# The Mathematics of Juggling and more from George Hart

[Dr. Chase guest blogging again]

You’re probably familiar with Vi Hart’s math videos. Less well-known are her father’s math videos. Although I was aware of his mathematical sculpture, I was not aware until today that since August 2012, he has been producing a mathematical video series called mathematical impressions for the Simons Foundation. The 10th in the series is The Mathematics of Juggling. Check it out!

# Progress Toward Twin-Prime Conjecture

This nice article came through on wired today:

# Unknown Mathematician Proves Surprising Property of Prime Numbers

By Erica Klarreich, Simons Science News

Image: bwright923/Flickr

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

(more)

This is very exciting news, and the whole story has a fantastic David & Goliath feel–“little known mathematician delivers a crushing blow to a centuries old problem” (not a fatal blow, but a crushing one). It’s such a feel-good story, almost like Andrew Wiles and Fermat’s Last Theorem. Here’s my favorite part of the article:

…during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said.

Just chillin’ in his friend’s backyard…and it came to him! Anyone who has worked on math problems or puzzles has had this experience, right? It seems like an experience common to all people. This has definitely happened to me lots of times–an insight hits me out of nowhere and unlocks a problem I’ve been working on for weeks. It’s one of the reasons we do mathematics!

# What is a Point of Inflection?

Simple question right?

This website, along with the Calc book we’re teaching from, define it this way:

A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.

No debate about there being an inflection point at x=0 on this graph.

There’s no debate about functions like $f(x)=x^3-x$, which has an unambiguous inflection point at $x=0$.

In fact, I think we’re all in agreement that:

1. There has to be a change in concavity. That is, we require that for $x we have $f''(x)<0$ and for $x>c$ we have $f''(x)>0$, or vice versa.*
2. The original function $f$ has to be continuous at $x=c$. That is, $f(x)=\frac{1}{x^2}$ does not have a point of inflection at $x=0$ even though there’s a concavity change because $f$ isn’t even defined here. If we then piecewise-define $f$ so that it carries the same values except at $x=0$ for which we define $f(0)=5$, we still don’t consider this a point of inflection because of the lack of continuity.

The point of inflection x=0 is at a location without a first derivative. A “tangent line” still exists, however.

But the part of the definition that requires $f$ to have a tangent line is problematic, in my opinion. I know why they say it this way, of course. They want to capture functions that have a concavity change across a vertical tangent line, such as $f(x)=\sqrt[3]{x}$. Here we have a concavity change (concave up to concave down) across $x=0$ and there is a tangent line ($x=0$) but $f'(0)$ is undefined.

Is x=0 a point of inflection? Some definitions say no, because no tangent line exists.

So It’s clear that this definition is built to include vertical tangents. It’s also obvious that the definition is built in such a way as to exclude cusps and corners. Why? What’s wrong with a cusp or corner being a point of inflection? I would claim that the piecewise-defined function $f(x)$ shown above has a point of inflection at $x=0$ even though no tangent line exists here. [Edit: Originally I had used $f(x)=x^{2/3}$ as my example, but as my dad so astutely pointed out, that does have a tangent line at zero. Doh! Thanks dad!]

I prefer the definition:

A point where the graph of a function is continuous and where the concavity changes is a point of inflection.

That is, I would only require the two conditions listed at the beginning of this post. What do you think?

Once you’re done thinking about that, consider this strange example that has no point of inflection even though there’s a concavity change. As my colleague Matt suggests, could we consider this a region of inflection? Now we’re just being silly, right?

A region/interval of inflection?

———————————————

Footnotes:

* When we say that a function is concave up or down on a certain interval, we actually mean $f''(x)>0$ or $f''(x)<0$ for the whole interval except at finitely many locations. If there are point discontinuities, we still consider the interval to have the same concavity.

** This source, interestingly, seems to require differentiability at the point. I think most of us would agree this is too strong a requirement, right?

# Friday fun from around the web

Here are two fun mathy things that came through my feed today. Many of you have probably already seen today’s math-themed xkcd:

And I also saw this today [on thereifixedit], which delighted the mathematician in me:

Happy Friday everyone!

I’ve been loving the videos that SpikedMathGames has been posting on youtube. Check out their channel here. In particular, I’ve enjoyed Paradox Tuesday. Here’s one from a few weeks ago which really interested me (if you go to the youtube page, you’ll see I’ve been active in the comments!):

They also cover some famous paradoxes like the Unexpected Hanging Paradox or the  Barber Paradox. If you’ve never heard of these, go watch these now.

I’m especially interested in paradoxes that deal with infinity, countability, and probability. Here’s another great paradox that deals with just those issues that my friend Matthew Wright shared with me a few months ago (thanks Matthew!). It’s called the Grim Reaper paradox (can’t link to the Wikipedia article–it doesn’t yet exist), proposed in 1964 by José Amado Benardete in his book Infinity: an essay in metaphysics, and I first read about it on Alexander Pruss’s blog here, and I quote:

Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive, and if you’re alive, it instantaneously kills you, and if you’re not alive, it doesn’t do anything. Suppose there are countably infinitely many Grim Reapers, and before they go to bed for the night, each sets his alarm for a time (not necessarily the same time as the other Reapers) strictly between 8 and 9 am. Suppose, also, that no other kind of death is available for you, and that you’re not going to be resurrected that day.

Then, you’re going to be dead at 9 am, since as long as at least one Grim Reaper wakes up during that time period, you’re guaranteed to be dead. Now whether there is a paradox here depends on how the Grim Reapers individually set their alarm clocks. Suppose now that they set them in such a way that the following proposition p is true:

(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.

Here’s a useful Theorem: If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.

Now, if p is true, then no Grim Reaper kills you. For suppose that a Grim Reaper who wakes up at some time t1, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t1, say at t0. But if so, then you’re going to be dead right after t0, and hence the Grim Reaper who woke up at t1 is not going to do anything, since you’re dead then. Hence, if p is true, no Grim Reaper kills you. On the other hand, I’ve shown that it is certain that a Grim Reaper kills you. Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

Go visit his blog post for a discussion of why this seems unresolvable, and how it may actually put forward a case for time being discrete rather than continuous. Crazy thought.

I sometimes ask my students this somewhat related question–perhaps you’ve heard it too:

How many positive integers have a 3 in them? (That is, in their decimal representation. 6850104302 has a 3 but 942009947 does not.)

If you haven’t ever considered this question, take the time to do it now.

Though I actually once worked out the result using limits (like Alexander Bogomolny does marvelously here), it’s easy enough to work out the result in our heads:

First ask yourself how many digits a randomly selected integer has. The number of digits is almost certainly greater than 2, right? There are only 90 two-digit positive integers, a finite number, and there are an infinite number of integers with more than two digits. It follows that if you were to pick one at random from among all positive integers*, it would be almost certain to contain more than two digits.

The same argument could be applied to a larger number of digits. By the same logic as above, we can convince ourselves that ‘most randomly selected integers have more than a trillion digits’. It’s a bit of an incredible statement, really. We rarely ever work with the ‘most-common’ kind of numbers (the big ones!).**

What is the probability that a number with a trillion digits has a 3 in it? Well, it’s almost certain. The probability approaches 100%. If we consider ALL numbers, the probability IS 100% (or is it?). This is a real dilemma. How can we say that 100% of numbers have a 3 in them when this is clearly not true?

We’ve been pretty sloppy here, but regardless, this kind of fast-and-loose infinite probability question is unsettling.

Do you want to try taking a crack at these? Feel free to comment below.

Oh, and Happy Birthday Euler!

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# Flat Donuts

[Guest blogger Dr. Gene Chase]

You know about flat donuts if you played a computer game in which when you go off the screen at the left, you return coming in at the right, and similarly when you go off the screen at the top, you return coming in at the bottom.

Mathematicians call the rectangle of your computer screen a flat torus. The word “torus” reminds us that we are only thinking of the surface of the donut.

What does that computer screen have to do with a torus? Stretch the screen around a torus as this picture begins to show.

Since the left and right sides of the screen are “the same” and the top and bottom sides of the screen are “the same,” the screen seamlessly takes the shape of the donut. Mathematicians say that the flat donut and the donut have the same topology, because we bent one into the other without cutting or pasting. (I remind you that opposite edges of the flat donut were already pasted by regarding them as “the same” before we began to bend it.)

Although a flat torus and a torus have the same topology, they do not have the same geometry. Geometry is about measuring space. In particular, on a flat torus, the shortest distance between two points is always a straight line. But on a torus, the shortest distance between two points staying in the torus is never a straight line.

Is it possible to paste the opposite edges of the flat torus together in such a way that the resulting thing in 3D has the same geometry? That is, such that straight lines are still straight?

It is easy to take one step, to create a cylinder, by pasting together just one pair of opposite edges. Notice that no stretching is involved at all.

How about pasting both pairs of edges? My intuition says that such a thing is impossible.

My intuition is wrong. In 1954, John Nash — yes, that John Nash of A Beautiful Mind — proved that it is possible, but without saying how. (He gave a so-called “existence proof.”)

But only 11 months ago did we have a picture of what the resulting flat torus would look like. Here’s a news article with a computer-generated picture to illustrate it.