat’s last theorem is so named because it was the last of his asserted theorems to be proved or disproved. But there are variations on another conjectures of Fermat that remain unresolved.

Fermat conjectured that numbers

are always prime. We now call these “Fermat numbers.” Fermat knew that the first five, F₀ through F₄, were all prime.

In some ways, this conjecture failed spectacularly. Euler showed in 1732 that the next number in the sequence, F₅, is not prime by factoring it into 641 × 6700417. So are all the Fermat numbers prime? No.

But that’s not the end of the story. Now we go back and refine Fermat’s conjecture. Instead of asking whether all F_n are prime, we could ask which F_n are prime.

The next several values, F₅ through F₃₂, are all known to be composite. So we might turn Fermat’s original conjecture around: are all F_n composite (for n > 4)? Nobody knows.

Well, let’s try weakening the conjecture. Is F_n composite for infinitely many values of n? Nobody knows. Is F_n prime for infinitely many values of n? Nobody knows that either, though at least one of these two statements must be true!

Here’s why I find all this interesting.

1. It shows how proof by example fails. Fermat knew that the first five numbers in his series were prime. It was reasonable to guess from this that the rest might be prime, though that turned out not to be the case.
2. It shows how what appears to be a dead end can be opened back up with a small refinement of the original question.
3. Like many questions in number theory, the revised question is easy to state but has defied proof for centuries.

The comments in the previous post touched on surprising applications of math, so I thought I’d expand this theme into it’s own post. Below I’ll give a couple general examples of surprising applications and then I’ll give a couple more personal applications I found surprising.

Number theory has traditionally been the purest of pure mathematics. People study number theory for the joy of doing so, not to make money. At least that was largely true until the advent of public key cryptography. The difficulty of solving certain number theory problems now ensures the difficulty of decrypting private communication, or so we hope. (By the way, I’ve always thought Euler deserved part of the credit for the RSA encryption scheme. Maybe it should be called RSAE encryption. R, S, and A came up with the brilliant idea to apply E’s theorem to cryptography.)

Non-euclidean geometry started as a pure mathematical abstraction. Of course the physical world is Euclidean, but let’s see what happens if we monkey with Euclid’s fifth postulate. Then along came Einstein and suddenly the real world is non-Euclidean.

One personal application of math that I found surprising was using Fibonacci numbers in practical computation. Computing Fibonacci numbers is a computer science cliché, but I actually needed to compute Fibonacci numbers for a numerical integration problem. I wrote up the details in Fibonacci numbers at work.

Another application that surprised me was using the trapezoid rule for real work. The trapezoid rule is a crude numerical integration technique. It’s good for teaching because it’s very simple, but it’s not very accurate. Or so I thought. It’s not very accurate in general, but in the right circumstances, it can be extraordinarily accurate. I explain more in Three surprises with the trapezoid rule.

One surprising non-application has been differential equations. For the past three centuries, differential equations have been at the heart of applied math. One could argue that to first approximation, applied math equals differential equations and supporting material. But I personally have not had nearly as much opportunity to use differential equations professionally as I expected, even though that was my specialization in grad school.

Related posts:

Ten surprises in numerical linear algebra
Impure math

This morning’s xkcd cartoon is “What your favorite map projection says about you.” It’s really funny, but poor Mercator comes off as the most boring projection.

Mercator is the most familiar projection, but it has some interesting properties. The most important is that lines of constant bearing on the Earth correspond to straight lines on the map, obviously a desirable property for navigation. More details here.

The inverse of the Mercator projection, going from maps onto the globe, is more interesting. Aside from its geographical importance, it gives a way of relating circular and hyperbolic functions without using complex numbers. More details here.

The Mercator projection is also historically interesting. The modern derivation of the Mercator projection uses logarithms and calculus, but Mercator came up with his projection in 1569 before logarithms or calculus had been discovered. (Update: See more historical detail in Thony C’s comment below.)

In 1989, Star Trek: The Next Generation aired The Royale. In this episode, we learn that Captain Picard tries his hand at proving Fermat’s Last Theorem (FLT) in his spare time. The writers must have believed that FLT would still be unresolved in the 24th century. Four years after The Royale, Andrew Wiles announced his proof of FLT. There was a flaw in Wiles’ first proof, but this was patched two years later in 1995.

Richard Feynman also tried his hand at FLT. He wrote a paper (unpublished) in which he gave a pseudo-proof of FLT based on probability. Feynman said that the probability of FLT being false was “certainly less than 10^-200.” The argument was high creative, sketchy, but ultimately nonsensical. Paul Nahin concludes

Feynman’s probabilistic analysis of Fermat’s Last Theorem would have no mathematical interest at all but for the fact it was Feynman who cooked it up.

Source: Number-Crunching by Paul Nahin.

I love this trig identity. I could imagine a student believing it for the wrong reason, a grader counting it wrong for the wrong reason, and a teacher counting it right for the right reason.

sin(x – y) sin(x + y) = (sin(x) – sin(y)) (sin(x) + sin(y))

Someone manipulating symbols unknowingly might think this is obviously true: of course you can replace sin(x + y) with sin(x) + sin(y) and replace sin(x – y) with sin(x) – sin(y). All the world is linear.

Someone with a little more experience would say that this identity obviously cannot be true. After all, sin(x ± y) clearly does not equal sin(x) ± sin(y).

But someone with a little more patience might get a pencil and paper and work out that it indeed is true. Even though naive symbol manipulation would be wrong-headed, in this case it happens to lead you to the right result.

For more examples of a novice and an expert agreeing but someone in between disagreeing, see Coming full circle.

Related posts:

How many trig functions are there?
When does the sum of three numbers equal their product?

From C. S. Lewis:

It has always therefore been one of my main endeavors as a teacher to persuade the young that firsthand knowledge is not only more worth acquiring than secondhand knowledge, but it usually much easier and more delightful to acquire.

This quote comes from the essay On the Reading of Old Books, part of the collection God in the Dock: Essays on Theology and Ethics. Lewis says here that it is easier to read Plato or St. Paul, for example, than to read books about Plato or St. Paul.  Lewis says that the fear of reading great authors

… springs from humility. The student is half afraid to meet one of the great philosophers face to face. He feels himself inadequate and thinks he will not understand him. But if he only knew, the great man, just because of his greatness, is much more intelligible than his modern commentators.

This does not only apply to literature. I see the same theme in math. Sometimes early math papers are easier to read because they are more concrete. When I was a postdoc at Vanderbilt I asked Richard Arenstorf about a theorem attributed to him in a book I was reading. He scoffed that he didn’t recognize it. He had done his work in a relatively concrete setting and did not approve of the fancy window dressing the author had placed around his theorem. I sat in on a few lectures by Arenstorf and found them amazingly clear.

The same theme appears in software development. Sometimes you can dive to the bottom of an abstraction hierarchy and find that things are simpler there than you would have supposed. The intervening layers obscure the substance of the program, making its core seem unduly mysterious. Like a mediocre mind commenting on the work of a great mind, developers who build layers of software around core functionality intend to make things easier but may do the opposite.

Related posts:

Endless preparation
Opening black boxes
Why Shakespeare is hard to read
C. S. Lewis on reading old books

Imagine a set of points in three dimensions. You want to connect the points with the shortest possible network. Sometimes you can make the network shorter by adding extra points. (These extra points are called Steiner points.) How much can extra points help? The answer is worth $1,000.

Here’s an example. Suppose our points are the corners of a unit cube. You can connect these points with a network of length 7. If you add a point in the center of the cube and connect every point to the center, you get a network of length 4 √ 3 = 6.928. So in this case, adding an extra point made it possible to reduce the size of the minimal spanning network by about 1%. You could do better by adding more points.

What is the most you can reduce the length of the minimum spanning network in three dimensions by adding extra points? The question concerns all possible sets of points, not just a particular set like the example above. It is conjectured that the most you can save is about 21.6%. That is, for any set of points, the ratio of the length of the shortest network with extra points to that of the shortest network without extra points is bounded below by

In their new book Magical Mathematics, Persi Diaconis and Ron Graham say “We currently offer a thousand dollars for a proof (or disproof) that this ratio is the best possible.”

As unwieldy as the number above appears, it makes some sense. It looks like the square roots come from repeated applications of the Pythagorean theorem. Someone may be able to reverse engineer the example the conjecture is based on by using the form of the proposed lower bound.

(Diaconis and Graham say that the corresponding problem in two dimensions have been solved and the optimal ratio is √ 3 / 2. However, this paper says that the conjecture is still unresolved, contrary to popular belief.)

Martin Gardner (1914 – 2010) was best known for his articles on recreational mathematics, especially his column in Scientific American which he wrote from 1956 to 1981. Once Gardner wrote a letter of recommendation for a young man applying to graduate school at Harvard.

I don’t know a lot about mathematics, but this kid invented two of the best card tricks of the last ten years. You ought to give him a chance.

The kid was Persi Diaconis. At the time, Diaconis, like Gardner, was something of a mathematical outsider, someone with more creativity than credentials. Diaconis went on to become a mathematics professor at Harvard and won two MacArthur genius awards. He is now a professor at Stanford.

Source: Magical Mathematics

“Nothing brings fear to my heart more than a floating point number.” — Gerald Jay Sussman

The context of the above quote was Sussman’s presentation We really don’t know how to compute. It was a great presentation and I’m very impressed by Sussman. But I take exception to his quote.

I believe what he meant by his quote was that he finds floating point arithmetic unsettling because it is not as easy to rigorously understand as integer arithmetic. Fair enough. Floating point arithmetic can be tricky. Things can go spectacularly bad for reasons that catch you off guard if you’re unprepared. But I’ve been doing numerical programming long enough that I believe I know where the landmines are and how to stay away from them. And even if I’m wrong, I have bigger worries.

Nothing brings fear to my heart more than modeling error.

The weakest link in applied math is often the step of turning a physical problem into a mathematical problem. We begin with a raft of assumptions that are educated guesses. We know these assumptions can’t be exactly correct, but we suspect (hope) that the deviations from reality are small enough that they won’t invalidate the conclusions. In any case, these assumptions are usually far more questionable than the assumption that floating point arithmetic is sufficiently accurate.

Modeling error is usually several orders of magnitude greater than floating point error. People who nonchalantly model the real world and then sneer at floating point as just an approximation strain at gnats and swallow camels.

In between modeling error and floating point error on my scale of worries is approximation error. As Nick Trefethen has said, if computers were suddenly able to do arithmetic with perfect accuracy, 90% of numerical analysis would remain important.

To illustrate the difference between modeling error, approximation error, and floating point error, suppose you decide that the probability of something can be represented by a normal distribution. This is actually two assumptions: that the process is random, and that as a random variable it has a normal distribution. Those assumptions won’t be exactly true, so this introduces some modeling error.

Next we have to compute something about a normal distribution, say the probability of a normal random variable being in some range. This probability is given by an integral, and some algorithm estimates this integral and introduces approximation error. The approximation error would exist even if the steps in the algorithm could be carried out in infinite precision. But the steps are not carried out with infinite precision, so there is some error introduced by implementing the algorithm with floating point numbers.

For a simple example like this, approximation error and floating point error will typically be about the same size, both extremely small. But in a more complex example, say something involving a high-dimensional integral, the approximation error could be much larger than floating point error, but still smaller than modeling error. I imagine approximation error is often roughly the geometric mean of modeling error and floating point error, i.e. somewhere around the middle of the two on a log scale.

In Sussman’s presentation he says that people worry too much about correctness. Often correctness is not that important. It’s often good enough to produce a correct answer with reasonably high probability, provided the consequences of an error are controlled. I agree, but in light of that it seems odd to be too worried about inaccuracy from floating point arithmetic. I suspect he’s not that worried about floating point and that the opening quote was just an entertaining way to say that floating point math can be tricky.

Related posts:

Floating point numbers are a leaky abstraction
Avoiding overflow, underflow, and loss of precision
Just an approximation

Here’s an elegant proof from Paul Erdős that there are infinitely many primes. It also does more, giving a lower bound on π(N), the number of primes less than N. [click to continue...]

The tautochrone is a curve such that a ball rolling down the curve takes the same amount of time to reach the bottom, no matter where along the curve it starts. (The name comes from the Greek tauto for same and chrono for time.) It doesn’t sound like such a curve should be possible because balls starting further up the curve have longer to travel. However, balls starting higher also have more potential energy, and so they travel further but faster. See the video below for a demonstration.

[The video is entitled "brachistochrone race" rather than "tautochrone race." The brachistochrone problem is to find the curve of fastest descent. But its solution is the same curve as the tautochrone. So different problems, same solution.]

I first heard of the tautochrone as a differential equation problem to find its equation. But someone could run into it in an American literature class.

Clifford Pickover’s new book The Physics Book has a chapter on the tautochrone. (In this book, “chapters” are only two pages: one page of prose and one full-page illustration.) Pickover points out a passage in Moby Dick that discusses a bowl called a try-pot that is shaped like a tautochrone in the radial direction.

[The try-pot] is a place also for profound mathematical meditation. It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.

Suppose you have a right triangle and every side has rational length. What can you say about the area? For example, is it possible that such a triangle could have area 1?

It turns out the answer is no, and here’s a proof. Suppose you had a right triangle with sides of length a, b, and c with c being the length of the hypotenuse. And suppose a, b, and c are all rational numbers.

Start with the equation

(a² – b²)² = (a² + b²)² – 4a²b²

Suppose the area of the triangle is 1. Then ab/2 = 1 and so ab = 2. Use this and the Pythagorean theorem to rewrite the right side of the equation above. Now we have

(a² – b²)² = c⁴ – 16

But this result contradicts a theorem of Fermat: in rational numbers, the difference of two fourth powers cannot be a square.

So a rational right triangle cannot have area 1. Also, it cannot have area r² for any rational number r. (If it did, you could divide each side by r and have a rational triangle with area 1.)

Now things are about to get a lot deeper.

What values can the area of a rational right triangle take on? Euler called such numbers congruent. Determining easily testable criteria for congruent numbers is an open problem. However, if the Birch and Swinnerton-Dyer conjecture is correct, then an algorithm of Jerrold Tunnell resolves the congruent number problem. (Incidentally, the Clay Mathematics Institute has placed a $1,000,000 bounty on the Birch and Swinnerton-Dyer conjecture.)

What if instead of limiting the problem to rational right triangles we considered all triangles with rational sides? See this paper for such a generalization.

Several people have asked me whether studying math changed my view of the world, and if so how.

I see applications of math everywhere. But more fundamentally, studying math has led me to believe that complex problems often have simple solutions.

Simple solutions may be hard or impossible to find. But you’re more likely to find a simple solution if you believe it exists because you’ll keep looking longer.

Related posts:

Forced to be simple
Rewarding complexity
Adding simplicity