1. τ-doku

I mentioned the possibility of tau-themed sudoku.  Thanks to juanmah, I know that at least one such exists, and here it is.  Kudos to the puzzle creator and thanks for the link.  I’ve solved that puzzle, so I know that it works.  It’s quite easy, so you could give it even to sudoku novices to promote Tau Day.  One quirk of the linked puzzle is that it uses two colors so that the repeated digits in 6.2831853 can count as distinct symbols.

I wondered whether a more interesting puzzle could be found by not distinguishing the 8s and 3s.  What follows is my take on that idea.  I offer for your consideration my first ever sudoku puzzle.

(Edit: I mistyped one of the givens when I first posted this.  This is corrected now.  The 1 in row six was incorrectly given as a dot.  My apologies for any frustration this may have caused anyone.)

taudoku 1 (fixed)

The central row emphasis is purely decorative.  Note that, unlike the sudoku linked to above, the 3 and 8 are not distinguished by color here; this is a sudoku puzzle with only seven symbols.  Each row, column, and block will have two 3s and two 8s (and one each of the other five symbols).

Near the end of the construction process, I was forced to sacrifice some of the difficulty level that I wanted in order to maintain symmetry of the givens.  I’m not happy about that, but I don’t think it turned out too easy to be interesting.  As I said, it was my first try at sudoku construction, but it won’t be my last.

2. τ as a natural period

We casually call tau the circle constant, but let’s not forget that its ubiquity extends well beyond geometry (would it really be ubiquity otherwise?).  Here’s a simple demonstration of that.  In the comment thread I formulated this in terms of differential equations, but here let me be more casual.

Imagine two particles (infinitesimal bugs, if you like) on the number line, call them Alice and Bob.  Their starting positions can be anywhere (but let’s avoid the trivial case where both bugs start at 0; having one but not the other start at 0 is fine).  Then the bugs start moving.  At any point in time, Alice’s velocity is given by Bob’s position, and Bob’s velocity is the negative of Alice’s position (imagine that they continuously look at each other to decide how to move).  Quite a simple dynamical system, but if you actually simulate it, you’ll see something potentially unexpected.  The bugs may wander far from where they started, but they always come back; indeed, they will always both come back to their starting position at exactly the same moment, and then the cycle will repeat.  How long will it take for this repetition to occur , that is, what is the period of the motion?  No matter where you start the bugs, the period is always exactly τ time units.

Why is this happening?  Because the problem setup encodes the differential equation , whose solutions are linear combinations of sine and cosine (in natural units), which are both periodic with period τ.

3. Is τ the right symbol?

Xamuel (whose blog is excellent, by the way) called into question the symbol choice, on the grounds that pi has two legs while tau has only one.  Since the circle constant is twice pi and not the other way around, what we should want is a symbol with four legs, the argument goes.  This made me smile, because it was the very same objection that occured to me when I first watched Vi Hart’s anti-pi video, and I even emailed her to that effect.  However, in the course of writing that email, I realized the natural counter.  Read the legs as Roman numerals.  Pi parses as “horizontal squiggle over two” while tau parses as “horizontal squiggle over one”.  Quite consistent with our usage.

(On a less whimsical note, τ is a good choice because it stands for turn; a right angle measures because it is a quarter-turn.)

My Tau Day started with a mass email from Michael Hartl (creator of the Tau Manifesto, newly updated today), which linked to What Tau Sounds Like, a terrifically timely video from Michael John Blake.  If you haven’t watched it yet, you should.  (This video made the internet rounds today, including this CNN article which follows up on their Pi Day Half Tau Day article.)

As the CNN article reminds us, Tau Day is currently a holiday without any particular traditions or practices.  I’ve already done my part by proposing the official dessert.   Tau Day is a holiday with delicious potential.

blackberry / apple taue baked for Tau Eve 2011

I made that one yesterday (lesson learned: blackberry filling is runny, should have been the bottom layer).  Today I made this one.

blueberry-cherry taue, baked for Tau Day 2011

We haven’t cut into that one yet.  The next one I plan to make will have a blueberry spice layer and an apricot layer, but it may be a while before I feel the need for more pie, taue, or the like.

As Michael Hartl put it in the email, this is “the best Tau Day yet. (OK, it’s only the second one, but still!)”.  This is a very young holiday.  There is a lot of room for the creative here.  And we should all want Tau Day 2012 to be better than Tau Day 2011.  Poems whose word lengths come from the digits of pi have been around for quite some time.  What about tau-digit poetry?  On Pi Day Half Tau Day, I saw a sudoku variant based on the digits and shape of pi (though I can’t seem to find it now).  If there can be pidoku, there can and should be taudoku.  And that can be just the beginning.

We have 366 days until Taue Day 2012.  What are you going to do?

If the Tau movement and its central tenet, that it should be tau and not pi which is foregrounded in our math curriculum, our discourse, our literature, our mathophile culture, are at all new to you, then you may  be filled with doubts and objections.

Is this really worth caring about?  Isn’t changing such a well-established notation totally infeasible? Won’t this just be more confusing for the children? And so on.  Every objection and problem I can think of is addressed in Michael Hartl’s Tau Manifesto.

Every problem except one.

The real unsolved problem is one of dessert.

Vi Hart lays the problem out with mouthwatering clarity in this video.  It seems like 1/2 pi radians should cover half a pie, but it doesn’t.  It only covers a quarter of a pie.  This captures the nonintuitive nature of pi very well (and a fortiori a reason why tau should be in the classroom).

But there’s a problem.  No one is suggesting that we change pi to mean 6.28and-so-on.  That would be horribly confusing.  And I doubt we can change the meaning of the word “pie”.  One solution might be to change units to double radians, or dradians.  Then a pie would contain pi dradians.  But remember, we have already seen that radians are the “right” units to use.

It’s a tricky problem.  Here is my solution.

I propose the taue, a double-decker pie.  (Just remember that a taue is like 2 pie, or if you’re really into this that a pie is like half a taue.)  Pi radians of a taue is the rough equivalent of a whole ordinary pie (on the basis of filling, of course; valuing the crust is subtler and ignored here).  Pi/4 radians (one eighth of a circle) contains a quarter of a pie in yummy fruit goodness.

It’s not obvious a priori that this will actually work.  Will the bottom layer of filling vent properly?  What will you actually bake the thing in, if you don’t have some kind of double-deep pie pan (which should be called a taue pan if it exists).

Well, I did my first official test of my proposed solutions to those problems yesterday (Saturday), and I’m here to tell you that progress tastes delicious.

• Solution to Problem #1: Use lattice crusts for the top and middle.
• Solution to Problem #2: I baked my concoction in a 10″ springform pan, because they have nice high walls.
• For the test run I used canned fillings.  One can of cherry on top, one can of apple on the bottom.  (Each layer had the amount of filling I’d ordinarily use in an 8″ or 9″ pie.)
• My crust was made from 3 3/4 cups of flour (I’m partial to King Arthur whole wheat), 3/4 C of ice water, 3/4 tsp salt, and 1 1/2 cups of organic salted butter, in the “usual” way.  Mix the dry, dice and cut in the butter, and add the water a little at a time
• I divided the dough into three balls (proportioned roughly 2:1:1 or maybe 5:2:2) and let chill a little less than 4 hours.
• I rolled out the  large ball of dough and used that to cover the bottom and sides of the pan.  Then the apple layer.  Then rolled out one of the small balls and made a lattice (three strips each way and two diagonals).  Then the cherry layer.  Then rolled out the other ball for the top lattice (five strips each way).

I wish I’d used less flour when rolling out the dough, and I could have been more careful making my lattice strips prettier, but I brought the taue to my parents’ house for tasting by myself, my wife, my parents, and my children.  Success.

With the test out of the way and successful, I’ll be making at least two “official” taues this Tau Day season: one on Tau Eve and another on Tau Day proper.  I’ll post pictures and recipes of those here.

P.S.

Do you have an objection or doubt not addressed in the Tau manifesto?  By all means raise it here.  I’ll address it if I can and forward it if I cannot.

My favorite, though, is that shirt … you’ve probably seen one … with the digits of pi formatted in the shape of pi.  (I’ve also seen the same for phi, the golden ratio, but I don’t own that.)  I wish I had the analogous thing for tau, but I don’t think anyone’s made that yet.

As a first step… here’s this, which I whipped up with Mathematica.

That was dashed off pretty quick, over lunch; I plan to make some prettier versions, including a mathematica notebook to automatically do the same thing for an image of your choice.

Last time I talked a little about why some conventions are better .. righter .. than others, in a sense more meaningful than mere personal preference.

Circle constants are not just found in geometry.  They are ubiquitous in mathematics.  Everywhere you find yourself tripping over 3.1415somethings and 6.2831and-so-ons, in applied and theoretical stuff.  You probably call those numbers and , and I’d rather call them and . The Tau Day claim is that the latter choice is better than the former.

There are lots of ways to make the case for the primacy of (and I’ll mention more as it gets closer), but the most concise is this: There are radians in a full circle.

This begs a question, of course.  How many angle-units are in a full circle depends on what units we use.  There are 360 degrees, or 4 right-angleses (much early Euclid-style geometric writing measured angles relative to the right angle), or 400 gradians, or 4 million myriogrades, if that’s your thing.  So we’ve traded “Why ?” for “Why radians?”.

But that’s a relatively easy question, and certainly a well-known one.  There are lots of reasons why radians are the most natural/desirable, and lots of places to read about them.  Even the wikipedia article does a decent job.  So let me just give you a few concise points.

1. The radian measure of an angle can be defined intrinsically as follows.  For any angle, draw a circle centered at the vertex, and compute the ratio of the arc length inside the angle to the radius of the circle.  (Hence the term radian ; the angle measure in radians tells you how many multiples of the radius are in the arc length contained in the angle.)  It seems like this might depend on how big you draw the circle, but a moment’s thought shows that it doesn’t, and you end up with a nonarbitrary angle measure.
2. The calculus-related behavior of the trigonometric functions is best when you work in radians.  For example, if you work in radians, the derivative of is and the derivative of is .  If you don’t work in radians, you get ugly correction factors.
3. (This is actually very closely related to the previous.)  If we work in radians, sine and cosine have the simplest possible power series, and it is easy to see (at least formally) the celebrated Euler’s formula , where that e is the base of the exponential function, the one we met yesterday.
4. If we use radians, then the best first-order approximation to is, if is small enough, simply itself.  (That is, if .)  For any other choice of units, there’s a correction factor.

A parting math-Zen koan, especially for those of you who haven’t thought about this much.  I have asserted (and do deeply believe) that there is a canonical, nonarbitrary right unit for measuring angles.  What is the analogous best unit for measuring length?

Tau Day is 6/28 (by analogy with Pi Day Half Tau Day, 3/14), so our countdown stands at 6 days ( days, if you will).

I won’t get into the details here, but the key contention is that τ, not π, is the circle constant we should focus on.  Instead of (or ), it would be better to write .  For a quick and cute introduction, check out the Pi is Still Wrong video from everybody’s favorite mathemusician, Vi Hart.  For the whole manifesto, check out Michael Hartl’s wonderful site (updated a few months ago for Pi Day Half Tau Day).

Especially for the non-mathematicians here, “should” and “better” might seem strange word choices.  Today I want to focus on what exactly they might mean in this context.

(Remark: If you’re more science-minded, this is a little like the oft-heard-in-some-circles assertion that the convention for positive and negative charges is “backwards”, the opposite of what it “ought to be”.  Apropos xkcd comic!)

Rather than circle constants, let’s look at the example of exponential functions and logarithms.

The exponential functions, such as , , and , are a family of continuous functions characterized by their continuity and the property (and let’s add non-constancy, to rule out the trivial ).  They have many interesting properties in common, and in many ways if you understand one you understand them all.  If is an exponential function and is another, then there is a nonzero constant so that .  (That is, the graphs of all exponential functions are the same, apart from horizontal rescaling and reflection across the y-axis.

The logarithm functions, such as , , and , are a family of continuous functions from the positive numbers to the real numbers characterized by their continuity and the property .  They are the inverse functions of the exponential functions, and vice versa.  Again, their properties are very similar to one another, and in many ways if you understand one you understand them all.  If is a logarithm function and is another, then there is a nonzero constant so that .  (That is, the graphs of all logarithm functions are the same, apart from vertical rescaling and reflection across the x-axis.

So there are lots of functions that have earned the right to be called an exponential function (or a logarithm function).  Indeed, for each positive number , there is an exponential function and a logarithm function ; different ‘s give different functions.  So when mathematicians refer to the exponential function or the logarithm function , which functions are those?  Does that even make sense?

Indeed it does, and mathematicians have with great consensus identified the exponential function, which is to say a canonical choice of , which is somehow the best or more natural or most beautiful choice.

This is a somewhat unusual and provocative assertion, not least because it occupies a strange place between the objective and the subjective.  Why is one choice best?  It’s certainly not a fact, like or .  But it’s also not just a “mere” opinion, like “Grape jelly isn’t as good as strawberry.” or “Lilacs are the best-smelling flower.”

There are lots of reasons to single out a particular function from each family, and they just about all lead to the same choice.  I’ll just give a couple here.

With basic calculus, it’s not hard to show that, if is an exponential function, then the derivative has the form for some (which would be different for different functions).  That is, if a population is growing exponentially, it’s rate of growth at any moment in time is proportional to its size at that time.  Well, what’s the constant?  As I said, it depends on the function.  If , then the constant is about 0.693147.  If , then the constant is more like 2.302585.  Whatever the constant is, it starts showing up all over the place once you start doing serious analysis or problem-solving with the function.  You know what would be a good constant?  1.  Then the rate of change of the function is precisely the function itself, there is no correction factor, and we have the simplest formulas.  The exponential function is characterized by the property that there is no correction factor.

On the logarithm side, if is a logarithmic function, then we can show that , i.e. the rate of change is proportional to .  Of all the functions proportional to , which is the most natural?  Yup, it’s .  Choosing the base so that the corrective scaling constant disappears again gives the base .  The logarithm function is , often called the natural logarithm (and confusingly written as $\latex \ln x$ in grade-school textbooks).

So it’s not an arbitrary choice.  You might say it’s the “most convenient” choice.  I like to think that I have more poetry in my soul than that, and say it’s the “most beautiful” choice.  Somehow that base just fits best.  The formulas come out cleaner.  Everything just works with a little less noise.  That’s (part of the reason) why the number (approximately 2.7182818284590452) has a reserved letter and a name (Euler’s constant; I’ll let you decide whether e stands for Euler, or exponential, or something else).

Subjective-objective-wise, it’s something like the situation in music theory, where it is said that certain combinations of notes or progressions of chords “sound good”.  This looks superficially like a personal value judgment, but it isn’t really.  It’s a shorthand for a statement about the ways sound waves combine which is hard or impossible to actually express in words, but you know it when you hear it.  It’s not that hard to imagine a person of highly idiosyncratic musical taste who just doesn’t like, just can’t stand the sound of the “standard” harmonies.  But such a person would probably still know what “that chord progression sounds good” means.

Likewise, you could imagine a person who loves the number 7 — how it looks, how it smells, how much fun it is to write — so much that she would prefer to use , and the aesthetic joy of all those 7s.  Such a person, whatever her country or planet or universe of origin, would still see what I see about the base e, would still know why it is called the exponential function.

It’s just right.

The book is You Can Count on Monsters by Richard Evan Schwartz (ISBN 1568815786), and it’s a picture book about prime number decompositions.  It’s much more colorful than you’re picturing, I promise.

One thing that makes this book so nice is that it doesn’t beat you over the head with anything.  There are some very basic remarks about primes and multiplication at the beginning  and some slightly deeper remarks at the very end, but the vast bulk of the pages have no text at all.

Each number from 1 to 100 gets a double page.  On the left is the the number and a configuration of that many dots (usually clustered into spirals or some such in an interesting way).  On the right is a whimsical (and strangely compelling) drawing of a monster.  For a prime number, the monster is some simple monster which smoehow embodies the nature of the number (the right number of teeth, or legs, or whatever).  For a composite number, the monster is in some way a conglomeration of the corresponding prime monsters.  (So the 70-monster incorporates the natures of the 2-, 5-, and 7-monsters.)

There’s plenty to stare at, plenty of patterns sitting right near the surface, and plenty more lurking underneath to be discovered over time.  An artistically-inclined child might try to invent prime monsters larger than 100, or to draw some composite monster.  (I’ll confess that when I bought the book, I entertained myself for quite some time trying to draw the 1001 monster.)  It’s just fun to look at, and it provides lots of interesting avenues for mathematical conversation with “grown-up” mathematicians.

Speaking as a number theorist, I love the way this book conveys the essential predictable/chaotic dual nature of prime numbers.  There are always more monsters, but you’re never quite sure when you’ll meet the next one.

My favorites from the bunch are two books on mathematics for a general audience by John Stillwell.  Stillwell is best known, I believe, for his excellent tome Mathematics and its History, which is an outstanding textbook for a course (or two or three) in the history of mathematics, expertly blending mathematical content, biographical information, and insight into the historical progression.  I can’t say enough good things about that text — it’s one of my favorite books — but it’s not something that a nonmathematician is likely to buy (because it’s so big and correspondingly expensive).

These two books, though, are short, affordable, accessible, and beautifully written.  Each tells a story of mathematicians dealing with a certain big theme, with a logical progression of increasingly sophisticated and deep mathematics.

The first is Yearning for the Impossible: The Surprising Truths of Mathematics (ISBN 156881254X).  Here the theme is the ongoing evolution of mathematics in response to questioning the impossibility of certain ideas.  There is no real solution to , so we could just throw up our hands and say “It’s impossible!”, but the results are much more interesting if we ask “Is there some other sense in which it is possible?”  This book, just over 200 pages, is really remarkable for the number of disparate and sophisticated ideas it manages to introduce to a general audience.  Let me illustrate this by simply listing the chapter headings.

1. The Irrational
2. The Imaginary
3. The Horizon
4. The Infinitesimal
5. Curved Space
6. The Fourth Dimension
7. The Ideal
8. Periodic Space
9. The Infinite

The follow-up is Roads to Infinity: The Mathematics of Truth and Proof (ISBN 1568814666).  The title makes it sound very profound, and it is.  This book goes deep into various notions of infinity, including a friendly but surprisingly thorough treatment of ordinal and cardinal numbers.  Godel’s Theorem(s) made accessible without being dumbed down.  Good stuff.

Here’s the takeaway.  These books are beautiful, they, make me happy, you should go buy them and read them.  Someday I want to be able to write like that.

P.S.

If anyone local wants to borrow any of the books mentioned, just drop me a line or stop by my (Ann Arbor) office.  I have a 2nd edition and a 3rd edition of Mathematics and its History, though I plan to award the 2nd ed to a worthy student at semester’s end.

Well, this was intriguing. I hadn’t told him which of the various proofs I knew was the one I intended to give. How was he so sure I would do it “the wrong way”? And what did that even mean?

I asked him (a bit skeptically at first) to show me “the right proof”, and now I will show you.

First, “the wrong proof”. Traditionally, the Pythagorean Theorem is phrased in terms of squares.

If you draw three squares, one based on each side of a right triangle, then the combined area of the smaller two squares equals the area of the largest square.

Simple and beautiful visual proofs of this fact are easily found.  (Here or here, for example.)

But we can give an even simpler visual proof of the Pythagorean Theorem, if we step back and understand better what it really says.

The crucial concept for us is the similarity ratio, which is something that most people understand on an intuitive level.  Suppose that I decide to build, in my backyard, a larger-than-life statue of me.  It will be exactly the same shape as me, but twice as tall.  Then, without knowing anything about the precise shape of me, you know that its left arm will be twice as long as mine, its belt will be twice as long as mine, etc.  But its footprint will be four times as large as mine, since that’s a two-dimensional measurement; likewise its shirt would take four times as much fabric as mine, etc.  And it would take up eight times as much space as I do.

Suppose that the sides of a right triangle are in the proportion , and suppose that I take three similar versions of the same shape (it could be a square, but it could be any shape: a semicircle, a stop sign, the outline of my face…) in the same proportions.  Then the areas of the three shapes will be in the proportions .

So the Pythagorean Theorem is really asserting the following.

If you draw three similar shapes, one based on each side of a right triangle, then the combined area of the smaller two shapes equals the area of the largest shapes.

Crucially, the statement is true for one shape if and only if it’s true for all shapes.  The key is to use the triangle itself.  Consider the following diagram.

Proof of the Pythagorean Theorem

Here an arbitrary right triangle is divided into two by an altitude.  By standard angle-chasing, the two smaller triangles are similar to each other and to the large triangle.  Furthermore, these three similar triangles are in the same proportions as the sides of the right triangle (look at the hypotenuses).

Bottom line: to prove the Pythagorean Theorem once and for all, all I have to do is show that the green triangle and the blue triangle together have the same area as the large triangle.  But this is obvious.

The Pythagorean Theorem has nothing to do with squares and everything to do with similarity ratios.

Clearly, I need to buy and read this book.  By analogy with Stephen Colbert’s “truthiness” (the quality of stuff that feels true in the gut), Charles Seife coins “proofiness” to describe statements that feel like evidence, that feel decisive.  The author’s story about the museum tour guide (by far the best story I know involving the number 65,000,058) is a personal favorite, one that I tell often.  This is really a story about how the human mind intuitively deals with numbers and numerical information, and the intuitive weaknesses that exposes.