1 The circle constant
The Tau Manifesto is dedicated to one of the most important numbers in mathematics, perhaps the most important: the circle constant relating the circumference of a circle to its linear dimension. For millennia, the circle has been considered the most perfect of shapes, and the circle constant captures the geometry of the circle in a single number. Of course, the traditional choice for the circle constant is
1.1 An immodest proposal
We begin repairing the damage wrought by
The number

It should be obvious that
Because the diameter of a circle is twice its radius, this number is numerically equal to

In “

The Tau Manifesto is based on the proposition that the proper response to “
Throughout the rest of this manifesto, we will see that the number
1.2 A powerful enemy
Before proceeding with the demonstration that

Meanwhile, some people memorize dozens, hundreds, even thousands of digits of this mystical number. What kind of sad sack memorizes even 40 digits of
Truly, proponents of
2 The number tau
We saw in Section 1.1 that the number
The upper limit of the
and again in the Fourier transform,
It recurs in Cauchy’s integral formula,
in the
and in the values of the Riemann zeta function for positive even integers:5
These formulas are not cherry-picked—crack open your favorite physics or mathematics text and try it yourself. 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 of circles, and especially the nature of angles. Although it’s likely that much of this material will be familiar, it pays to revisit it, for this is where the true understanding of
2.1 Circles and angles
There is an intimate relationship between circles and angles, as shown in Figure 6. Since the concentric circles in Figure 6 have different radii, the lines in the figure cut off different lengths of arc (or arc lengths), but the angle

Perhaps the most elementary angle system is degrees, which breaks a circle into 360 equal parts. One result of this system is the set of special angles (familiar to students of trigonometry) shown in Figure 7.

A more fundamental system of angle measure involves a direct comparison of the arc length
This suggests the following definition of radian angle measure:
This definition has the required property of radius-invariance, and since both
Naturally, the special angles in Figure 7 can be expressed in radians, and when you took high-school trigonometry you probably memorized the special values shown in Figure 8. (I call this system of measure


Now, a moment’s reflection shows that the so-called “special” angles are just particularly simple rational fractions of a full circle, as shown in Figure 9. This suggests revisiting Eq. (4), rewriting the arc length
Notice how naturally

Although there are many other arguments in
Finally, by comparing Figure 8 with Figure 10, we see where those pesky factors of
The ramifications
The unnecessary factors of
2.2 The circle functions
Although radian angle measure provides some of the most compelling arguments for the true circle constant, it’s worth comparing the virtues of

Let’s examine the graphs of the circle functions to better understand their behavior.6 You’ll notice from Figure 12 and Figure 13 that both functions are periodic with period


Of course, since sine and cosine both go through one full cycle during one turn of the circle, we have
That’s right: I was astonished to discover that I had already forgotten that
2.3 Euler’s identity
I would be remiss in this manifesto not to address Euler’s identity, sometimes called “the most beautiful equation in mathematics”. This identity involves complex exponentiation, which is deeply connected both to the circle functions and to the geometry of the circle itself.
Depending on the exact route chosen, the following equation can either be proved as a theorem or taken as a definition; either way, it is quite remarkable:
Known as Euler’s formula (after Leonhard Euler), this equation relates an exponential with imaginary argument to the circle functions sine and cosine and to the imaginary unit
Evaluating Eq. (6) at
which simplifies to Euler’s identity:7
In words, Eq. (8) makes the following fundamental observation:
The complex exponential of the circle constant is unity.
Geometrically, multiplying by
A rotation by one turn is 1.
Since the number
As in the case of radian angle measure, we see how natural the association is between
Not the most beautiful equation
Of course, the traditional form of Euler’s identity is written in terms of
and simplifes to
But that minus sign is so ugly that Eq. (10) is almost always rearranged immediately, giving the following “beautiful” equation:
At this point, the expositor usually makes some grandiose but purely numerological statement about how Eq. (11) relates
In this context, it’s remarkable how many people complain that Eq. (8) relates only four of those five numbers. Fine:
Indeed, we saw in Eq. (7) that there is actually a
Eq. (13), without rearrangement, actually does relate the so-called “five most important numbers in mathematics”:
Eulerian identities
Since you can add zero anywhere in any equation, the introduction of
Geometrically, this says that a rotation by half a turn is the same as multiplying by
Written in terms of
Rotation angle | Eulerian identity | ||
We can take this analysis a step further by noting that, for any angle
Polar form | Rectangular form | Coordinates |

3 Circular area: the coup de grâce
If you arrived here as a
No, wait. The area formula is always written in terms of the radius, as follows:
We see here
3.1 Quadratic forms
Let us examine this putative paragon of
Falling in a uniform gravitational field
Galileo Galilei found that the velocity of an object falling in a uniform gravitational field is proportional 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:8
Potential energy in a linear spring
Robert Hooke found that the external force required to stretch a spring is proportional to the distance stretched:
The constant of proportionality is the spring constant
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 to velocity
3.2 A sense of foreboding
Having seen several examples of simple quadratic forms in physics, you may by now have a sense of foreboding as we return to the geometry of the circle. This feeling is justified.

As seen in Figure 15, the area of a circle can be calculated by breaking it down into circular rings of length
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
There is simply no avoiding that factor of a half (Table 3).
Quantity | Symbol | Expression |
Distance fallen | ||
Spring energy | ||
Kinetic energy | ||
Circular area |
Quod erat demonstrandum
We set out in this manifesto to show that
4 Conflict and resistance
Despite the definitive demonstration of the superiority of
4.1 One turn
The true test of any notation is usage; having seen
There are two main reasons to use
The second reason is that
Since the original launch of The Tau Manifesto, I have learned that Peter Harremoës independently proposed using
Ambiguous notation
Of course, with any new notation there is the potential for conflict with present usage. As noted in Section 1.1, “
But getting a new symbol accepted is difficult: it has to be given a name, that name has to be popularized, and the symbol itself has to be added to word processing and typesetting systems. Using an existing symbol allows us to route around the mathematical establishment.12
Rather than advocating a new symbol, The Tau Manifesto opts for the use of an existing Greek letter. As a result, since
Despite these arguments, potential usage conflicts have proven to be the greatest source of resistance to
One example of such easily tolerated ambiguity occurs in quantum mechanics, where we encounter the following formula for the Bohr radius, which (roughly speaking) is the “size” of a hydrogen atom in its lowest energy state (the ground state):
where
where
Have you noticed the problem yet? Probably not, which is just the point. The “problem” is that the
which has an
I have no doubt that if a separate notation for Euler’s number did not already exist, anyone proposing the letter
By the way, the
At first glance, this appears to be more natural than the version with
As usual, appearances are deceiving: the value of
which shows that the circle constant enters the calculation through
4.2 The Pi Manifesto
Although most objections to
While we can certainly consider the appearance of the Pi Manifesto a good sign of continuing interest in this subject, it makes several false claims. For example, it says that the factor of
This is wrong: the factor of
A related claim is that the gamma function evaluated at
where
But
The Pi Manifesto also examines some formulas for regular
This issue was dealt with in “
calling it “clearly… another win for
which is just
In other words, the area of an
In this context, we should note that the Pi Manifesto makes much ado about
so there’s no way to avoid the factor of
In short, the difference between angle measure and area isn’t arbitrary. There is no natural factor of
5 Getting to the bottom of pi and tau
It is time now to determine exactly what is wrong with
The resulting section is more advanced than the rest of this manifesto and can be skipped without loss of continuity; if you find it confusing, I recommend proceeding directly to the conclusion in Section 6. But if you’re up for a mathematical challenge, you are invited to proceed…
5.1 Surface areas and volumes in dimensions
We start our investigation with the generalization of circles and circular disks to arbitrary dimensions. These objects are known as hyperspheres or
A
which consists of the two points
This is a line segment from
A
This figure forms the boundary of a
This is a closed disk of radius
This is the boundary of a
The generalization to arbitrary dimension
which forms the boundary of the corresponding
The “volume of a hypersphere (or
The subscripts on
Now, The Pi Manifesto (discussed in Section 4.2) includes a formula for the volume of a unit
where the gamma function is given by Eq. (17). Eq. (21) is a special case of the formula for general radius, which is also typically written in terms of
Because
Rather than simply take these formulas at face value, let’s see if we can untangle them to shed more light on the question of
Seen this way,
In the
(Here we write
Let’s examine Eq. (25) in more detail. Notice first that MathWorld uses the double factorial function
(By definition,
To solve this mystery, we’ll start by taking a closer look at the formula for odd
Upon examining the expression
we notice that it can be rewritten as
and here we recognize our old friend
Now let’s look at the even case in Eq. (25). We noted above how strange it is to use the ordinary factorial in the even case but the double factorial in the odd case. Indeed, because the double factorial is already defined piecewise, if we unified the formulas by using
So, is there any connection between the factorial and the double factorial? Yes—when
(This can be verified using mathematical induction.) Substituting this into the volume formula for even
which bears a striking resemblance to
and again we find a factor of
Putting these results together and setting
Eq. (27) is definitely an improvement on Eq. (22). Like Eq. (22), Eq. (27) depends explicitly on the parity of the dimension (i.e., whether
Eq. (27) appears to be yet another argument in favor of
We thus see that the volume of an
Looking at the even case in Eq. (27), we see that pulling out a factor of
One might reasonably wonder whether this is really worth the trouble, but amazingly the same trick works in the odd case as well. We see from Eq. (27) that there is already a factor of
Note that the extra factor of
Substituting
Finally, note that, as
We see that this is exactly the same pattern as in the powers of
and
In other words, we have
which precisely matches the exponents in the even and odd cases of Eq. (30). This means that we can rewrite Eq. (30) as
At this point, we see that something astonishing has happened: the expressions in the even and odd cases of Eq. (31) are now identical! In the process of expressing the volume in terms of the
without breaking it into explicit even and odd cases.20 The corresponding surface-area formula then follows by differentiating Eq. (32) with respect to
The formula Eq. (32) is the product of three terms, two of which are conceptually quite simple. We saw in Eq. (28) that the factor of
where
Considering the case of a unit
Thus, we see that
To my knowledge, Eq. (32) and Eq. (33) are the most compact expressions for the spherical volumes and surface areas in terms of elementary functions and fundamental constants. In the case of
and
where we have used the relation
5.2 Three families of constants
Equipped with the tools developed in Section 5.1, we’re now ready to get to the bottom of
We saw in Section 5.1 that the most natural constant in the context of
We can define this family of constants, which we’ll call
Second, we’ll define a family of “volume constants”
With the two families of constants defined in Eq. (34) and Eq. (35), we can write the surface area and volume formulas (Eq. (33) and Eq. (32)) compactly as follows:
and
Because of the relation
Thus, when discussing
which is the generalization of
Let us make some observations about these two families of constants. The family
Here
Meanwhile, the
This shows that
So, to which family of constants does
We thus see that
We can express this in terms of the family
We are now finally in a position to understand exactly what is wrong with
It’s true that this happens to equal
But this equality is a coincidence: it occurs only because
6 Conclusion
Over the years, I have heard many arguments against the wrongness of
6.1 Frequently Asked Questions
Are you serious? Of course. I mean, I’m having fun with this, and the tone is occasionally lighthearted, but there is a serious purpose. Setting the circle constant equal to the circumference over the diameter is an awkward and confusing convention. Although I would love to see mathematicians change their ways, I’m not particularly worried about them; they can take care of themselves. It is the neophytes I am most worried about, 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 circle—one slice of pizza—is . Wait, I meant . See what I mean? It’s madness—sheer, 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 them to. If only someone would write, say, a manifesto on the subject…) The way to get people to start using is to start using it yourself.Isn’t it too late to switch? Wouldn’t 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 Franklin’s choice for the signs of electric charges leads to the most familiar example of electric current (namely, free electrons in metals) being positive when the charge carriers are negative, and vice versa—thereby cursing beginning physics students with confusing negative signs ever since.21 To change this convention would require rewriting all the textbooks (and burning the old ones) since it is impossible to tell at a glance which convention is being used. In contrast, while redefining
is effectively impossible, we can switch from to on the fly by using the conversionIt’s 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 not happen all at once.Won’t 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 about saying “Let ”; understood narrowly, it’s just a simple substitution. Finally, we can embrace the situation as a teaching opportunity: the idea that might be wrong is interesting, and students can engage with the material by converting the equations in their textbooks from to to see for themselves which choice is better.Does any of this really matter? Of course it matters. The circle constant is important. People care enough about it to write entire books on the subject, to celebrate it on a particular day each year, and to memorize tens of thousands of its digits. I care enough to write a whole manifesto, and you care enough to read it. It’s precisely because it does matter that it’s hard to admit that the present convention is wrong. Since the circle constant is important, it’s important to get it right, and we have seen in this manifesto that the right number is
. Although is of great historical importance, the mathematical significance of is that it is one-half .Why did anyone ever use
in the first place? The origins of -the-number are probably lost in the mists of time. I suspect that the convention of using instead of arose simply because it is easier to measure the diameter of a circular object than it is to measure its radius. But that doesn’t make it good mathematics, and I’m surprised that Archimedes, who famously approximated the circle constant, didn’t realize that is the more fundamental number. As notation, was popularized around 300 years ago by Leonhard Euler, based on the work of William Jones. For example, in his hugely influential two-volume work Introductio in analysin infinitorum, Euler uses to denote the semicircumference (half-circumference) of a unit circle or the measure of a arc.22 Unfortunately, Euler doesn’t explain why he introduces this factor of , though it may be related to the occasional importance of the semiperimeter of a polygon. In any case, he immediately notes that sine and cosine have periodicity , so he was certainly in a position to see that he was measuring angles in terms of twice the period of the circle functions, making his choice all the more perplexing. He almost got it right, though: somewhat incredibly, Euler actually used the symbol to mean both and at different times!23 What a shame that he didn’t standardize on the more convenient convention.What is the strongest argument in favor of
?The strongest argument in favor of the number
is that it is the area of a unit disk, though we saw in Section 5.2 that the equality between this constant and the number is a coincidence. As for proper, thinking in terms of can be convenient in certain applications, such as water pipes or agricultural fields, where the diameter plays an especially important role.What is the strongest argument against
?Based on Eq. (32) and Eq. (33), one might reasonably argue that
—which we could call the hypersphere constant—is the more fundamental number. On the other hand, the choice ( -spheres and -balls, i.e., “circles” broadly defined) is by far the most important special case, with ( -spheres and -balls) running a distant second, and the importance of higher-dimensional hyperspheres being negligible by comparison. It thus makes sense to standardize on even if the general case is arguably more natural in terms of . Indeed, the vast majority of the time that enters mathematical expressions, it is via , so using in all cases would introduce factors of everywhere, just as using introduces factors of .In any case, with respect to the circle constant that is the subject of this manifesto,
is clearly a lost cause, and applies to more naturally to general hyperspheres, so undoubtedly deserves the name. As the legendary mathematician John Conway once observed: “ is obviously the correct constant!”Why does this subject interest you? First, as a truth-seeker I care about correctness of explanation. Second, as a teacher I care about clarity of exposition. Third, as a hacker I love a nice hack. Fourth, as a student of history and of human nature I find it fascinating that the absurdity of
was lying in plain sight for centuries before anyone seemed to notice. Moreover, many of the people who missed the true circle constant are among the most rational and intelligent people ever to live. What else might be staring us in the face, just waiting for us to discover it?Are you like a crazy person? That’s really none of your business, but no. Like everyone, I do have my idiosyncrasies, but I am to all external appearances normal in practically every way. You would never guess from meeting me that, far from being an ordinary citizen, 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 a piece of —one-half , to be exact. The identity says: “Be one with the .” And though the observation that “A rotation by one turn is 1” may sound like a -tology, it is the true nature of the . As we contemplate this nature to seek the way of the , we must remember that ism is based on reason, not on faith: ists are never ous.
6.2 Embrace the tau
We have seen in The Tau Manifesto that the natural choice for the circle constant is the ratio of a circle’s circumference not to its diameter, but to its radius. This number needs a name, and I hope you will join me in calling it
The usage is natural, the motivation is clear, and the implications are profound. Plus, it comes with a really cool diagram (Figure 16). We see in Figure 16 a movement through yang (“light, white, moving up”) to

6.28 Tau Day
The Tau Manifesto first launched on Tau Day: June 28 (6/28), 2010. Tau Day is a time to celebrate and rejoice in all things mathematical.25 If you would like to receive updates about
Share the τ Manifesto
6.283 Acknowledgments
I’d first like to thank Bob Palais for writing “
I’ve been thinking about The Tau Manifesto for a while now, and many of the ideas presented here were developed through conversations with my friend Sumit Daftuar. Sumit served as a sounding board and occasional Devil’s advocate, and his insight as a teacher and as a mathematician influenced my thinking in many ways.
I have also received encouragement and helpful feedback from several readers. I’d like to thank Vi Hart and Michael Blake for their amazing
I got several good suggestions from Christopher Olah, particularly regarding the geometric interpretation of Euler’s identity, and Section 2.3.2 on Eulerian identities was inspired by an excellent suggestion from Timothy “Patashu” Stiles. Don Blaheta anticipated and inspired some of the material on hyperspheres, and John Kodegadulo put it together in a particularly clear and entertaining way. Then Jeff Cornell suggested a wonderful refinement with the introduction of
I’d also like to acknowledge my appreciation for the volunteer translators who have made The Tau Manifesto available in so many different languages: Juan Guijarro Ferreiro (Spanish); Daniel Rosen and Alexis Drai (French); Andrea Laretto (Italian); Gustavo Chaves (Portuguese); Axel Scheithauer, Jonas Wagner, and Johannes Clemens Huber, with helpful notes from Caroline Steiblin (German); Aleksandr Alekseevich Adamov (Russian); and Daniel Li Qu (simplified Chinese).
Finally, I’d like to thank Wyatt Greene for his extraordinarily helpful feedback on a pre-launch draft of the manifesto; among other things, if you ever need someone to tell you that “pretty much all of the [now deleted] final section is total crap”, Wyatt is your man.
6.28318 Copyright
The Tau Manifesto. Copyright © 2010–2023 Michael Hartl. Please feel free to distribute The Tau Manifesto PDF for educational purposes, and consider buying one or more copies of the print edition for distribution to students and other interested parties.
6.283185 Dedication
The Tau Manifesto is dedicated to Harry “Woody” Woodworth, my eighth-grade science teacher. Although I gratefully received support from many teachers over the years, Woody believed in my potential to an extraordinary, even irrational (dare I say transcendental?) degree—confidently predicting that “someday they’ll be teaching the ‘Hartl theory’ in schools.” Given how many teachers have reached out indicating their support for and teaching of the material in The Tau Manifesto, I suppose in a sense Woody’s prediction has now come true.