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The Tau ManifestoMichael HartlTau Day, 2010updated Half Tau Day,
2011Official Tau Day T-shirts are now available! Proceeds benefit
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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
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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
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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
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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
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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
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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!
1,194
Ethan Sawyer and 11,077 others like this. Unlike
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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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