manifesto tau

8/6/2019 Manifesto Tau

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

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

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, The Tau Manifestoproposes that this name should be the Greek letter (tau):

C r

= 6 .283185307179586. . .

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

1.2 A powerful enemy

Before proceeding with the demonstration that is the natural choice for the circle constant, let us rst ac-knowledge what we are up againstfor there is a powerful conspiracy, centuries old, determined to propagatepro- propaganda. Entire books are written extolling the virtues of . (I mean, books !) And irrational devo-tion to has spread even to the highest levels of geekdom; for example, on Pi Day 2010 Google changed its logo to honor (Figure 2 ).

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

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

3The video in Figure 3 (available at http://vimeo.com/12914981 ) is an excerpt from a lecture given by Dr. Sarah Greenwald , aprofessor of mathematics at Appalachian State University . Dr. Greenwald uses math references from The Simpsons and Futurama toengage her students interest and to help them get over their math anxiety. She is also the maintainer of the Futurama Math Page .

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

2 The number tau

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

2

0

0f (r, ) rdrd.

The upper limit of the integration is always 2 . The same factor appears in the denition of the Gaussian(normal) distribution ,

12 e

( x ) 2

2 2 ,

and again in the Fourier transform ,

f (x) =

F (k) e2ikx dk

F (k) =

f (x) e2ikx dx.

It recurs in Cauchys integral formula ,

f (a) =1

2i f (z)z a dz,in the n th roots of unity ,

zn = 1 z = e2i/n ,

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s 1

s 2

r 1

r 2

Figure 4: An angle with two concentric circles.

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

(2n) =

k =1

1k2n

=Bn

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

There are many more examples , and the conclusion is clear: there is something special about 2 .To get to the bottom of this mystery, we must return to rst principles by considering the nature of circles,

and especially the nature of angles . Although its likely that much of this material will be familiar, it pays torevisit it, for this is where the true understanding of begins.

2.1 Circles and angles

There is an intimate relationship between circles and angles, as shown in Figure 4 . Since the concentriccircles in Figure 4 have different radii, the lines in the gure 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 angle does not depend on theradius of the circle used to dene the arc. The principal task of angle measurement is to create a system thatcaptures this radius-invariance.

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 in Figure 5 .

4Here B n is the n th Bernoulli number .

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Figure 5: Some special angles, in degrees.

A more fundamental system of angle measure involves a direct comparison of the arclength s with theradius r . Although the lengths in Figure 4 differ, the arclength grows in proportion to the radius, so the ratioof the arclength to the radius is the same in each case:

s

r

s1

r 1=

s2

r 2.

This suggests the following denition of radian angle measure :

sr

.

This denition has the required property of radius-invariance, and since both s and r have units of length,radians are dimensionless by construction. The use of radian angle measure leads to succinct and elegantformulas throughout mathematics; for example, the usual formula for the derivative of sin is true only when is expressed in radians:

dd

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

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

Now, a moments reection shows that the so-called special angles are just particularly simple rational fractions of a full circle, as shown in Figure 7 . This suggests revisiting the denition of radian angle measure,rewriting the arclength s in terms of the fraction f of the full circumference C , i.e., s = fC :

=sr

=fC r

= f C r f.

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Figure 6: Some special angles, in -radians.

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

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0,

Figure 8: Some special angles, in radians.

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

Although there are many other arguments in s favor, Figure 8 may be the most striking. Indeed, uponcomparing Figure 8 with Figure 7 , I consider it decisive. We also see from Figure 8 the genius of Bob Palaisidentication of the circle constant as one turn : is the radian angle measure for one turn of a circle.Moreover, note that with there is nothing to memorize : a twelfth of a turn is / 12, an eighth of a turn is / 8, and so on. Using gives us the best of both worlds by combining conceptual clarity with all the concretebenets of radians; the abstract meaning of, say, / 12 is obvious, but it is also just a number:

a twelfth of a turn = 12

6.28318512

= 0 .5235988.

Finally, by comparing Figure 6 with Figure 8 , we see where those pesky factors of 2 come from: one turnof a circle is 1 , but 2 . Numerically they are equal, but conceptually they are quite distinct.

2.1.1 The ramications

The unnecessary factors of 2 arising from the use of are annoying enough by themselves, but far moreserious is their tendency to cancel when divided by any even number. The absurd results, such as a half fora quarter circle, obscure the underlying relationship between angle measure and the circle constant. To thosewho maintain that it doesnt matter whether we use or when teaching trigonometry, I simply ask you

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(cos , sin )

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

to view Figure 6 , Figure 7 , and Figure 8 through the eyes of a child. You will see that, from the perspectiveof a beginner, using instead of is a pedagogical disaster.

2.2 The circle functions

Although radian angle measure provides some of the most compelling arguments for the true circle constant,its worth comparing the virtues of and in some other contexts as well. We begin by considering theimportant elementary functions sin and cos . Known as the circle functions because they give the coor-dinates of a point on the unit circle (i.e., a circle with radius 1), sine and cosine are the fundamental functionsof trigonometry ( Figure 9 ).

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

Of course, since sine and cosine both go through one full cycle during one turn of the circle, we haveT = ; 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 / 4, a half-period is / 2, etc. In fact, when making Figure 10 , at one

5These graphs were produced with the help of Wolfram|Alpha .

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sin

3 T 4

T 4

T 2

T

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

3 T 4

T 4

T 2

T

cos

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

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point I found myself wondering about the numerical value of for the zero of the sine function. Since thezero occurs after half a period, and since

6.28, a quick mental calculation led to the following result:

zero = 2 3.14.

Thats right: I was astonished to discover that I had already forgotten that / 2 is sometimes called .Perhaps this even happened to you just now. Welcome to my world.

2.3 Eulers identity

I would be remiss in this manifesto not to address Eulers identity , sometimes called the most beautifulequation in mathematics. This identity involves complex exponentiation , which is deeply connected both tothe 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 a

denition; either way, it is quite remarkable:

ei = cos + i sin .

Known as Eulers formula (after Leonhard Euler ), this equation relates an exponential with imaginary argu-ment to the circle functions sine and cosine and to the imaginary unit i. Although justifying Eulers formula isbeyond the scope of this manifesto, its provenance is above suspicion, and its importance is beyond dispute.

Evaluating Eulers formula at = yields Eulers identity :6

ei = 1 .

In words, this equation makes the following fundamental observation:

The complex exponential of the circle constant is unity.

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

A rotation by one turn is 1.

Since the number 1 is the multiplicative identity , the geometric meaning of ei = 1 is that rotating a point inthe 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 turn of acircle. Indeed, the identication of with one turn makes Eulers identity sound almost like a tautology. 7

2.3.1 Not the most beautiful equation

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

ei = 1.6Here Im implicitly dening Eulers identity to be the complex exponential of the circle constant , rather than dening it to be

the complex exponential of any particular number. If we choose as the circle constant, we obtain the identity shown. As well seemomentarily, this is not the traditional form of the identity, which of course involves , but the version with is the most mathematicallymeaningful statement of the identity, so I believe it deserves the name.

7Technically, all mathematical theorems are tautologies, but lets not be so pedantic.

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Rotation angle Eulerian identity0 ei 0 = 1

/ 4 ei / 4 = i / 2 ei / 2 = 13 / 4 ei (3 / 4) = i

ei = 1

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

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

ei + 1 = 0 .

At this point, the expositor usually makes some grandiose statement about how Eulers identity relates 0, 1,e, i, and sometimes called the ve most important numbers in mathematics.

Alert readers might now complain that, because its missing 0, Eulers identity with relates only four of those ve. We can address this objection by noting that, since sin = 0 , we were already there:

ei = 1 + 0 .

This formula, without rearrangement, actually does relate the ve most important numbers in mathematics:0, 1, e, i , and .

2.3.2 Eulerian identities

Since you can add zero anywhere in any equation, the introduction of 0 into the formula ei = 1 + 0 is a

somewhat tongue-in-cheek counterpoint to ei

+ 1 = 0 , but the identity ei

= 1 does have a more seriouspoint to make. Lets see what happens when we rewrite it in terms of :ei / 2 = 1.

Geometrically, this says that a rotation by half a turn is the same as multiplying by 1. And indeed this is thecase: under a rotation of / 2 radians, the complex number z = a + ib gets mapped to a ib, which is infact just 1 z.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, ei = 1 can be interpreted as a rotationby radians, but the near-universal rearrangement to form ei + 1 = 0 shows how using distracts fromthe identitys natural geometric meaning.) The quarter-angle identities have similar geometric interpretations:ei / 4 = i says that a quarter turn in the complex plane is the same as multiplication by i, while ei (3 / 4) = isays that three-quarters of a turn is the same as multiplication by

i. A summary of these results, which we

might reasonably call Eulerian identities , appears in Table 1 .We can take this analysis a step further by noting that, for any angle , ei can be interpreted as a point

lying on the unit circle in the complex plane. Since the complex plane identies the horizontal axis withthe real part of the number and the vertical axis with the imaginary part, Eulers formula tells us that eicorresponds to the coordinates (cos , sin ). Plugging in the values of the special angles from Figure 8then gives the points shown in Table 2 , and plotting these points in the complex plane yields Figure 12 . Acomparison of Figure 12 with Figure 8 quickly dispels any doubts about which choice of circle constant betterreveals the relationship between Eulers formula and the geometry of the circle.

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Polar form Rectangular form Coordinatesei cos + i sin (cos , sin )ei 0 1 (1, 0)

ei / 12 32 +12 i (

32 ,

12 )

ei / 8 1 2 +1

2 i (1

2 ,1

2 )

ei / 6 12 + 32 i (

12 ,

32 )

ei / 4 i (0, 1)

ei / 3 12 + 32 i (12 ,

32 )

ei / 2 1 (1, 0)ei (3 / 4)

i (0,

1)

ei 1 (1, 0)

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

ei 0

, ei

e i / 2

e i(3 / 4)

ei / 4

e i / 3 e i / 6

e i / 8

e i / 12

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

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3 Circular area: the coup de grce

If you arrived here as a believer, you must by now be questioning your faith. is so natural, its meaning sotransparentis there no example where shines through in all its radiant glory? A memory stirsyes, thereis such a formulait is the formula for circular area! Behold:

A = r 2 .

We see here , unadorned, in one of the most important equations in mathematicsa formula rst proved byArchimedes himself. Order is restored! And yet, the name of this section sounds ominous. . . If this equationis s crowning glory, how can it also be the coup de grce ?

3.1 Quadratic forms

Let us examine this paragon of , A = r 2 . We notice that it involves the diameterno, wait, the radius raised to the second power. This makes it a simple quadratic form . Such forms arise in many contexts; as aphysicist , my favorite examples come from the elementary physics curriculum. We will now consider severalin turn.

3.1.1 Falling in a uniform gravitational eld

Galileo Galilei found that the velocity of an object falling in a uniform gravitational eld is proportional tothe time fallen:

vt.

The constant of proportionality is the gravitational acceleration g:

v = gt.

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

y = v dt = t

0gtdt = 12 gt

2 .

3.1.2 Potential energy in a linear spring

Robert Hooke found that theexternal force required to stretch a spring is proportional to thedistance stretched:

F x.

The constant of proportionality is the spring constant k:8

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

U = F dx = x

0kxdx = 12 kx

2 .

8You may have seen this written as F = kx . In this case, F refers to the force exerted by the spring . By Newtons third law, theexternal force discussed above is the negative of the spring force.

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3.1.3 Energy of motion

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

F a.

The constant of proportionality is the mass m :

F = ma.

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

K = F dx = madx = m dvdt dx = m dxdt dv = v

0mvdv = 12 mv

2 .

3.2 A sense of forebodingHaving seen several examples of simple quadratic forms in physics, you may by now have a sense of fore-boding as we return to the geometry of the circle. This feeling is justied.

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

dA = C dr.

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

C r.

The constant of proportionality is :

C = r.The area of the circle is then the integral over all rings:

A = dA = r

0C dr =

r

0rdr = 12 r

2 .

If you were still a partisan at the beginning of this section, your head has now exploded. For we see thateven in this case, where supposedly shines, in fact there is a missing factor of 2. Indeed, the original proof by Archimedes shows not that the area of a circle is r 2 , but that it is equal to the area of a right triangle withbase C and height r . Applying the formula for triangular area then gives

A = 12 bh =12 Cr =

12 r

2 .

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

3.2.1 Quod erat demonstrandum

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

9This is a physicists diagram. A mathematician would probably use r , limits, and little-o notation , an approach that is morerigorous but less intuitive.

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r

dr

dA = C dr

Figure 13: Breaking down a circle into rings.

Quantity Symbol ExpressionDistance fallen y 12 gt

2

Spring energy U 12 kx 2Kinetic energy K 12 mv

2

Circular area A 12 r2

Table 3: Some common quadratic forms.

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4 Why tau?

The true test of any notation is usage; having seen used throughout this manifesto, you may already beconvinced that it serves its role well. But for a constant as fundamental as it would be nice to have somedeeper reasons for our choice. Why not , for example, or ? Whats so great about ?

4.1 One turn

There are two main reasons to use for the circle constant. The rst is that visually resembles : aftercenturies of use, the association of with the circle constant is unavoidable, and using feeds on thisassociation instead of ghting it. (Indeed, the horizontal line in each letter suggests that we interpret thelegs as denominators , so that has two legs in its denominator, while has only one. Seen this way, therelationship = 2 is perfectly natural. 10 ) The second reason is that corresponds to one turn of a circle,and you may have noticed that and turn both start with a t sound. This was the original motivationfor the choice of , and it is not a coincidence: the root of the English word turn is the Greek word forlathe, tornos or, as the Greeks would put it,

oo.

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

4.2 Conict and resistance

Of course, with any new notation there is the potential for conicts with present usage. As noted in Sec-tion 1.1 , Is Wrong! avoids this problem by introducing a new symbol ( Figure 1 ). There is precedentfor this; for example, in the early days of quantum mechanics Max Planck introduced the constant h, whichrelates a light particles energy to its frequency (through E = h ), but physicists soon realized that it is oftenmore convenient to use h (read h-bar)where h is just h divided by...um... 2and this usage is nowstandard. But getting a new symbol accepted is difcult: it has to be given a name, that name has to be pop-ularized, and the symbol itself has to be added to word processing and typesetting systems. That may havebeen possible with h, at a time when virtually all mathematical typesetting was done by a handful of scienticpublishers, but today such an approach is impractical, and the advantages of using an existing symbol are toolarge to ignore.

Fortunately, although the letter appears in some current contexts, there are surprisingly few commonuses. is used for certain specic variablese.g., shear stress in mechanical engineering, torque in rotationalmechanics, and proper time in special and general relativitybut there is no universal conicting usage.Moreover, we can route around the few present conicts by selectively changing notation, such as using N for torque 11 or p for proper time.

Despite these arguments, potential usage conicts are the greatest source of resistance to . Some cor-respondents have even atly denied that (or, presumably, any other currently used symbol) could possibly

10 Thanks to Tau Manifesto reader Jim Porter for pointing out this interpretation.11 This alternative for torque is already in use; see, for example, Introduction to Electrodynamics by David Grifths, p. 162.

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overcome these issues. But scientists and engineers have a high tolerance for notational ambiguity, and claimsthat -the-circle-constant cant coexist with other uses ignores considerable evidence to the contrary. For ex-ample, in a single chapter (Chapter 9) in a single book ( An Introduction to Quantum Field Theory by Peskinand Schroeder), I found two examples of severe conicts that, because of context, are scarcely noticeable tothe trained eye. On p. 282, for instance, we nd the following integral:

dpk2 exp i( pk (qk+1 qk ) p2k / 2m .Note the presence of (or, rather, 2 ) in the denominator of the integrand. Later on the same page we ndanother expression involving :

H = d3x 12 2 + 12 ( )2 V () .But this second occurrence of is not a number; it is a conjugate momentum and has no relationshipto circles. An even more egregious conict occurs on p. 296, where we encounter the following ratherformidable expression:

det1e

2 D DA e iS [A] ( A (x)) .Looking carefully, we see that the letter e appears twice in the same expression , once in a determinant ( det )and once in an integral ( ). But e means completely different things in the two cases: the rst e is the chargeon an electron, while the second e is the exponential number . As with the rst example, to the expert eye it isclear from context which is which. Such examples are widespread, and they undermine the view that currentusage 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 conicts with present usage. Indeed, based on these arguments

(put forward by me and by Peter Harremos), Bob Palais himself has thrown his support behind .

4.3 The formulas revisited

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

Integral over all space in polar coordinates:

0

0f (r, ) rdrd

Normal distribution: 1 e

( x ) 2

2 2

Fourier transform:

f (x) =

F (k) eikx dk

F (k) =

f (x) e ikx dx

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Cauchys integral formula:

f (a) =1i

f (z)z a dz

n th roots of unity:ei/n

The Riemann zeta function for positive even integers:

(2n) =

k =1

1k2n

=Bn 2n

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

4.4 Frequently Asked Questions

Over the years, I have heard many arguments against the wrongness of and against the correctness of , so

before concluding our discussion allow me to address some of the most frequently asked questions.

Are you serious?Of course. I mean, Im having fun with this, and the tone is occasionally lighthearted, but there is aserious purpose. Setting the circle constant equal to the circumference over the diameter is an awkwardand confusing convention. Although I would love to see mathematicians change their ways, Im notparticularly worried about them; they can take care of themselves. It is the neophytes I am most worriedabout, for they take the brunt of the damage: as noted in Section 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 / 8. Wait, I meant / 4. 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 = 2 , and then proceed as usual. (Of course, this might just prompt the question, Why wouldyou want to do that?, and I admit it would be nice to have a place to point them to. If only someonewould write, say, a manifesto on the subject. . . ) The way to get people to start using is to start usingit yourself.

Isnt it too late to switch? Wouldnt all the textbooks and math papers need to be rewritten?No on both counts. It is true that some conventions, though unfortunate, are effectively irreversible.For example, Benjamin Franklins choice for the signs of electric charges leads to electric current beingpositive, even though the charge carriers themselves are negativethereby cursing electrical engineerswith confusing minus signs ever since. 12 To change this convention would require rewriting all thetextbooks (and burning the old ones) since it is impossible to tell at a glance which convention is beingused. In contrast, while redening is effectively impossible, we can switch from to on the y byusing the conversion

12 .Its purely a matter of mechanical substitution, completely robust and indeed fully reversible. Theswitch from to can therefore happen incrementally; unlike a redenition, it need not happen all atonce.

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

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Wont using confuse people, especially students?If you are smart enough to understand radian angle measure, you are smart enough to understand and why is actually less confusing than . Also, there is nothing intrinsically confusing aboutsaying Let = 2 ; understood narrowly, its just a simple substitution. Finally, we can embracethe situation as a teaching opportunity: the idea that might be wrong is interesting , and studentscan engage with the material by converting the equations in their textbooks from to to see forthemselves 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 write entirebooks on the subject, to celebrate it on a particular day each year, and to memorize tens of thousandsof its digits. I care enough to write a whole manifesto, and you care enough to read it. Its preciselybecause it does matter that its hard to admit that the present convention is wrong. (I mean, how do youbreak it to Lu Chao , the current world-record holder, that he just recited 67,890 digits of one half of the true circle constant?) 13 Since the circle constant is important, its important to get it right, and wehave seen in this manifesto that the right number is . Although is of great historical importance, themathematical signicance of is that it is one-half .

Why does this subject interest you?First, as a truth-seeker I care about correctness of explanation. Second, as a teacher I care about clarityof exposition. Third, as a hacker I love a nice hack. Fourth, as a student of history and of humannature I nd it fascinating that the absurdity of was lying in plain sight for centuries before anyoneseemed to notice. Moreover, many of the people who missed the true circle constant are among themost rational and intelligent people ever to live. What else might be staring us in the face, just waitingfor 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 external appear-ances normal in every way. You would never guess that, far from being an ordinary citizen, I am in facta notorious mathematical propagandist.

But what about puns?We come now to the nal 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 a piece of one-half , to be exact. The identity ei = 1 says: Be 1 with the . And though the observationthat A rotation by one turn is 1 may sound like a -tology, it is the true nature of the . As wecontemplate this nature to seek the way of the , we must remember that ism is based on reason, noton faith: ists are never ous.

5 Embrace the tau

We have seen in The Tau Manifesto that the natural choice for the circle constant is the ratio of a circlescircumference not to its diameter, but to its radius. This number needs a name, and I hope you will join mein calling it :

circle constant = C r

= 6 .283185307179586. . .

The usage is natural, the motivation is clear, and the implications are profound. Plus, it comes with a reallycool diagram ( Figure 14 ). We see in Figure 14 a movement through yang (light, white, moving up) to / 2

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

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0,

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

and a return through yin (dark, black, moving down) back to .14 Using instead of is like having yangwithout yin .

5.1 Tau Day

The Tau Manifesto rst launched on Tau Day: June 28 (6/28), 2010. Tau Day is a time to celebrate and rejoicein all things mathematical and true. 15 If you would like to receive updates about , including noticationsabout possible future Tau Day events, please join the Tau Manifesto mailing list below. And if you think thatthe circular baked goods on Pi Day are tasty, just waitTau Day has twice as much pi(e)!

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

The signup form is available online at http://tauday.com/#sec:tau_day .

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

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

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5.1.1 Acknowledgments

Id rst like to thank Bob Palais for writing Is Wrong!. I dont remember how deep my suspicions about ran before I encountered that article, but Is Wrong! denitely opened my eyes, and every section of The Tau Manifesto owes it a debt of gratitude. Id also like to thank Bob for his helpful comments on thismanifesto, 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 here weredeveloped through conversations with my friend Sumit Daftuar. Sumit served as a sounding board and occa-sional Devils advocate, and his insight as a teacher and as a mathematician inuenced my thinking in manyways.

I also received helpful feedback from several readers. The pleasing interpretation of the yin-yang symbolused in The Tau Manifesto is due to a suggestion by Peter Harremos , who (as noted above) has the raredistinction of having independently proposed using for the circle constant. I also got several good sug-gestions from Christopher Olah , particularly regarding the geometric interpretation of Eulers identity, andSection 2.3.2 on Eulerian identities was inspired by an excellent suggestion from Timothy Patashu Stiles.The site for Half Tau Day beneted from suggestions by Evan Dorn , Wyatt Greene , Lynn Noel , ChristopherOlah , and Bob Palais . Finally, Id like to thank Wyatt Greene for his extraordinarily helpful feedback on apre-launch draft of the manifesto; among other things, if you ever need someone to tell you that pretty muchall of [now deleted] section 5 is total crap, Wyatt is your man.

5.1.2 About the author

The Tau Manifesto author Michael Hartl is a physicist, educator, and entrepreneur. He is the creator of the Ruby on Rails Tutorial book and screencast series , which teach web development with Ruby on Rails . Pre-viously, he taught theoretical and computational physics at Caltech , where he received the Lifetime Achieve-ment Award for Excellence in Teaching and served as Caltechs editor for The Feynman Lectures on Physics:The Denitive and Extended Edition . He is a graduate of Harvard College , has a Ph.D. in Physics from theCalifornia Institute of Technology , and is an alumnus of the Y Combinator entrepreneur program.

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

5.1.3 Copyright and license

The Tau Manifesto . Copyright c 2010 by Michael Hartl. Please feel free to share The Tau Manifesto , whichis available under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License . Thismeans that you cant alter it or sell it, but you do have permission to distribute copies of The Tau ManifestoPDF , print it out, use it in classrooms, and so on. Go forth and spread the good news about !

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