Tau

Tau

τ

Devotional panel, egg tempera and gold leaf on wood; gilder unknown. The disc is the circle and the line from its centre to the rim is the radius, so that the panel states the definition without writing it out. The banner gives seven places. Nine hundred and ninety-three more are printed below.

Value	6.28318 53071 79586…
Often misspelled	2π
Defined as	Circumference divided by radius
The radius, in practice	The measurement a circle is drawn with
Provenance
Argued for	Bob Palais, 2001[1]
Named	Michael Hartl, 28 June 2010[2]
Quantity's own age	As old as the circle
How it comes out
A full turn	τ
A quarter turn	τ/4
An eighth of a turn	τ/8
Once round the unit circle	eiτ = 1
Area of a circle	½τr²
Improvement on the alternative	A factor of two, everywhere
Recommended	Yes

Tau, written τ, is the true circle constant: the ratio of a circle's circumference to its radius, approximately 6.28319. It is exactly twice π, carries precisely the same information, and is the constant the subject has been reaching for since it began.

The case for it is not a discovery and does not pretend to be. Both numbers have always been available, the mathematics is identical either way, and no result changes. What changes is how much of a student's attention is spent carrying a factor of two that the subject never asked for. Once the substitution has been seen it cannot be unseen, which is the usual sign that a notation was wrong.

Readers who wanted the other one may proceed to π, where it is kept.

The radius is the measurement you have

A circle is defined as the set of points at a fixed distance from a centre, and that distance is the radius. It is what a compass is set to, what a wheel turns about, what an orbit is quoted in, and what appears in every equation of a circle ever written: $x^2 + y^2 = r^2$.

The diameter is a derived quantity. Nothing constructs a circle from its diameter; the diameter is a line you can draw across one afterwards. Defining the circle constant against it is therefore a choice to measure the figure by a length its own definition does not mention, and the factor of two that follows is the price of that choice, paid in full, every time, by everybody.

Angles come out with the right numbers on them

This is the argument that persuades people, and it takes about ten seconds.

In the τ column the number beneath the line is the number of parts the turn has been divided into, so that τ/8 is an eighth of a turn and can be read as one. In the π column it is half that, and the reader must halve it silently while thinking about something else. The whole of the difficulty that first-year students have with radians is in that column, and it is a difficulty of notation and not of mathematics.^[3]

The half in the area

The one formula that is shorter in π is the area of a circle, πr² against ½τr². It is also the formula that shows what π has been doing.

The area is obtained by integrating the circumference outward from the centre:

and the ½ arrives from the integration, exactly as it does in $\tfrac{1}{2}at^2$ for a distance under constant acceleration, $\tfrac{1}{2}mv^2$ for kinetic energy, $\tfrac{1}{2}kx^2$ for a spring and $\tfrac{1}{2}LI^2$ for an inductor. Every one of those is the integral of a linear quantity, and every one wears its ½ where the integration put it.

πr² is the only member of the family that has had its ½ absorbed into the constant, and it looks simpler because the working has been hidden rather than done. That is the whole of the argument for π, and it is an argument for a shorter formula rather than a clearer one.^[4]

Euler, and the rest of physics

The most quoted equation in mathematics is $e^{i\pi} + 1 = 0$, which is presented as beautiful because it contains five constants.

It is worth being scrupulous here, because the τ case is often put carelessly. Transcribed into the true circle constant, that identity becomes $e^{i\tau/2} + 1 = 0$, which is no tidier and is not the argument. The argument is that the formula underneath it, $e^{i\theta} = \cos\theta + i\sin\theta$, is more telling when taken at a whole turn than at a half:

Both are true and neither is a rearrangement of the other. The π form says that going half way round the unit circle puts you at −1, which is so. The τ form says that going all the way round puts you back where you began, which is the property the formula exists to express.^[5]

Away from Euler, the two turns up so persistently beside π that mathematics has repeatedly given it a name of its own rather than write it. ħ exists because Planck's constant is nearly always wanted as h/2π. The normal distribution carries 1/√(2π). The Fourier kernel is $e^{2\pi i x}$. The Cauchy integral formula, the Gaussian integral, the period of a pendulum, the resonance of a circuit: 2π, 2π, 2π. Each is a whole turn, written as two halves, in a subject that had a symbol available for the whole turn and did not use it.

The digits

τ is irrational and transcendental, so the expansion neither terminates nor repeats, and no digit of it can be worked out from the digits before it. It is also of no practical use: thirty-seven places give the circumference of the observable universe to within the width of a hydrogen atom, and everything past that is kept for pleasure.^[6]

It is printed here to a thousand places on account of what stands at the seven hundred and sixty-first.

Beginning at the 761st place, the expansion gives seven nines together. It is the longest run of any repeated digit in the first thousand places, and it is near enough the front that a determined person could reach it. Nobody did until September 1947, when D. F. Ferguson passed it at the 808th place with a desk calculator, in the last computation of these digits anybody made by hand.^[7]

π has a shorter version of the same thing and has made a great deal of it: six nines, beginning one place later, at the 762nd. This is the Feynman point, so called from the story that Feynman said he should like to learn π that far, so as to finish "nine nine nine nine nine nine, and so on" and leave the room to suppose the rest was more of the same. The story is not in his memoirs and was not known to his biographer; the earliest telling of it is Douglas Hofstadter's, in a book of 1985, where he is describing an ambition of his own.^[8]

The most celebrated feature of π's expansion is therefore a run one digit shorter, one place later, named after a man who is not recorded as having mentioned it, for a wish first set down by somebody else.

The one objection

It is that π is entrenched, and this is true and is not a small thing. Every textbook, every table, every piece of software and every person now living who was taught mathematics uses π, and a notation's value is largely in everyone using the same one. A change would cost more than it saved, and nobody serious proposes forcing one. Working the cost and then declining is what distinguishes a proposal from an assertion; for one advanced without any costing at all, see one times one equals two.

The encyclopedia's position is accordingly settled. τ is the constant; π is the habit; and the reason we have the habit is that a Welsh mathematician in 1706 picked a letter for the perimeter and measured it against the diameter.^[9] The subject has been paying the two ever since, in small instalments, cheerfully.

It is worth being exact about what is defended when π is defended. Not a result, not a theorem, not one line of mathematics: a letter, chosen once, set against a length that appears nowhere else in the subject, and not looked at again for two hundred and ninety-five years.

Tau Day

Tau Day falls on 28 June, from the American rendering 6/28, and is observed with twice the pastry of the rival festival on 14 March. Hartl proposed both the constant and the day in the same document, which is more than most reformers manage.^[2]

The encyclopedia keeps it, and regards the observance on 14 March as a pleasant enough occasion for people who have not yet had the matter explained to them.

See also

• Pi – one half of this article's subject, in general use
• Radian – the angle whose arc equals its radius; there are τ of them in a turn, which is the point
• Value of one radian – where the count of radians is mistaken for a distance
• The doctrinal value of pi – a different complaint about the same constant, and a worse one
• Hairy ball theorem – another subject the literature named badly and then kept
• Circle – the figure this is the circumference of, measured against the length it was drawn with

References

1. ^ B. Palais, "π is wrong!", The Mathematical Intelligencer 23 (3), 7–8, 2001. Palais proposed a three-legged symbol rather than τ, and the argument rather than the letter is what survived.
2. ^ M. Hartl, The Tau Manifesto, published on 28 June 2010, which fixed both the letter and the date. τ was chosen partly because it resembles π with one leg removed, which is the correct relation between them in more than one sense.
3. ^ The complaint is pedagogical rather than mathematical and is the strongest form of the case. Nothing is easier to prove in τ; a good deal is easier to teach.
4. ^ The sphere is untidy in both and no honest account claims otherwise; see pi, where the objection is set out.
5. ^ Both are Euler's formula, e^iθ = cos θ + i sin θ, evaluated at different angles, and neither follows from the other by substitution. Euler's identity proper is the π form; the τ form is a different instance of the same formula and the tidier one. The five-constants argument additionally requires the +1 and the 0 to be written out, which is a presentational choice rather than a property of the mathematics.
6. ^ Thirty-eight significant figures, so thirty-seven places after the point. The observable universe has a comoving radius of about 4.4 × 10²⁶ m and so a circumference of about 2.8 × 10²⁷ m; a hydrogen atom is about 10⁻¹⁰ m across; the ratio is near 3 × 10³⁷. The remaining places have been computed to many trillions and are wanted by nobody.
7. ^ William Shanks reached 707 places by hand in 1873 and was correct to 527 of them, so the nines were twice out of his reach: fifty-four places beyond where he stopped, and two hundred and thirty-four beyond where he last had it right. Ferguson found that error and passed the run in September 1947; the ENIAC gave 2,037 places two years later, in seventy hours, and no one has computed them by hand since.
8. ^ The run at π's 762nd place was named for Feynman long after the fact, and the naming is the whole of his connection to it: it is absent from his memoirs and his biographer James Gleick did not know of it. The earliest known account is Douglas Hofstadter's, in Metamagical Themas of 1985, and it is not about Feynman at all: Hofstadter is describing a schoolboy ambition of his own, to learn the expansion as far as the six nines. A constant's best-known digits are thus named for a man who did not mention them, after a wish belonging to somebody who did, which is Stigler's law of eponymy operating on a decimal expansion.
9. ^ See pi. William Jones, 1706, from περιφέρεια. He is not otherwise blamed for anything in this encyclopedia and did a good deal of useful work.

External links

• B. Palais, "π is wrong!" (2001) – the paper the argument descends from
• The Tau Manifesto – Michael Hartl, 2010, which named the constant and the day