Tau Before It Was Cool
Joseph Lindenberg
(email = name at yahoo.com)

        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

To Derive By Summing Skinny Triangles:    ϕ r + ϕ r + ϕ r + ϕ r + ϕ r + ...  =  (ϕ + ϕ + ϕ + ϕ + ϕ + ... ) r  =  τr or θr
½ ϕ r2 + ½ ϕ r2 + ½ ϕ r2 + ½ ϕ r2 + ½ ϕ r2 + ...  =  ½ (ϕ + ϕ + ϕ + ϕ + ϕ + ... ) r2  =  ½ τ r2 or ½ θ r2

        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
, 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?

1 circle ► 2 π radians   
1 π radians ► 1/2 circle
1 circle ◄► 1 τ radians
1/2 circle ► 1 π radians      
1/2 π radians ► 1/4 circle   
1/2 circle ◄► 1/2 τ radians
3/8 circle ► 3/4 π radians   
3/8 π radians ► 3/16 circle 
3/8 circle ◄► 3/8 τ radians
2/3 circle ► 4/3 π radians   
2/3 π radians ► 1/3 circle   
2/3 circle ◄► 2/3 τ radians
turns ► 7½ π radians  
π radians ► 1⅞ turns    
turns ◄► τ radians
6.87 turns ► 13.74 π radians 6.87 π radians ► 3.435 turns  
6.87 turns ◄► 6.87 τ radians


60 hertz ► sin 120πt
sin 60πt ► 30 hertz  
60 hertz ◄► sin 60τt
440 hertz ► sin 880πt  
sin 440πt ► 220 hertz  
440 hertz ◄► sin 440τt
876.89 hertz ► sin 1753.78πt sin 876.89πt ► 438.445 hertz 876.89 hertz ◄► sin 876.89τt

        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 60
τt instead of sin 120πt?  Or 440 hertz "A" music notes written as sin 440τt instead of sin 880πt?  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•106πt?  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 440
πt 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 107.91062πt 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?

        In 2 dimensions, the set of points a distance r away from a center point is a circle.  The size of its interior is the area πr2, and the size of that area's boundary is the circumference 2πr.  In 3 dimensions, the set of points a distance r away from a center point is a sphere.  The size of its interior is the volume 4/3 πr3 and the size of that volume's boundary is the surface area 4πr2.  What about in more than 3 dimensions?  Though trying to visualize it might make your brain hurt, it's not much different mathematically from 2 or 3 dimensions.  You just use a longer list of coordinates to describe where something is located.  We use (x,y) coordinates in 2 dimensions and (x,y,z) coordinates in 3 dimensions.  Don't let a little thing like running out of alphabet letters stop you.  In 4 dimensions, we can use (w,x,y,z) or, more often, (a1,a2,a3,a4).  Additional dimensions just mean additional numbers in the coordinate list.
        The distance from point A (a1,a2,a3,...,an) to point B (b1,b2,b3,...,bn) is simply (b1-a1)2 + (b2-a2)2 + (b3-a3)2 + ... + (bn-an)2.  So what happens then to "the set of points a distance r away from a center point"?  The size of its interior?  The size of that interior's boundary?  There's a different pair of formulas for every number of dimensions n.  But as the red arrows show, all those different formulas are just /n multiplied over and over again (or its reciprocal multiplied over and over again)

The leftward direction of the bottom-row arrows was chosen because it makes what's inside the arrowheads slightly simpler and creates a striking symmetry with the top-row arrows.  Normally though, we start off knowing formulas for small n values and, from those, calculate formulas for bigger n values.  So the bottom-row arrows would probably be more useful if we reversed their direction and wrote /n-2 in their arrowheads instead.  Then, using our simple "multiply by /n" and "multiply by /n-2" recurrence relations, we could write general formulas for all n:
Interior size is 2(/3)(/5)(/7)(/9)...(/n)rn for odd n; and (/2)(/4)(/6)(/8)...(/n)rn for even n.
Boundary size is 2()(/3)(/5)(/7)...(/n-2)rn-1 for odd n; and ()(/2)(/4)(/6)...(/n-2)rn-1 for even n.
There are absolutely no single π's in these formulas, only 's.  From here, the formulas can be manipulated algebraically into various different forms.  One approach is to first separate the 2's from the π's, as if the arrowheads had actually said to multiply by π/n/2 and π/(n-2)/2
Interior size is 2(π/3/2)(π/5/2)(π/7/2)(π/9/2)...(π/n/2)rn for odd n; and (π/2/2)(π/4/2)(π/6/2)(π/8/2)...(π/n/2)rn for even n.
Boundary size is 2()(π/3/2)(π/5/2)(π/7/2)...(π/(n-2)/2)rn-1 for odd n; and ()(π/2/2)(π/4/2)(π/6/2)...(π/(n-2)/2)rn-1 for even n.
Then, merge all numerators and denominators:
Interior size is π(n-1)/2rn/(1/2)(3/2)(5/2)(7/2)(9/2)...(n/2) for odd n; and πn/2rn/(2/2)(4/2)(6/2)(8/2)...(n/2) for even n.
Boundary size is 2π(n-1)/2rn-1/(1/2)(3/2)(5/2)(7/2)(9/2)...((n-2)/2) for odd n; and 2πn/2rn-1/(2/2)(4/2)(6/2)(8/2)...((n-2)/2) for even n.
In both expressions for odd n, move π-1/2 down to the denominator as π.  In both boundary size expressions, rewrite ((n-2)/2) as (n/2)-1.  The resulting formulas can then be written compactly by sweeping all the mess of their denominators into gamma functions:
Interior size = πn/2rn / Г (n/2 + 1)            Boundary size = n/2rn-1 / Г(n/2)
Replacing π with τ/2 in these formulas now would make them more complicated, not less.  While that's unfortunate, it doesn't show pi is more fundamental.  For starters, consider that these same two formulas would also be simpler if written using the number of "dimension pairs" m = n/2
Interior size = πmr2m / Г (m+1)            Boundary size = mr2m-1 / Г(m)
So before you start shopping for a new 1.5-Dimension-Pair television, remember that we've only derived one possible set of formulas here, and as Michael Hartl shows in section 5.2 of The Tau Manifesto, an alternative set is simpler using tau.  One of Hartl's readers, Jeffrey Cornell, came up with a third set, which is actually simpler using π/2:
Interior size = (π/2)⌊n/2⌋ (2r)n/n!!            Boundary size = (π/2)⌊n/2⌋ 2nrn-1/(n-2)!!
where ⌊n/2⌋ is the floor function and n!! is the double factorialHartl's and Cornell's formula sets have the advantage of not containing a gamma function, which has a disturbingly complex definition as the (Calculus) integral  e-t tm-1 dt taken from 0 to infinity.  (This was supposed to be simple geometry, just circles and spheres but with a few more dimensions.  What's something like that doing in here?)
No single constant can simplify all three formula setsSo which set should determine if pi, tau, or even pi/2 is the "fundamental" number?  Ultimately, none of these are the fundamental formulas that we must consider.  The simplest, most basic underlying formulas are the recursive ones represented by the arrows in the chart above, and they all use (tau).  Finally, just to prove that those 2's really do belong with the π's and not dividing the n's, consider the other arrows that can be drawn on the chart.  A "multiply by " arrow can be drawn from every top row box to the bottom row box 2 dimensions to its right.  From that box, a "multiply by 1/n" arrow can be drawn to the top row box directly above it.  Also notice that no arrows have just π without a 2 attached.  The does in fact arrive in the formulas as a single entity.  The derivation of the recurrence relations involves an integral in polar coordinates from 0 to .

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 '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 , 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 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:

        For n = 2, we get Euler's Identity.  (It was BORN with that 2.)  For larger values of n, we get longer, more complicated identities.  Those also contain 0, 1, e, i, and π ( actually), but mathematicians like the n = 2 identity better because it is shorter and simpler.  Why, then, not look for an even more simple identity by making n smaller than 2?  We really can't make n = 0, because in a sense, every number is a 0th root of 1.  (Except 0.)  Adding all those numbers together would produce an infinitely long identity.
        But we can make n = 1.  There is exactly 1 1th root of 1.  It's 1.  So if we add all of the 1th roots together, we get 1.  We wanted a simpler identity, and we got it:  1 = 0.  How could that happen?  Why didn't we at least get a correct identity like 0 = 0?  Well, the 1th root  can't sum to zero alone by itself.  (Unless it is zero.  It's not.)  You need, at minimum, 2 equal but opposite roots for them to cancel each other out when added together.  Fewer than 2 won't work.  More than 2 just makes things more complicated than necessary.
        Those same themes are behind many examples in The Importance of the Number 2.  Binary numbers.  Merge sort.  Subdividing units of measurement.  Mitosis.  Even male and female genders.  (
Imagine if reproduction required 3, 4, or 5 different genders to all get together and blend their genes.  That would be more complicated than necessary.)  So amazingly, the number 2 gains entrance to Euler's Identity, math's exclusive club of important numbers, for the very same reason that it appears prominently so many other places.  To see the similarity another way, notice in the drawings above how every set of n nth roots subdivides the unit circle into n equal sectors.  The smallest number of sectors you can divide it into (and actually be dividing it) is 2, which requires 2 roots.

An Euler's Identity more amazing than  1 + e = 0  and  e = 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 '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 , 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.

TIP:  To find links only about the number tau with Google, search using the three words tau pi circle.

The original π Is Wrong! by Bob Palais and some of his more recent thoughts.
Tau Day website, home of The Tau Manifesto by Michael Hartl

Al-Kashi’s constant τ by Peter Harremoës  (See the bottom of his site for a lot more links.)
Clocks that Run Backwards (and other innovations) by Brian Dickens at Hostile Fork
Conquest of the Plane by Thomas Colignatus
Eric Raymond (ESR) blogs about Tau versus Pi
MIT's "Pi Day, Tau Time" cartoon, which established March 14, 6:28pm as the moment future students learn they've been accepted
My Conversion to Tauism by Math Horizons editor Stephen Abbott
The Pi Manifesto by Michael Cavers (a.k.a. MSC) and The Pi Manifesto Discussion Forum at Spiked Math Forums
Radian Measurement: What It Is, and How to Calculate It More Easily Using τ Instead of π by Stanley Max
Tau Tracts from Spiked Math Comics
Tau vs Pi: Fixing a 250-year-old Mistake, an upcoming University of Oxford seminar by Robin Whitty
Wikipedia's old Tau_(2π) article just before it was destroyed by π loyalists in the Pi Day Massacre of 2012

Vi Hart's classic pie-making video Pi Is (still) Wrong and her Song About A Circle Constant
Michael Hartl's talk at CalTech on Tau Day 2011
Kevin Houston's Pi is wrong! Here comes Tau Day
Khan Academy founder Salman Khan on Tau versus Pi
Numberphile:  Phil Moriarty being logical and Phil Moriarty being musical
Numberphile:  Steve Mould and Matt Parker go head-to-head in Tau vs Pi Smackdown
Numberphile:  James Grime chooses tau over pi
Phillip Bascom's presentation to the Society of Physics Students at UMass Amherst
Robert Dixon's Pi ain't all that

Tau 2000! website of Ethan Brown, the world record holder for most digits of tau recited from memory (2,012 digits)
What Tau Sounds Like by Michael Blake
The Sound of Tau by Luke Harrald
Tau and Pi music video (Parody of Kid Cudi's Day N Nite)
Pilish(6) is(2) writing(7) where(5) the(3) word(4) lengths(7) follow(6) the(3) digits(6) of(2) pi(2), so(2) now(3) there(5) is(2) taulish(7).
Stargate: SG-1 fan fiction, The Argument For Tau
Students picketing for tau inside their college math building on Pi Day
Viva La Resistance: The Story of Tau, a movie created by a high school Calculus class
That same class had matching "τ > π" shirts made up and picketed through their high school on Pi Day.

How to make Tau-nados, delicious two-pie twisters
Cadence Watch Company's Tau Circle Watch, in alloy or stainless steel, has the unit circle marked in base-12 fractions of tau
T-shirts, mugs, and all sorts of other Cafe Press merchandise bearing the slogan "τ > π"
Zazzle merchandise:  "τ is the new 2π" (black print), "τ is the new 2π" (white print), "2π or not 2π, tau is the question" (black print), "2π or not 2π, tau is the question" (white print), "TAU DAY 6.28 / THE π IS A LIE", "happy tau day!", "I'm a τist 2π", "I'm a τist 6.28"

The Importance of the Number 2

Universally Significant Numbers  (Incomplete draft from Fall 1991)

                                            Joseph M. Lindenberg

                                              University of Maryland
                                              College Park, MD 20742
                                              (301) ███-████

                                              Columbia, MD 21045
                                              (301) ███-████

                   Universally Significant Numbers                 

For the purposes of this text, τ ≡ 2π

    In the fields of science and mathematics, certain numbers appear
again and again in the equations describing many diverse topics.  Some
such numbers, like the speed of light and Planck's constant have units
of measurement attached to them.  So, their exact value depends on the
system of measurement you are working in.  The value of the speed of
light in meters per second, for example, looks nothing like its value
in miles per hour.  Other numbers though, perhaps even more
interestingly, have no units. The most famous of these numbers are π
(= 3.14159...) and e (= 2.71828...).  They appear in the equations of
all variety of fields.  They are somehow "universally significant". 
Though they are the most obvious of the universally significant
numbers (USN's), they are not the only ones.  Zero, one, and the
imaginary number i
= √-1 are other examples of such numbers that seem
to have an inherent importance in the universe.

    My thesis is two-fold:
         1. τ (≡ 2π), not π, is a USN
         2. The number 2 is a USN

                   Part 1 - Why τ, not π, is a USN                 

In examining the many equations where π appears, you begin to notice
that the π's are often accompanied by 2's.  To the practically-minded,
this presents no problem.  We have simply slapped an extra factor of 2
in where it is needed to make the equation correct.  But let's observe
what we have really done.  We have given the number 3.14159... a
special distinction by giving it a universally-recognized symbol, π. 
We have said that we find this particular number to be so important as
to warrant its own symbol.  I contend that if we are going to do this,
we should at least make sure we have chosen the right number.

The following are some examples of how τ makes for simpler, more
symmetric equations than π does:

1. One cycle or one wavelength would correspond to τ, instead of to
2π.  There are many quantities that have "normal" and "cyclic" forms. 
For example, frequency comes in two forms, ƒ and ω.  Planck's constant
has forms h and ħ. In each case, the two forms differ by a factor of

2. Radius is generally a more significant dimension than diameter. 
So, an important number defined as the circumference divided by the
radius of a circle (as τ is) makes more sense than one defined as the
circumference divided by the diameter (as π is).  In other words, C =
τr is simpler than C = 2πr.

In response to this, one might point out that the area of a circle is
πr2 and would be 1/2τr2, which is a more complicated formula.  But take
a second look at this formula.  1/2τr2  This is a very important form
that shows up in many areas.  1/2mv2 is classical kinetic energy.  1/2εE2
is electric field energy density.  (As a side note, the importance of
this form is that it is the integral of τr or mv or εE.)  In any case,
it is clear that τ fits into the structure better than π.

And extending this treatment to spheres, the surface area of a sphere
would be 2τr2 rather than 4πr2.  This difference is more than just the
aesthetics of having the coefficient equal the exponent, for this same
number is also the number of parameters required to specify a point on
a sphere.  It takes two parameters, θ and φ, to specify a point on a
sphere.  Similarly, it takes one parameter, θ, to specify a point on a
circle.  And the coefficient and exponent in the equation for the
circumference of a circle is 1.

Finally, as a side note, in response to one person's comments that
many physics formulas have 4π's in them, this is generally the result
of using the formula for the surface area of a sphere (4πr2).

3. Gaussian distribution - has √2π

   Stirling's formula - has √2π

                       Part 2 - Why 2 is a USN                     

One might gather, from the growing importance of digital, that there
is something universally significant or efficient about base 2.

Dividing something into 2 equal pieces is generally easier than
dividing it into any other number of equal pieces.  Take a look at
your average set of wrenches.  What sizes do you see?  1/2", 1/4",
3/4", 3/8" -- all based on halving the inch repeatedly.  A quarter is
just half a half.  An eighth is half a half a half.  You see the same
thing with measuring cups, or many other systems of measurements. 
When you go to the deli, you order half a pound of this, a quarter of
a pound of that.  In telling the time, we say half-past, quarter til. 
Look at the financial pages.  IBM is up 21/4.  AT&T is down 11/8.

Why is a given number a USN?  Why does the number seem to show up
everywhere?  With some USN's, like 0 and 1, it is somewhat obvious. 
With others, like τ and e, we don't know.  I believe that I understand
what makes 2 a USN.  2 is the smallest (cardinal) number with which
you have choice.  The need for choice exists in many places.  You need
a choice between digits to have a numbering system (e.g. base 0 and
base 1 don't work)  Another example of the need for choice is the
merge sort algorithm.  (The merge sort is a recursive algorithm for
sorting a list of items.  Briefly, the sort involves dividing the list
in half, sorting each half by merge sort, and then merging the two
sorted sublists back together.)  Why not break the list into three
sublists, or four or five?  Two is the minimum number needed to
actually be breaking the list.  Why are binary trees the most common
type of tree?  Why not have 3-ary trees?  Two is the minimum needed to
give a choice between branches.

The major point I'm trying to make here is that the importance of
binary does not have to do with the particular electrical properties
of silicon.  Binary is the fundamental, universal numbering system.

There are many examples of duality in the universe:

   1. Wave-particle duality (in quantum mechanics)

   2. The Heisenberg Uncertainty Principle relates the uncertainty in
      2 variables.

   3. Logic - true or false (2 possibilities).

   4. Existence of sets of dual equations in logic and in set theory.

   5. 2 types of charge - positive and negative.

   6. 2 types of energy - kinetic and potential.

   7. 2 types of motion - linear and rotational.

   8. Positive or negative (2 possibilities).

   9. Greater than or less than (2 possibilities).

   10. 2 sexes - male & female (alright, I know that's pushing it).

Other interesting facts about 2:

   1. In base 2, the only digits are 0 and 1, two already-acknowledged

   2. Inverse square laws - power is a 2.

   3. 2 is the smallest prime and the only even prime.

   4. Fermat's Last Theorem.  There are integer solutions to the
      equation an + bn = cn for n = 2, but not for any greater

              The Connection Between Part 1 and Part 2             

    Euler's equivalence provides us with an interesting equation:

                             eπi + 1 = 0                           

What has been found so interesting about this particular equation is
that it relates what, up til now, have been believed to be all 5 USN's
(π, e, 0, 1, and i).  This phenomenon is preserved, though, if both τ
and 2 are USN's, for:
                             e2 + 1 = 0                           

So, the equation still relates all known USN's.  Only now, 2 and τ,
but not π, are included.

       Appendix A - The Advantages of Hexadecimal over Decimal     

    You may ask, if binary is the fundamental, universal numbering
system, then why don't we use it?  The main disadvantage of binary is
the long length of numbers represented in base 2.  Consider a typical
price for a new car, $XX,XXX in decimal.  In binary, the price would
be written $XXXXXXXXXXXXXX.  (Imagine the sticker shock.)  In decimal,
the number is shorter, but each of the number's digits can be any one
of 10 possibilities.  The human brain, because of the particular way
it works, has a much easier time remembering the shorter number, even
though each of its digits has more possibilities .  Hexadecimal
provides an excellent solution to this problem.  Numbers written in
hexadecimal are just as easy to remember as those written in decimal;
but hexadecimal is easily converted into binary by splitting each
hexadecimal digit into four binary digits.
    Hexadecimal also provides some extra advantages, which combined
with those previously mentioned, make it the optimal numbering system
for human brains.  The ability to convert numbers into binary by
simply splitting each digit into multiple binary digits is a feature
of any numbering system that is based on a power of 2 (e.g. base 8 or
base 32).  So, you might ask, why not use one of these?  These other
bases don't work well in computers because when you convert numbers
from these bases to binary, each digit becomes some abnormal number of
binary digits.  In octal, for example, each octal digit becomes three
binary digits.  This does not work well in computers, which like
numbers of bits that are powers of 2.  So the next highest base that
might work is base 256, in which each digit can be broken up into 8
binary digits.  Hexadecimal is still superior, since 8 = 2^3 while 4 =
2^2.  3 is not a power of 2; 2 is.  Base 256 is impractical anyway
since it requires 256 separate digits.  (By the way, the next base
that works as well as hexadecimal is base 65536 (2^16)).  Base 4 also
works, but as in binary, the numbers get quite long.  So, hexadecimal
seems to provide a uniquely excellent compromise.

The only reason we work in base 10 is because we happen to have 10
fingers and 10 toes.  This particular fact is just an engineering
result (that more fingers would get in the way, and having fewer
fingers would reduce our manual dexterity), not some sign that 10 is a
good numeric base to use.  This does, however, point out the one
disadvantage of hexadecimal compared to decimal - we couldn't count on
our fingers.  I contend that the advantages of hexadecimal outweigh
this one inconvenience.

In a world where digital machines are so widespread, working in
hexadecimal all the time would make more sense.  No arduous
conversions between base 10 and base 2.  No decimal numbers that in
base 10 are neat but in base 2 are nonterminating.  (Many such numbers
we only have a use for because we work in base 10.)  In computers, one
single format would have both the ease of translation of binary coded
decimal and the ease of mathematic manipulation of binary.

No one has to convince people who work with computers of the
advantages of hexadecimal.  But you might ask, what's in it for the
rest of us?  I came across one example while repairing my bike.  You
have a bolt you need to take off, so you find a wrench that looks
about the right size.  You try it and find that it's a little too big
or a little too small.  Now, what's the next size wrench?  You find
yourself finding common denominators.  In hexadecimal, it would be
practical to print the sizes in radix-point form.  In decimal, 5/16 is
written .3125.  In hexadecimal, it is written .5.  7/32 is written
.21875 in decimal, .38 in hex.

If we're going to be living with binary from now on, and hexadecimal
is the best way for humans to deal with binary numbers, we might as
well legitimize it with its own set of digit symbols, instead of using
the 0-9, A-F notation.  We should probably just create 16 completely
new symbols, rather than using the current 0-9 and adding extra
symbols for 10-15.

If we converted to hexadecimal, we would want SI unit extensions for
164, 168, 1612, etc.  That way, each successive extension would
represent moving the hexadecimal point four places over.  Also, we
would put commas between every set of four digits instead of every
three like we do now (e.g. instead of 87,264 we would write 1,54E0)

I do not make any pretence of believing that the world could be
convinced to switch to hexadecimal.  (The U.S. can't even be convinced
to switch to metric.)  Nor do I believe it should - the world has more
and larger problems than to be disrupted by this.  I am simply
asserting that given a choice, hexadecimal works better.

          Appendix B - Could Any Other USN's be Incorrect?         

With the problem I have pointed out in π, one might wonder whether any
other numbers we currently believe to be USN's are off?  Could the
number 1/e actually be a USN rather than e itself, for example?  Well,
it is fairly clear that 0 and 1 are correct.  The cases for τ and 2
are made in this paper.  I believe that e is correct because of the
many symmetries that show up between e and 2 in binary analysis.  As
for i, I really don't have an opinion, but I see no evidence that it
is incorrect.

So, if we believe in that equation we derived from Euler's
equivalence, we may be able to conclude that there are only 6 USN's in
the universe:  0, 1, 2, τ, e, and i.

There are, of course, still the "significant numbers" with units.  I
believe that knowledge of these numbers (how many there are, what they
are, how they are related, etc.) is still rudimentary.  One might
consider 0, 1, 2, τ, e, and i to be the "math USN's" while numbers
like the speed of light and Planck's constant to be the "physics

                   Appendix C - Choice of a Symbol                 

The symbol, presumably, should be a Greek letter.  (Personally, I
think the numbers 2.71828... and √-1 should have been assigned to
Greek letters, not to letters of the English alphabet like e and i
The symbol for a USN should unambiguously represent that USN, like π
currently does.)  Also, since the number is a scalar constant, the
symbol should be lower-case.  Below, I have listed the lower-case
letters of the Greek alphabet.

An "X" next to a letter means that letter could not be used; it would
cause too much confusion with current uses of the letter.  Every Greek
letter is used in some field of study, so I have blocked out only
those letters that are widely used in situations where π commonly
occurs.  Sorry, but someone is going to have to make room.  π cannot
be re-used because it would cause too much confusion during the
changeover period.

A "-" means the letter is not in the standard IBM ASCII.  Don't laugh. 
With the penetration of IBM-compatible machines in the scientific
community, this is a serious consideration.

    α   X   Used for angles
    β   X   Used for angles
    γ   X   Used for angles
    δ   X   delta, or change, used throughout science and math
    ε   -
    ζ   -
    η   -
    θ   X   Used for angles
    ι   X   Looks too much like the letter 'i'
    κ   X   Looks too much like the letter 'k'
    λ   X   Used for wavelength
    ν   X   Used for frequency
    ξ   -
    ο   X   Looks too much like the letter 'o', or the digit '0'
    π   X   Would cause confusion during transition period
    ρ   X   Used for radius in cylindrical and/or spherical coord.
    υ   -
    φ   X   Used for angles
    χ   -
    ψ   -
    ω   X   Used for angular frequency

Of the remaining letters without any strikes against them, I have
chosen τ because it seems to be used the least, and also because it
bears a close resemblance to π, which should help during the
transition period.

Euler's Pizza Cutter

This Euler's Pizza Cutter section is pretty half-baked, but it's good for a chuckle.
        To illustrate that Euler's Identity really is just like those other situations where we subdivide units of measurement by repeatedly dividing them in 2, and because no tau versus pi debate is complete without circle-shaped foods, I present Euler's Pizza Cutter.  No, this isn't some overpriced product I'm trying to sell on my website... yet.  But it represents Euler's Identity perfectly.  Imagine a unit pizza on the complex plane.  Recall that vectors drawn to the n nth roots of unity divide the unit circle into n sectors.  For n = 2 (the Euler's Identity case), there are just two roots, 1 and eiπ (= -1).  So together their vectors divide the unit circle in half along the horizontal axis.  Just like Euler's Pizza Cutter is doing to that pizza.
        But what if you want the n = 4, 8, 16, ... roots of unity identity, or put another way, what if 4, 8, 16, ... people want pizza?  We could memorize an identity for every possible n value and buy a pizza cutter for every possible number of people who might want to share a pizza.  The model here on the right is designed for n = 8.  How much would you pay for...  No, wait!  It's not necessary!  Though incredibly simple, Euler's Identity / Pizza Cutter can be applied repeatedly to get the exact same results as all those complicated, expensive n = 4, 8, 16, ... identities / pizza cutters.

        For example, consider the n = 4 case.  Just start with the n = 2 case:
                                                                                                       1 + eiπ = 0     

        Now rotate the pizza cutter a QUARTER turn, which of course is ONE HALF pi:
eiπ/2 ( 1 + eiπ = 0 )    ==>    eiπ/2 + ei3π/2 = 0      
        Combine the results:
1 + eiπ/2 + eiπ + ei3π/2 = 0    

        (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.)