What is τ (Tau)?
τ (tau) equals 2π ≈ 6.28318. Its defining property is simple: one full revolution of a circle is exactly τ radians. Half a turn is τ/2 = π radians. A quarter turn is τ/4. For those who find this more natural than π, the circle constant is τ, not π.
One full revolution = τ radians. τ/4 = 90°. τ/2 = 180° = π radians. The circumference of a circle is C = τr.
The case for τ: the circumference formula becomes C = τr (circumference = tau × radius), and any fraction of a turn is that fraction times τ. sin(τ) = 0, cos(τ) = 1 (returning to start). Euler's identity in terms of τ: e^(iτ) = 1, a full rotation. The case against: π is established in every textbook and formula for centuries.
| Formula | with π | with τ |
|---|---|---|
| Circumference | 2πr | τr |
| Area of circle | πr² | τr²/2 |
| Full turn | 2π rad | τ rad |
| Euler identity | eⁱπ+1=0 | eⁱτ=1 |
| Gaussian integral | √(2π) | √τ |
τ = 2π is transcendental (since π is transcendental). Whether it is the better circle constant is a matter of taste, not mathematics. The Tau Manifesto (Michael Hartl, 2010) makes the pedagogical argument. τ to 20 digits: 6.28318530717958647692…
With π, a quarter turn is π/2: half of the full-turn constant. With τ, a quarter turn is τ/4: literally one quarter. Every fraction of a turn maps directly to the same fraction of τ.
Tau is exactly 2 times pi, approximately 6.28318530717958647692. It is irrational and transcendental. One tau radian equals one full circle, making it arguably more natural than pi as the circle constant. Proposed by Bob Palais in 2001 and popularised by Michael Hartl's Tau Manifesto. Tau Day is June 28 (6.28). Euler's identity with tau reads e^(iτ) = 1: a full rotation of the complex plane returns to the start.
Tau τ is irrational. Its decimal expansion never ends and never repeats. The digits shown below are verified against the circle definition.
FAQ
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Who proposed \u03C4 (tau)?
Mathematician Bob Palais first argued for using 2\u03C0 as a fundamental circle constant in his 2001 article Pi Is Wrong. Michael Hartl formalized the proposal as The Tau Manifesto, published on June 28, 2010, chosen because 6.28 matches \u03C4's leading digits. -
Why June 28 for Tau Day?
\u03C4 is approximately 6.28318, so the date 6/28 (June 28) mirrors these digits, just as Pi Day is March 14 (3/14) for \u03C0. Tau Day was first celebrated on June 28, 2010, the day The Tau Manifesto was published. -
Is \u03C4 mathematically different from \u03C0?
No. \u03C4 = 2\u03C0 exactly - it is the same constant scaled by 2. The debate is purely about notation: which constant makes formulas cleaner? For angles and oscillations, \u03C4 wins; for area formulas, \u03C0 is slightly simpler. -
Has \u03C4 been adopted anywhere officially?
Python's math module has included math.tau since version 3.6 (2016). Several universities hold Tau Day events. The BBC and other science communicators have covered the debate. Mainstream mathematical convention still uses \u03C0. -
Does \u03C4 appear in serious physics?
Yes. The reduced Planck constant h-bar = h/\u03C4 is ubiquitous in quantum mechanics. Angular frequency omega = \u03C4f appears in every wave equation. The Fourier transform naturally involves 2\u03C0 (i.e., \u03C4) throughout.
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