Tau Before It Was Cool
Learning one set of formulas for full circles (C = 2πr, A = πr2) and a different set for partial circles (s = rθ, A = 1/2θr2) is not an elegant solution. Using tau instead of pi makes the formula sets identical, differing only by what greek letter represents the angle. θ for any angle. τ for full circles (because θ is τ for full circles). They also match a related third formula set, for so-called "skinny triangles", which I discuss in the next section. This uniformity makes all three formula sets easier to remember. And it eliminates the "formula mash-up" errors students sometimes make when they misremember, for example, partial circle area as θr2 because their memory mashes together the area formulas for partial and full circles. (Credit to Bob Palais for the observation about what I'm calling "formula mash-up" errors, to Michael Cavers for the correspondence of 1/2τr2 to 1/2θr2, and to Peter Harremoës for skinny triangles.)
Either Way You Slice It, A Circle's Area Formula Comes From Triangles
Finally, I should point out one more famous area formula with a 1/2 in it:
Area of a regular polygon = 1/2 perimeter ∙ apothem
Notice that if we increase the number of sides to infinity, the polygon becomes a circle, the perimeter becomes its circumference, and the apothem becomes its radius.
Examples of the Same Pattern in Physics Formulas
A Half-Massed Approach
Maybe instead, the physics formulas should all be changed to be more like the traditional circle formulas involving π. Maybe physicists just don't realize that multiplying by 2 is easier than dividing by 2. We could replace the constants a, α, m, I, k, κ, ε, C, μ, and L in the formulas above with constants having 1/2 those values. For example, everyone could describe objects by their "halfmass" instead of their mass m, so that what we now call a "1 kilogram" object, we would instead call a "1/2 kilogram" object. The formulas in box 3 would become:
momentum = 2 ∙ halfmass ∙ velocity (like circumference = 2∙π∙r)
kinetic energy = halfmass ∙ velocity2 (like area = π∙r2 )
We've gotten rid of the 1/2 in the kinetic energy formula, but at the cost of an extra 2 in the momentum formula. Still, most people would rather multiply by 2 than divide by 2, so it's a net improvement. But consider how describing objects by their halfmass would complicate the rest of physics. For starters, Newton's Second Law would become:
force = 2 ∙ halfmass ∙ acceleration
This would make Newton spin in his grave (though at only half the rate you would expect). In our halfmass-based physics system, a gravitational field of strength 1 Newton/kilogram would exert a force of 2 Newtons on (what would be called) a 1 kilogram object. Even if we fixed that by redefining the strength of a gravitational field to be force/halfmass, we would just create a new oddity. Namely, that a field strength of 9.8 meter/second2 would cause an acceleration of only 4.9 meter/second2.
We Still Love Pi Day
You might expect tau supporters to be against celebrating Pi Day on March 14 now that we have Tau Day on June 28. Nothing could be further from the truth. For one thing, it's still Einstein's birthday. I should point out however, that while he was born on Pi Day, Einstein was conceived on Tau Day of the previous year. Don't believe me? Do the math. (The normal length of time from conception to birth is 38 weeks.) Einstein was a man ahead of his time, but only one week ahead in this case. Besides, which event sounds like the basis for a more fun holiday?
Nonetheless, Pi Day has done a lot of good in getting children to like math class. Since most kids are on summer vacation when Tau Day comes around, it can't take over that role. So while the name could use updating (Half-Pie Pi Day?), the 3.14 holiday stays! But instead of serving one whole, circle-shaped pie, be mathematically correct. Cut it in half – along its diameter – and serve two semicircle-shaped pi's.
At the end of the day on March 14, if you just can't take any more of this absurdity where one pie contains two pi, join fellow tau supporters in celebrating a new Pi Day tradition – Tau Hour. When is Tau Hour, you ask? Why, "SIX(6) TO(2) EIGHT(8)", of course. Does that seem ridiculous? After all, it's two hours, not one. Yeah, that's kind of the point. And fortunately, Tau Hour overlaps with another celebration at many neighborhood establishments – Happy Hour. But wherever you gather to celebrate Tau Hour, don't forget to raise one stein to Einstein. Or mug, if you don't happen to have any steins handy. Just remember, Einstein didn't drink alcohol, and we don't want Tau Hour accused of promoting it. So that means two things. One, you should really put something non-alcoholic in that stein, and two, we'll have to find a different explanation for his hair. I'm still looking for clever suggestions about what drink might be appropriate. (And yes, I've already thought of Dos Equis – 2 X – but as I said, let's find something that doesn't make the room spin. It's still Pi Day after all, and semicircles don't spin smoothly.) Soft drinks (in mugs) for the kids. Milk or juice (in sippy cups) for the littlest ones that still use diameter in math class. As always, if you do drink things that could make the room spin, be safe and have a designated driver. However there is an exception to this rule for Pi Day. You can try to drive yourself home, but only if your car has semicircle-shaped wheels.
Enough silliness. Let's get back to the mathematics.
TO MULTIPLY BY 2 OR DIVIDE BY 2 ─ That is the question
What was the very first unit of measurement you learned for angles? Degrees? Radians? Nope. Long before you learned those, you learned a much simpler one. It's every bit as genuine a unit for describing angles as degrees and radians, but you probably learned it playing Simon Says as a child. Any preschooler who knows what to do when told, "Simon Says spin around three times" has mastered it. This unit goes by many different names – cycle, revolution, complete rotation, full circle – but they all describe the special angle (equal to 360°) that brings you right back where you started. Most discussions about tau use "turn" as the formal term for the angle unit, so I will too, but feel free to mentally replace it with whatever term you usually use.
Once children understand, "Simon Says spin around three-and-a-half times," they're well on their way to mastering fractional turns. As they learn more fractions, like ⅓, ¼, and ¾, they easily apply them to turns. And because of all the things around them that arrive shaped as circles – compared to very few that arrive shaped as semicircles – a natural association develops between those fractions and those fractions of a circle. Half a cookie looks like half a circle. A fourth a pizza looks like a fourth a circle. It's no surprise then that as adults, we use pie charts, where one full circle represents 100%. If we took advantage of this natural association instead of fighting against it, radians would be less awkward to learn and use.
Furthermore, the calculation issue with π and radians isn't just that you have to multiply by 2. Sometimes you have to multiply by 2, sometimes you have to divide by 2, and you always have to stop first and decide which one to do. Those extra steps interrupt a student's concentration on the real mathematics to perform what are essentially mindless circle-to-semicircle and semicircle-to-circle conversions. Using τ instead of π removes these mental speed bumps entirely. FILL-IN-THE-BLANK circles equals FILL-IN-THE-BLANK τ radians. Don't multiply. Don't divide. Don't stop to decide. (Catchy, isn't it?)
No, this change won't result in failing students suddenly getting A's in math. But why leave such an easily-removed impediment in place if τ really is the true fundamental constant and π was just a historical mistake? For the sake of tradition?
SHOW ME THE FREQUENCY! SHOW ME THE FREQUENCY! (in Jerry Maguire voice)
Wouldn't it be helpful to see the actual frequencies in our equations, instead of twice the number? To see 60 hertz alternating current written as sin 60τ•t instead of sin 120π•t? Or 440 hertz "A" music notes written as sin 440τ•t instead of sin 880π•t? Although these example numbers are easy to multiply/divide by 2 in your head, most frequencies aren't so simple. Do you immediately recognize the significance of the frequency of sin 215.8•106π•t? If you were scanning through the stations on your car radio and still hadn't found anything to listen to, you would. It's 107.9 megahertz, the highest/last FM radio station frequency.
Not only does using π hide the frequency. It invites the mistake of thinking the frequency is twice what it actually is. Seeing sin 440π•t and thinking that's 440 hertz is an easy mistake to make. Even for people who know better, since we're often forced to write expressions like sin 107.9•106•2π•t to show a frequency more explicitly. The unexpected absence of the 2 can easily get overlooked.
Consider the needless snare in Fourier series we set by writing ... + (●) sin 6πƒt + (●) sin 8πƒt + (●) sin 10πƒt + (●) sin 12πƒt + ... and then asking students to pick out the 6th harmonic. Or the even harmonics.
Though I used the sine function in all these examples, the same issue applies to other common functions like cosine and eiωt. (It's actually very common, for example, to represent alternating current using imaginary numbers. So 60 hertz shows up in the equations as ei120πt. It gets worse when the frequency's not such a nice round number. 876.89 hertz shows up as ei1753.78πt. Wouldn't ei876.89τt be clearer?)
The sum of the internal angles of a polygon – A rebuttal
The sum of the internal angles of a simple n-sided polygon is (n–2)π, and in the specific case of a triangle, it is π. This is sometimes offered as evidence that pi is a more fundamental number than tau. But as someone (I can't find the quote) has written, when you see a formula with π instead of 2π in it, you should immediately suspect that it's only measuring half of something. And that is indeed the case here. Notice the word "internal". What about the sum of the angles on the outside of the polygon? And remember, that is not the same thing as the sum of the external angles. In a confusing mismatch of terms, internal angles are the angle measurements on the inside of a polygon, but external angles are not the angle measurements on the outside. In fact, the angle measurements on the outside of a polygon have no standard name, which may explain why they've been forgotten in the tau/pi debate. I'll call them the "uninternal angles" to highlight that they are not the same as the external angles. But really, the external angles would have been better named "deflection angles" or "pivot angles", since they are the angles someone would pivot when walking along the polygon's perimeter.
Setting aside polygons for just a moment, consider that around any point, the total angle measure is 360°, or τ radians. That's fundamental. But if you draw a line through that point, you split the angle measure into two 180° (= π radians) angles. Polygons are only a little more complicated. The sum of the internal angles is (n–2)π. The sum of the uninternal angles is (n+2)π. These two added together are nπ + nπ = nτ. So each +1 increase in n increases this total angle sum by τ radians, and the increase is split evenly between the internal angle sum and the uninternal angle sum. Each gets half, π radians.
But what about the extra –2π in the internal angle sum and the extra +2π in the uninternal angle sum? That is actually the external angle sum. (Described better as the pivot angle sum.) It causes a transfer of 2π radians from the internal angle sum to the uninternal angle sum. And in fact, for polygons that cross over themselves, that number can be any multiple of 2π: 0, 2π, 4π, 6π, etc. The (n–2)π formula only applies to what are called "simple" polygons, which don't cross over themselves. The general formula is actually internal angle sum = nπ – external angle sum.
A different pair of formulas for every dimension. Their common link?
Yes, Multiplying by 2 is easier than Dividing by 2
This is the one complaint about tau that I can't dispute. Multiplication is easier to perform, it's easier to write, and as some of the awkwardly formatted formulas on this website demonstrate, it's easier to display on computers. But that's a separate issue from whether pi or tau is more fundamental. To use a slightly silly analogy, even though many people choose to buy dill pickle halves, cucumbers don't naturally grow that way. Furthermore, if avoiding division operations is our goal, we can do better than pi. It only cuts the size of the constant we use in half. As long as we're going to use some symbol that represents a fraction of the fundamental constant, we could use a smaller fraction and eliminate even more division operations. Our formulas are going to be messed up by any such fraction, so why not use 1/4 of tau? Think of all the π/2 's we could avoid writing if we used η = π/2 in mathematics instead of pi. Yes, there would be 2η's in equations where there used to be π's, but that's not really much worse. I would much rather multiply than divide. (This is the exact same rationale people give for using pi instead of tau, so if it applies there, it applies here.) If we went even smaller and used a symbol to represent 1/24 of tau, we could eliminate the fractions in all the "standard" angles like π/6 (30°), π/4 (45°), π/2 (90°), and 2π/3 (120°). You know, I might just have to start a new website.
This Web Page Used to Start Here Before I Added the Sections Above
The idea that 2π, not π, is the special number that deserves its own symbol, first occurred to me in the fall of 1988, when I was an undergraduate student in electrical engineering. After having seen 2's next to π's for years in countless equations, the reason for all those hangers-on 2's finally sank in one evening. Mathematics was using the wrong constant! π should be 6.28..., not 3.14... The idea really shocked me, that there was this glaring mistake in something as basic to mathematics as π. I got up from working on my homework and took a walk outside to think about it. Whether I was really shocked, or just looking for an excuse to ditch my homework, I don't know. But by the time I returned to my dorm room, I was convinced. I enthusiastically described my brand new idea to my roommate, who was also an engineering student. His reaction? The blankest stare imaginable.
Over the following years, I saw that same blank stare again and again when I described my idea to people. It's not that they didn't understand the math. They just didn't see why it mattered. Yeah, if we made π = 6.28..., then maybe we'd save ourselves writing a few extra 2's here and there. What's the big deal? Pencil lead is cheap. But I continued to develop the idea. As I thought up reasons why 2π was really the fundamental constant, I began writing them down and organized them into a paper.
At some point, I realized that redefining π to be 6.28... was the wrong approach. Yes, that's how it should have been defined in the first place. But trying to redefine it now would really confuse things. Instead, we could just pick a new symbol to represent 6.28... That symbol, and the symbol for π, could be used side-by-side, without confusion, for years until people eventually stopped using π.
So the symbol I chose way back then (circa 1990) was, believe it or not, tau. The same symbol as Michael Hartl and Peter Harremoës each, independently, chose 20 years later. It's not completely amazing because, as you can read in Appendix C of my paper, a lot of greek letters get rejected because of their existing uses, and those uses haven't changed in 20 years. Also we could all see that τ looks a lot like π, plus or minus a leg. However, the main reason I chose tau over greek letters with even fewer common uses was because it was available in the limited 256-character ASCII of DOS computers back then. Lesser-used greek letters weren't, and I didn't realize that problem would disappear with later computers. (They were available in some programs, like the word processor for scientists ChiWriter, which I used to write my paper. But in my other word processor, for example, I had to use stand-in characters on the screen, plus modify my dot matrix printer's driver file to switch into graphics mode for just those characters and slowly print them as columns of dots, which I had made look like the greek letters I wanted.)
Euler's Identity was BORN with a 2 in it
A large part of my paper grew out of seeing how Euler's Identity,
, would change with tau, and trying to justify it. Math enthusiasts cherish this spartan formula with an almost religious reverence because only the 5 most important numbers of mathematics appear in it, each only once. Writing Euler's Identity with
in place of π would desecrate their altar. Underneath itself, in the immaculate equation, τ would introduce the profane number 2. But couldn't 2 also be considered an important number, worthy of a place in Euler's Identity? This idea grew to become the largest part of my paper about tau. (So much so that I broadened its title to Universally Significant Numbers.) Later I developed the idea a little further in its own separate paper titled The Importance of the Number 2. The final version, produced sometime between 1992 and 1994, is posted below.
After reexamining it in 2011, I realized I could go beyond some mere fuzzy notion that 2 is an important number and thus should be in Euler's Identity. Euler's Identity is just the shortest, simplest member of an entire family of identities that result from taking the nth root of the number 1. If we allow imaginary numbers, then 1 actually has n different nth roots, each of which, when raised to the nth power, produces 1. These numbers are
. If you add all of them together, a surprising thing happens. They cancel each other out, and you get zero. For almost any value of n:
An Euler's Identity more amazing than 1 + eiπ = 0 and eiτ = 1
Math enthusiasts also admire how
contains three major math operations each exactly once - addition, multiplication, and exponentiation. To excuse the absence of subtraction and division, they reply those are really just the inverses of addition and multiplication. Although
includes division, it still lacks subtraction. We can fix that. There is another entire family of identities, formed by going around the unit circle in the reverse direction. Like mirror images of the roots and identities above, they simply have each i replaced by a -i. The shortest, simplest identity in that family is
, or swapping sides, . This remarkable little formula contains: in standard order, addition, subtraction, multiplication, and division; in numerical order, 0, 1, and 2; in alphabetical order, e, i, and τ; and exponentiation -- ALL occuring exactly once.
In his original 2001 article, Bob Palais imagined if π had been correctly defined as circumference over radius. He wrote "Euler's [Identity] should be eiπ = 1 (or eiπ/2 = -1, in which case it involves one more fundamental constant, 2, than before)". Unfortunately, the part in parentheses has been ignored since then. While I have seen how effective the utter simplicity of
is in convincing many people about tau, we need to develop the second argument for those many other people who find too simple. Too simple without addition and the number 0. (Not a surprising reaction considering how many times we've heard that
is incredibly amazing for having them.) Too simple with no ability to show the complex exponential is antiperiodic with antiperiod
(which also logically implies the complex exponential is periodic with period τ). Too simple to have the same strong aesthetic or spiritual appeal for them. We can still convince these people about tau, but we need to use the other argument. It doesn't really matter whether they cherish
or or as the most convincing proof that God exists, so long as they write it using tau.
However, I do have a proposal for naming these new identities. (It could even be used with the old identities involving π.) Just as reusing the letter π to represent 6.28... would cause unnecessary confusion, so would reusing the name "Euler's Identity" to represent
. People already mix up Euler's Identity with Euler's Formula (
), which is actually more like what mathematicians think of as an identity since it contains a variable. So why not eliminate both problems and use the following more descriptive names:
Euler's Full-turn Euler's Reverse Full-turn
Euler's Half-turn Euler's Reverse Half-turn
The word "Identity" could be added onto the end when you're being extremely formal, but the rest of the time, saying "Use Euler's Full-turn" or "Use Euler's Half-turn" gives more information in fewer syllables than "Use Euler's Identity". And it wouldn't get confused with "Use Euler's Formula" (which I was taught to call "Euler's Equivalence", as you'll see in my paper).
When you can't convince people, wait 19 years and try again
I want to thank John Strebe and Charles Koppelman for their feedback as I was finishing my Universally Significant Numbers paper in early 1992. And the former also for recommending it to a university math professor he knew. (No, it wasn't Bob Palais.) Later in 1992, when I hoped to extend the pattern I thought I saw in the circumference of circles (2-dimensional) and the surface area of spheres (3-dimensional), Dena Morton and her father Michael Cowen helped me locate the formulas for n-spheres (n-dimensional). Unfortunately those formulas didn't follow my pattern, and I didn't notice the (very simple) recursive pattern involving tau that they did follow.
I didn't make an actual decision in 1992 to stop developing the paper. I just stopped finding new arguments for tau. Once that happens, you can only spend so much time polishing the wording. Ultimately, the paper failed to convince people who read it. So gradually, the paper, and the whole idea of tau, faded from my attention. Years later, when I no longer worked with all those equations full of 2π's, it faded from my memory too.
So much so, unfortunately, that I lost track of the final version of my paper. The last version I have found so far (in my basement in a box of floppy disks, which I'm amazed I didn't throw out years ago) is from fall 1991. I have posted it below. It surprised me to see how Bob Palais and I ended up venturing down several of the same paths with this idea. Even some seemingly remote ones like contemplating 1/e as a fundamental constant instead of e. Noticing the similarity of 1/2τr2 to 1/2mv2 and other similar quadratics that result from integrals. Defending the inclusion of the number 2 in Euler's Identity as giving it a sixth fundamental constant. Michael Hartl and Peter Harremoës then (each independently) picked the symbol τ. Consider how surreal it was for me June 29th of 2011, nineteen years after putting this whole issue on the shelf, to come across a news article saying: Mathematicians have determined that 2π, not π, is the true fundamental constant; they have decided to give this number its own special symbol; and the symbol they have chosen to use is... TAU. Then to read respected math papers and see all my old arguments -- those arguments which always got me blank stares from people.
Most of all though, I felt thrilled to finally see other people agreeing with all those arguments. Other than my long defense of the number 2 in Euler's Identity, they came up with everything I did and more, and have developed those arguments further than I did in even my final 1992 version. So if the 1991 version below doesn't convince you about tau, please look at Pi Is Wrong!, The Tau Manifesto, Al-Kashi's constant, Radian Measurement, Kevin Houston's video, Vi Hart's video, or Wikipedia's Tau_(2π) article before making up your mind. They do a much better job explaining tau's advantages. But because I developed my Universally Significant Numbers paper from scratch while between the ages of 17 and 20, I still feel proud of it.
The Importance of the Number 2
Universally Significant Numbers (Incomplete draft from Fall 1991)
Euler's Pizza Cutter
(To cut halves into quarters, we rotate Euler's Identity a quarter turn (half π) by multiplying it by ei (1/4) 2π to get ei (1/4) 2π + ei (3/4) 2π = 0. Add this identity and the unrotated Euler's Identity together. You get the exact same 4th roots of unity identity as earlier. To cut the quarters into eighths, just rotate the quarters identity an eighth turn (quarter π) by multiplying it by ei (1/8) 2π and add the result to the unrotated quarters identity. At each stage, you can think of the rotated identity as cutting the sectors of the unrotated identity in half.)
More Web Page Dregs
Using the true circle constant ─ circumference divided by radius ─ is a more elegant solution than duct taping two π's together like we have done for the last 300 years. Two halves don't always work as well as one whole:
Describing a book you've read as "written by a wit" is a compliment. Describing it as "written by two half-wits" is not.
In the American Revolution, Nathan Hale did not say, "I only regret that I have but two half-lives to give for my country." (If you don't get this one, ask a nuclear physicist or a pharmacist.)
The word "TAU" can be written upside down using only formal math symbols ("set intersection", "for all/any", "is perpendicular to").
The word "tau", written just the right way, has the word "fan" as its reflection underneath. (Yeah, I know it's not as impressive as the reflection of "PIE" looking like "314".)
Maybe we could just pronounce the greek letter π as "SEMITAU". Like semicircle. Think the Greeks would mind? Ah, what would they care? The Greeks actually pronounce their letter π as "PEE". (Completely true. Look it up. Then imagine what they must think when they read that Americans celebrate March 14 by consuming π.)