Tau Manifesto (Against Pi)

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The Tau ManifestoMichael HartlTau Day, 2010updated Half Tau Day, 2011Official Tau Day T-shirts are now available! Proceeds benefit DonorsChoose.org.1,194

1 The circle constantWelcome to The Tau Manifesto. This manifesto is dedicated to one of the most important numbers inmathematics, perhaps the most important: the circle constant relating the circumference of a circle toits linear dimension. For millennia, the circle has been considered the most perfect of shapes, and thecircle constant captures the geometry of the circle in a single number. Of course, the traditional choiceof circle constant is but, as mathematician Bob Palais notes in his delightful article IsWrong!,1 is wrong. Its time to set things right.

1.1 An immodest proposalWe begin repairing the damage wrought by by first understanding the notorious number itself. Thetraditional definition for the circle constant sets (pi) equal to the ratio of a circles circumference toits diameter:2

The number has many remarkable propertiesamong other things, it is transcendental, which

Ethan Sawyer and 11,077 others like this. Unlike

= 3.14159265 CD

No, really, pi is wrong: The Tau Manifesto by Michael Hartl | T... http://tauday.com/

1 of 25 3/15/11 12:20 PM

means that it is also irrationaland its presence in mathematical formulas is widespread.

It should be obvious that is not wrong in the sense of being factually incorrect; the number isperfectly well-defined, and it has all the properties normally ascribed to it by mathematicians. Whenwe say that is wrong, we mean that is a confusing and unnatural choice for the circle constant.In particular, since a circle is defined as the set of points a fixed distancethe radiusfrom a givenpoint, a more natural definition for the circle constant uses in place of :

Because the diameter of a circle is twice its radius, this number is numerically equal to . Like , itis transcendental and hence irrational, and (as well see in Section 2) its use in mathematics issimilarly widespread.

In Is Wrong!, Bob Palais argues persuasively in favor of the second of these two definitions forthe circle constant, and in my view he deserves principal credit for identifying this issue and bringingit to a broad audience. He calls the true circle constant one turn, and he also introduces a newsymbol to represent it (Figure 1). As well see, the description is prescient, but unfortunately thesymbol is rather strange, and (as discussed in Section 4.2) it seems unlikely to gain wide adoption.

Figure 1: The strange symbol for the circle constant from Is Wrong!.

The Tau Manifesto is dedicated to the proposition that the proper response to is wrong is No,really. And the true circle constant deserves a proper name. As you may have guessed by now, TheTau Manifesto proposes that this name should be the Greek letter (tau):

Throughout the rest of this manifesto, we will see that the number is the correct choice, and we willshow through usage (Section 2 and Section 3) and by direct argumentation (Section 4) that the letter is a natural choice as well.

1.2 A powerful enemyBefore proceeding with the demonstration that is the natural choice for the circle constant, let usfirst acknowledge what we are up againstfor there is a powerful conspiracy, centuries old,determined to propagate pro- propaganda. Entire books are written extolling the virtues of . (Imean, books!) And irrational devotion to has spread even to the highest levels of geekdom; forexample, on Pi Day 2010 Google changed its logo to honor (Figure 2).

r D

circle constant .Cr2

= 6.283185307179586 Cr


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Figure 2: The Google logo on March 14 (3/14), 2010 (Pi Day).

Meanwhile, some people memorize dozens, hundreds, even thousands of digits of this mysticalnumber. What kind of sad sack memorizes even 50 digits of (Figure 3)?3

Figure 3: Michael Hartl proves Matt Groening wrong by reciting to 50 decimal places.

Truly, proponents of face a mighty opponent. And yet, we have a powerful allyfor the truth is onour side.

2 The number tauWe saw in Section 1.1 that the number can also be written as . As noted in Is Wrong!, it istherefore of great interest to discover that the combination occurs with astonishing frequencythroughout mathematics. For example, consider integrals over all space in polar coordinates:

The upper limit of the integration is always . The same factor appears in the definition of theGaussian (normal) distribution,

2 2

f(r,) rdrd.2

0

0

2


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and again in the Fourier transform,

It recurs in Cauchys integral formula,

in the th roots of unity,

and in the values of the Riemann zeta function for positive even integers:4

There are many more examples, and the conclusion is clear: there is something special about .

To get to the bottom of this mystery, we must return to first principles by considering the nature ofcircles, and especially the nature of angles. Although its likely that much of this material will befamiliar, it pays to revisit it, for this is where the true understanding of begins.

2.1 Circles and anglesThere is an intimate relationship between circles and angles, as shown in Figure 4. Since theconcentric circles in Figure 4 have different radii, the lines in the figure cut off different lengths of arc(or arclengths), but the angle (theta) is the same in each case. In other words, the size of the angledoes not depend on the radius of the circle used to define the arc. The principal task of anglemeasurement is to create a system that captures this radius-invariance.

,12 e

(x)2

22

f(x) = F(k) dk

e2ikx

F(k) = f(x) dx.

e2ikx

f(a) = dz,12i f(z)z a

n

= 1 z = ,z n e2i/n

(2n) = = (2 . n = 1, 2, 3, k=1

1k 2n

B n2(2n)! )

2n

2


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Figure 4: An angle with two concentric circles.

Perhaps the most elementary angle system is degrees, which breaks a circle into 360 equal parts. Oneresult of this system is the set of special angles (familiar to students of trigonometry) shown inFigure 5.

Figure 5: Some special angles, in degrees.

A more fundamental system of angle measure involves a direct comparison of the arclength with theradius . Although the lengths in Figure 4 differ, the arclength grows in proportion to the radius, so

sr


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the ratio of the arclength to the radius is the same in each case:

This suggests the following definition of radian angle measure:

This definition has the required property of radius-invariance, and since both and have units oflength, radians are dimensionless by construction. The use of radian angle measure leads to succinctand elegant formulas throughout mathematics; for example, the usual formula for the derivative of

is true only when is expressed in radians:

Naturally, the special angles in Figure 5 can be expressed in radians, and when you took high-schooltrigonometry you probably memorized the special values shown in Figure 6. (I call this system ofmeasure -radians to emphasize that they are written in terms of .)

Figure 6: Some special angles, in -radians.

s r = .s 1r 1s 2r 2

.srs r

sin

sin = cos . (true only when is in radians)dd


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Figure 7: The special angles are fractions of a full circle.

Now, a moments reflection shows that the so-called special angles are just particularly simplerational fractions of a full circle, as shown in Figure 7. This suggests revisiting the definition ofradian angle measure, rewriting the arclength in terms of the fraction of the full circumference ,i.e., :

Notice how naturally falls out of this analysis. If you are a believer in , I fear that the resultingdiagram of special anglesshown in Figure 8will shake your faith to its very core.

s f Cs = fC

= = = f( ) f.srfCr

Cr


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Figure 8: Some special angles, in radians.

Although there are many other arguments in s favor, Figure 8 may be the most striking. Indeed,upon comparing Figure 8 with Figure 7, I consider it decisive. We also see from Figure 8 the genius ofBob Palais identification of the circle constant as one turn: is the radian angle measure for oneturn of a circle. Moreover, note that with there is nothing to memorize: a twelfth of a turn is ,an eighth of a turn is , and so on. Using gives us the best of both worlds by combiningconceptual clarity with all the concrete benefits of radians; the abstract meaning of, say, isobvious, but it is also just a number:

Finally, by comparing Figure 6 with Figure 8, we see where those pesky factors of come from: oneturn of a circle is , but . Numerically they are equal, but conceptually they are quite distinct.

The ramifications

The unnecessary factors of arising from the use of are annoying enough by themselves, but farmore serious is their tendency to cancel when divided by any even number. The absurd results, suchas a half for a quarter circle, obscure the underlying relationship between angle measure and thecircle constant. To those who maintain that it doesnt matter whether we use or when teachingtrigonometry, I simply ask you to view Figure 6, Figure 7, and Figure 8 through the eyes of a child.

/12

/8 /12

a twelfth of a turn = = 0.5235988.126.283185

122

1 2

2


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You will see that, from the perspective of a beginner, using instead of is a pedagogical disaster.

2.2 The circle functionsAlthough radian angle measure provides some of the most compelling arguments for the true circleconstant, its worth comparing the virtues of and in some other contexts as well. We begin byconsidering the important elementary functions and . Known as the circle functionsbecause they give the coordinates of a point on the unit circle (i.e., a circle with radius ), sine andcosine are the fundamental functions of trigonometry (Figure 9).

Figure 9: The circle functions are coordinates on the unit circle.

Lets examine the graphs of the circle functions to better understand their behavior.5 Youll noticefrom Figure 10 and Figure 11 that both functions are periodic with period . As shown in Figure 10,the sine function starts at zero, reaches a maximum at a quarter period, passes through zero at ahalf period, reaches a minimum at three-quarters of a period, and returns to zero after one full period.Meanwhile, the cosine function starts at a maximum, has a minimum at a half period, andpasses through zero at one-quarter and three-quarters of a period (Figure 11). For reference, bothfigures show the value of (in radians) at each special point.

sin cos

1

Tsin

cos


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Figure 10: Important points for in terms of the period .

Figure 11: Important points for in terms of the period .

Of course, since sine and cosine both go through one full cycle during one turn of the circle, we have; i.e., the circle functions have periods equal to the circle constant. As a result, the special

values of are utterly natural: a quarter-period is , a half-period is , etc. In fact, when makingFigure 10, at one point I found myself wondering about the numerical value of for the zero of thesine function. Since the zero occurs after half a period, and since , a quick mentalcalculation led to the following result:

Thats right: I was astonished to discover that I had already forgotten that is sometimes called . Perhaps this even happened to you just now. Welcome to my world.

2.3 Eulers identity

sin T

cos T

T = /4 /2

6.28

= 3.14.zero 2/2


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I would be remiss in this manifesto not to address Eulers identity, sometimes called the mostbeautiful equation in mathematics. This identity involves complex exponentiation, which is deeplyconnected both to the circle functions and to the geometry of the circle itself.

Depending on the route chosen, the following equation can either be proved as a theorem or taken as adefinition; either way, it is quite remarkable:

Known as Eulers formula (after Leonhard Euler), this equation relates an exponential with imaginaryargument to the circle functions sine and cosine and to the imaginary unit . Although justifyingEulers formula is beyond the scope of this manifesto, its provenance is above suspicion, and itsimportance is beyond dispute.

Evaluating Eulers formula at yields Eulers identity:6

In words, this equation makes the following fundamental observation:

The complex exponential of the circle constant is unity.

Geometrically, multiplying by corresponds to rotating a complex number by an angle in thecomplex plane, which suggests a second interpretation of Eulers identity:

A rotation by one turn is 1.

Since the number is the multiplicative identity, the geometric meaning of is that rotating apoint in the complex plane by one turn simply returns it to its original position.

As in the case of radian angle measure, we see how natural the association is between and one turnof a circle. Indeed, the identification of with one turn makes Eulers identity sound almost like atautology.7

Not the most beautiful equation

Of course, the traditional form of Eulers identity is written in terms of instead of . To derive it, westart by evaluating Eulers formula at , which yields

But that minus sign is so ugly that the formula is almost always rearranged immediately, giving thefollowing beautiful equation:

= cos + i sin .e i

i

= = 1.e i

e i

1 = 1e i

=

= 1.e i


11 of 25 3/15/11 12:20 PM

At this point, the expositor usually makes some grandiose statement about how Eulers identity relates, , , , and sometimes called the five most important numbers in mathematics.

Alert readers might now complain that, because its missing , Eulers identity with relates onlyfour of those five. We can address this objection by noting that, since , we were alreadythere:

This formula, without rearrangement, actually does relate the five most important numbers inmathematics: , , , , and .

Eulerian identities

Since you can add zero anywhere in any equation, the introduction of into the formula is a somewhat tongue-in-cheek counterpoint to , but the identity does have amore serious point to make. Lets see what happens when we rewrite it in terms of :

Geometrically, this says that a rotation by half a turn is the same as multiplying by . And indeedthis is the case: under a rotation of radians, the complex number gets mapped to

, which is in fact just .

Written in terms of , we see that the original form of Eulers identity has a transparent geometricmeaning that it lacks when written in terms of . (Of course, can be interpreted as arotation by radians, but the near-universal rearrangement to form shows how using distracts from the identitys natural geometric meaning.) The quarter-angle identities have similargeometric interpretations: says that a quarter turn in the complex plane is the same asmultiplication by , while says that three-quarters of a turn is the same asmultiplication by . A summary of these results, which we might reasonably call Eulerian identities,appears in Table 1.

Rotation angle Eulerian identity

Table 1: Eulerian identities for half, quarter, and full rotations.

We can take this analysis a step further by noting that, for any angle , can be interpreted as a

+ 1 = 0.e i

0 1 e i 0

sin = 0

= 1 + 0.e i

0 1 e i

0 = 1 + 0e i+ 1 = 0e i = 1e i

= 1.e i/2

1/2 z = a + ib

a ib 1 z

= 1e i + 1 = 0e i

= ie i/4i = ie i(3/4)i

0 e i0 = 1/4 e i/4 = i/2 e i/2 = 13/4 e i(3/4) = i e i = 1

e i


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point lying on the unit circle in the complex plane. Since the complex plane identifies the horizontalaxis with the real part of the number and the vertical axis with the imaginary part, Eulers formulatells us that corresponds to the coordinates . Plugging in the values of the specialangles from Figure 8 then gives the points shown in Table 2, and plotting these points in the complexplane yields Figure 12. A comparison of Figure 12 with Figure 8 quickly dispels any doubts aboutwhich choice of circle constant better reveals the relationship between Eulers formula and thegeometry of the circle.

Polar form Rectangular form Coordinates

Table 2: Complex exponentials of the special angles from Figure 8.

e i (cos , sin )

e i cos + i sin (cos , sin )e i0 1 (1, 0)e i/12 + i32 12 ( , )

32

12

e i/8 + i1212 ( , )

12

12

e i/6 + i1232 ( , )12

32

e i/4 i (0, 1)e i/3 + i12

32 ( , )12

32

e i/2 1 (1, 0)e i(3/4) i (0,1)e i 1 (1, 0)


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Figure 12: Complex exponentials of some special angles, plotted in the complex plane.

3 Circular area: the coup de grceIf you arrived here as a believer, you must by now be questioning your faith. is so natural, itsmeaning so transparentis there no example where shines through in all its radiant glory? Amemory stirsyes, there is such a formulait is the formula for circular area! Behold:

We see here , unadorned, in one of the most important equations in mathematicsa formula firstproved by Archimedes himself. Order is restored! And yet, the name of this section soundsominous If this equation is s crowning glory, how can it also be the coup de grce?

3.1 Quadratic formsLet us examine this paragon of , . We notice that it involves the diameterno, wait, theradiusraised to the second power. This makes it a simple quadratic form. Such forms arise in manycontexts; as a physicist, my favorite examples come from the elementary physics curriculum. We willnow consider several in turn.

Falling in a uniform gravitational field

A = .r 2

A = r 2


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Galileo Galilei found that the velocity of an object falling in a uniform gravitational field isproportional to the time fallen:

The constant of proportionality is the gravitational acceleration :

Since velocity is the derivative of position, we can calculate the distance fallen by integration:

Potential energy in a linear spring

Robert Hooke found that the external force required to stretch a spring is proportional to the distancestretched:

The constant of proportionality is the spring constant :8

The potential energy in the spring is then equal to the work done by the external force:

Energy of motion

Isaac Newton found that the force on an object is proportional to its acceleration:

The constant of proportionality is the mass :

The energy of motion, or kinetic energy, is equal to the total work done in accelerating the mass tovelocity :

3.2 A sense of foreboding

v t.g

v = gt.

y = vdt = gtdt = g .t

012 t 2

F x.

kF = kx.

U = Fdx = kxdx = k .x

012 x 2

F a.mF = ma.

v

K = Fdx = madx = m dx = m dv = mvdv = m .dvdtdxdt

v

012 v 2


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Having seen several examples of simple quadratic forms in physics, you may by now have a sense offoreboding as we return to the geometry of the circle. This feeling is justified.

Figure 13: Breaking down a circle into rings.

As seen in Figure 13,9 the area of a circle can be calculated by breaking it down into circular rings oflength and width , where the area of each ring is :

Now, the circumference of a circle is proportional to its radius:

The constant of proportionality is :

The area of the circle is then the integral over all rings:

If you were still a partisan at the beginning of this section, your head has now exploded. For we seethat even in this case, where supposedly shines, in fact there is a missing factor of . Indeed, theoriginal proof by Archimedes shows not that the area of a circle is , but that it is equal to the areaof a right triangle with base and height . Applying the formula for triangular area then gives

C dr CdrdA = Cdr.

C r.

C = r.

A = dA = Cdr = rdr = .r

0

r

012 r 2

2

r 2C r

A = bh = Cr = .12 12 12 r 2


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There is simply no avoiding that factor of a half (Table 3).

Quantity Symbol ExpressionDistance fallenSpring energyKinetic energyCircular areaTable 3: Some common quadratic forms.

Quod erat demonstrandum

We set out in this manifesto to show that is the true circle constant. Since the formula for circulararea was just about the last, best argument that had going for it, Im going to go out on a limb hereand say: Q.E.D.

4 Why tau?The true test of any notation is usage; having seen used throughout this manifesto, you may alreadybe convinced that it serves its role well. But for a constant as fundamental as it would be nice tohave some deeper reasons for our choice. Why not , for example, or ? Whats so great about ?

4.1 One turnThere are two main reasons to use for the circle constant. The first is that visually resembles :after centuries of use, the association of with the circle constant is unavoidable, and using feedson this association instead of fighting it. (Indeed, the horizontal line in each letter suggests that weinterpret the legs as denominators, so that has two legs in its denominator, while has only one.Seen this way, the relationship is perfectly natural.10) The second reason is that correspondsto one turn of a circle, and you may have noticed that and turn both start with a t sound. Thiswas the original motivation for the choice of , and it is not a coincidence: the root of the Englishword turn is the Greek word for lathe, tornosor, as the Greeks would put it,

Since the original launch of The Tau Manifesto, I have learned that physicist Peter Harremosindependently proposed using to Is Wrong! author Bob Palais, for essentially the same reasons.Dr. Harremos has emphasized the importance of a point first made in Section 1.1: using gives thecircle constant a name. Since is an ordinary Greek letter, people encountering it for the first time canpronounce it immediately. Moreover, unlike calling the circle constant a turn, works well in bothwritten and spoken contexts. For example, saying that a quarter circle has radian angle measure onequarter turn sounds great, but turn over four radians sounds awkward, and the area of a circle isone-half turn squared sounds downright odd. Using , we can say tau over four radians and thearea of a circle is one-half tau squared.

y g12 t 2U k12 x 2K m12 v 2A 12 r 2

= 2

o.o

r r


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4.2 Conflict and resistanceOf course, with any new notation there is the potential for conflicts with present usage. As noted inSection 1.1, Is Wrong! avoids this problem by introducing a new symbol (Figure 1). There isprecedent for this; for example, in the early days of quantum mechanics Max Planck introduced theconstant , which relates a light particles energy to its frequency (through ), but physicistssoon realized that it is often more convenient to use (read h-bar)where is just divided byum and this usage is now standard. But getting a new symbol accepted is difficult: it has to begiven a name, that name has to be popularized, and the symbol itself has to be added to wordprocessing and typesetting systems. That may have been possible with , at a time when virtually allmathematical typesetting was done by a handful of scientific publishers, but today such an approach isimpractical, and the advantages of using an existing symbol are too large to ignore.

Fortunately, although the letter appears in some current contexts, there are surprisingly few commonuses. is used for certain specific variablese.g., shear stress in mechanical engineering, torque inrotational mechanics, and proper time in special and general relativitybut there is no universalconflicting usage. Moreover, we can route around the few present conflicts by selectively changingnotation, such as using for torque11 or for proper time.

Despite these arguments, potential usage conflicts are the greatest source of resistance to . Somecorrespondents have even flatly denied that (or, presumably, any other currently used symbol) couldpossibly overcome these issues. But scientists and engineers have a high tolerance for notationalambiguity, and claims that -the-circle-constant cant coexist with other uses ignores considerableevidence to the contrary. For example, in a single chapter (Chapter 9) in a single book (AnIntroduction to Quantum Field Theory by Peskin and Schroeder), I found two examples of severeconflicts that, because of context, are scarcely noticeable to the trained eye. On p. 282, for instance,we find the following integral:

Note the presence of (or, rather, ) in the denominator of the integrand. Later on the same page wefind another expression involving :

But this second occurrence of is not a number; it is a conjugate momentum and has norelationship to circles. An even more egregious conflict occurs on p. 296, where we encounter thefollowing rather formidable expression:

Looking carefully, we see that the letter appears twice in the same expression, once in a determinant( ) and once in an integral ( ). But means completely different things in the two cases: the first

h E = h h

2

N p

exp[i( ( ) /2m].dpk2 pk qk+1 qk p2k

2

H = x[ + ( V()].d3 122 12 ) 2

det( )( D) DA ( (x)).1e 2 e iS[A] A

edet e e


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is the charge on an electron, while the second is the exponential number. As with the first example,to the expert eye it is clear from context which is which. Such examples are widespread, and theyundermine the view that current usage precludes using for the circle constant as well.

In sum, is a natural choice of notation because it references the typographical appearance of , hasetymological ties to one turn, and minimizes conflicts with present usage. Indeed, based on thesearguments (put forward by me and by Peter Harremos), Bob Palais himself has thrown his supportbehind .

4.3 The formulas revisitedThus convinced of its suitability to denote the true circle constant, we are free to use in all theformulas of mathematics and science. In particular, lets rewrite the examples from Section 2 andwatch the factors of melt away.

Integral over all space in polar coordinates:

Normal distribution:

Fourier transform:

Cauchys integral formula:

th roots of unity:

The Riemann zeta function for positive even integers:

e

2

f(r,) rdrd

0

0

1 e

(x)2

22

f(x) = F(k) dk

e ikx

F(k) = f(x) dx

eikx

f(a) = dz1i f(z)z a

n

e i/n


19 of 25 3/15/11 12:20 PM

4.4 Frequently Asked QuestionsOver the years, I have heard many arguments against the wrongness of and against the correctnessof , so before concluding our discussion allow me to address some of the most frequently askedquestions.

Are you serious?Of course. I mean, Im having fun with this, and the tone is occasionally lighthearted, but thereis a serious purpose. Setting the circle constant equal to the circumference over the diameter isan awkward and confusing convention. Although I would love to see mathematicians changetheir ways, Im not particularly worried about them; they can take care of themselves. It is theneophytes I am most worried about, for they take the brunt of the damage: as noted inSection 2.1, is a pedagogical disaster. Try explaining to a twelve-year-old (or to a thirty-year-old) why the angle measure for an eighth of a circleone slice of pizzais . Wait, Imeant . See what I mean? Its madnesssheer, unadulterated madness.How can we switch from to ?The next time you write something that uses the circle constant, simply say For convenience,we set , and then proceed as usual. (Of course, this might just prompt the question,Why would you want to do that?, and I admit it would be nice to have a place to point themto. If only someone would write, say, a manifesto on the subject) The way to get people tostart using is to start using it yourself.Isnt it too late to switch? Wouldnt all the textbooks and math papers need to berewritten?No on both counts. It is true that some conventions, though unfortunate, are effectivelyirreversible. For example, Benjamin Franklins choice for the signs of electric charges leads toelectric current being positive, even though the charge carriers themselves are negativethereby cursing electrical engineers with confusing minus signs ever since.12 To change thisconvention would require rewriting all the textbooks (and burning the old ones) since it isimpossible to tell at a glance which convention is being used. In contrast, while redefining iseffectively impossible, we can switch from to on the fly by using the conversion

Its purely a matter of mechanical substitution, completely robust and indeed fully reversible.The switch from to can therefore happen incrementally; unlike a redefinition, it need nothappen all at once.Wont using confuse people, especially students?If you are smart enough to understand radian angle measure, you are smart enough tounderstand and why is actually less confusing than . Also, there is nothing intrinsicallyconfusing about saying Let ; understood narrowly, its just a simple substitution.Finally, we can embrace the situation as a teaching opportunity: the idea that might be wrongis interesting, and students can engage with the material by converting the equations in their

(2n) = = n = 1, 2, 3, k=1

1k 2n

B n 2n2(2n)!

/8

/4

= 2

.12

= 2


20 of 25 3/15/11 12:20 PM

textbooks from to to see for themselves which choice is better.Who cares whether we use or ? It doesnt really matter.Of course it matters. The circle constant is important. People care enough about it to writeentire books on the subject, to celebrate it on a particular day each year, and to memorize tensof thousands of its digits. I care enough to write a whole manifesto, and you care enough to readit. Its precisely because it does matter that its hard to admit that the present convention iswrong. (I mean, how do you break it to Lu Chao, the current world-record holder, that he justrecited 67,890 digits of one half of the true circle constant?)13 Since the circle constant isimportant, its important to get it right, and we have seen in this manifesto that the right numberis . Although is of great historical importance, the mathematical significance of is that it isone-half .Why does this subject interest you?First, as a truth-seeker I care about correctness of explanation. Second, as a teacher I care aboutclarity of exposition. Third, as a hacker I love a nice hack. Fourth, as a student of history and ofhuman nature I find it fascinating that the absurdity of was lying in plain sight for centuriesbefore anyone seemed to notice. Moreover, many of the people who missed the true circleconstant are among the most rational and intelligent people ever to live. What else might bestaring us in the face, just waiting for us to discover it?Are you, like, a crazy person?Thats really none of your business, but no. Apart from my unusual shoes, I am to all externalappearances normal in every way. You would never guess that, far from being an ordinarycitizen, I am in fact a notorious mathematical propagandist.But what about puns?We come now to the final objection. I know, I know, in the sky is so very clever. And yet, itself is pregnant with possibilities. ism tells us: it is not that is a piece of , but that is apiece of one-half , to be exact. The identity says: Be 1 with the . And thoughthe observation that A rotation by one turn is 1 may sound like a -tology, it is the true natureof the . As we contemplate this nature to seek the way of the , we must remember that ism isbased on reason, not on faith: ists are never ous.

5 Embrace the tauWe have seen in The Tau Manifesto that the natural choice for the circle constant is the ratio of acircles circumference not to its diameter, but to its radius. This number needs a name, and I hope youwill join me in calling it :

The usage is natural, the motivation is clear, and the implications are profound. Plus, it comes with areally cool diagram (Figure 14). We see in Figure 14 a movement through yang (light, white, movingup) to and a return through yin (dark, black, moving down) back to .14 Using instead of is like having yang without yin.

= 1e i

circle constant = = 6.283185307179586 Cr

/2


21 of 25 3/15/11 12:20 PM

Figure 14: Followers of ism seek the way of the .

5.1 Tau DayThe Tau Manifesto first launched on Tau Day: June 28 (6/28), 2010. Tau Day is a time to celebrate andrejoice in all things mathematical and true.15 If you would like to receive updates about , includingnotifications about possible future Tau Day events, please join the Tau Manifesto mailing list below.And if you think that the circular baked goods on Pi Day are tasty, just waitTau Day has twice asmuch pi(e)!

Thank you for reading The Tau Manifesto. I hope you enjoyed reading it as much as I enjoyed writingit. And I hope even more that you have come to embrace the true circle constant: not , but . HappyTau Day!

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Acknowledgments

Id first like to thank Bob Palais for writing Is Wrong!. I dont remember how deep my suspicionsabout ran before I encountered that article, but Is Wrong! definitely opened my eyes, and everysection of The Tau Manifesto owes it a debt of gratitude. Id also like to thank Bob for his helpfulcomments on this manifesto, and especially for being such a good sport about it.

Ive been thinking about The Tau Manifesto for a while now, and many of the ideas presented herewere developed through conversations with my friend Sumit Daftuar. Sumit served as a soundingboard and occasional Devils advocate, and his insight as a teacher and as a mathematician influencedmy thinking in many ways.

I also received helpful feedback from several readers. The pleasing interpretation of the yin-yangsymbol used in The Tau Manifesto is due to a suggestion by Peter Harremos, who (as noted above)has the rare distinction of having independently proposed using for the circle constant. I also gotseveral good suggestions from Christopher Olah, particularly regarding the geometric interpretation ofEulers identity, and Section 2.3.2 on Eulerian identities was inspired by an excellent suggestion fromTimothy Patashu Stiles. The site for Half Tau Day benefited from suggestions by Evan Dorn, WyattGreene, Lynn Noel, Christopher Olah, and Bob Palais. Finally, Id like to thank Wyatt Greene for hisextraordinarily helpful feedback on a pre-launch draft of the manifesto; among other things, if youever need someone to tell you that pretty much all of [now deleted] section 5 is total crap, Wyatt isyour man.

About the author

The Tau Manifesto author Michael Hartl is a physicist, educator, and entrepreneur. He is the creator ofthe Ruby on Rails Tutorial book and screencast series, which teach web development with Ruby on


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Rails. Previously, he taught theoretical and computational physics at Caltech, where he received theLifetime Achievement Award for Excellence in Teaching and served as Caltechs editor for TheFeynman Lectures on Physics: The Definitive and Extended Edition. He is a graduate of HarvardCollege, has a Ph.D. in Physics from the California Institute of Technology, and is an alumnus of theY Combinator entrepreneur program.

Michael is ashamed to admit that he knows to 50 decimal placesapproximately 48 more thanMatt Groening. To make up for this, he is currently memorizing 52 decimal places of .

Copyright and license

The Tau Manifesto. Copyright 2010 by Michael Hartl. Please feel free to share The Tau Manifesto,which is available under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 UnportedLicense. This means that you cant alter it or sell it, but you do have permission to distribute copies ofThe Tau Manifesto PDF, print it out, use it in classrooms, and so on. Go forth and spread the goodnews about !

Palais, Robert. Is Wrong!, The Mathematical Intelligencer, Volume 23, Number 3, 2001,pp. 78. Many of the arguments in The Tau Manifesto are based on or are inspired by IsWrong!. It is available online at http://bit.ly/pi-is-wrong.

1.

The symbol means is defined as. 2.The video in Figure 3 (available at http://vimeo.com/12914981) is an excerpt from a lecturegiven by Dr. Sarah Greenwald, a professor of mathematics at Appalachian State University. Dr.Greenwald uses math references from The Simpsons and Futurama to engage her studentsinterest and to help them get over their math anxiety. She is also the maintainer of the FuturamaMath Page.

3.

Here is the th Bernoulli number. 4.These graphs were produced with the help of Wolfram|Alpha. 5.Here Im implicitly defining Eulers identity to be the complex exponential of the circleconstant, rather than defining it to be the complex exponential of any particular number. If wechoose as the circle constant, we obtain the identity shown. As well see momentarily, this isnot the traditional form of the identity, which of course involves , but the version with is themost mathematically meaningful statement of the identity, so I believe it deserves the name.

6.

Technically, all mathematical theorems are tautologies, but lets not be so pedantic. 7.You may have seen this written as . In this case, refers to the force exerted by thespring. By Newtons third law, the external force discussed above is the negative of the springforce.

8.

This is a physicists diagram. A mathematician would probably use , limits, and little-onotation, an approach that is more rigorous but less intuitive.

9.

Thanks to Tau Manifesto reader Jim Porter for pointing out this interpretation. 10.This alternative for torque is already in use; see, for example, Introduction to Electrodynamicsby David Griffiths, p. 162.

11.

The sign of the charge carriers couldnt be determined with the technology of Franklins time,so this isnt his fault. Its just bad luck.

12.

On the other hand, this could be an opportunity: the field for recitation records is wide open. 13.

B n n

F = kx F

r


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The interpretations of yin and yang quoted here are from Zen Yoga: A Path to Enlightenmentthough Breathing, Movement and Meditation by Aaron Hoopes.

14.

Since 6 and 28 are the first two perfect numbers, 6/28 is actually a perfect day. 15.


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Tau Manifesto (Against Pi)

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Transcript of Tau Manifesto (Against Pi)