The battle between τ and π

Introduction:

The history of circle constants is an important story on how mathematics evolved over time, specifically addressing the relationship between a circle's circumference and diameter. One of the most renowned constants in this context is Pi (π), a venerable figure with an enduring history that traces back to ancient civilizations.

The use of approximations for π dates back to the ancient Babylonians, who ingeniously approximated π as 25/8 (dec) = 3.125 (dec), while the ancient Egyptians ventured with a value of 256/81 (dec) = 3.1604938... (dec). The Greeks, most notably Archimedes, made significant strides in approximating π with remarkable precision. Archimedes calculated π to be between 223/71 (dec) ≅ 3.1408451... (dec) and 22/7 (dec) ≅ 3.1428571... (dec). Additionally, ancient Chinese mathematicians achieved accurate approximations of π, notably π ≅ 22/7 (dec) and π ≅ 355/113 (dec).

The journey of approximating π slowed down until Sir Isaac Newton's groundbreaking discovery of a rapid algorithm, utilizing calculus, to calculate the first digits of π accurately in various numerical bases.

In contrast, Tau (τ) emerged on the mathematical scene much later, with its first known usage attributed to the brilliant mathematician Leonhard Euler in the year 1746 (dec). Interestingly, Euler employed the letter π to represent this circle constant initially. It was not until the year 2001 (dec) that Robert Palais proposed the concept of measuring angles in radians per turn as the circle constant. He suggested utilizing the letter π, but with a distinct visual modification, incorporating three legs instead of the conventional two.

Finally, in the year 2010 (dec), Michael Hartl advocated for the adoption of the letter τ as the symbol for the circle constant. He justified this choice by emphasizing τ's resemblance to the letter π, the established symbol for another circle constant, and the fact that τ corresponded to the Latin letter 't.' Additionally, Hartl drew attention to the English language, where the word for 'turn' commences with the letter 't.' Much like the rationale behind selecting π, the selection of τ aimed to provide a symbol uniquely associated with this new circle constant.

Advantages and Disadvantages of Pi (π) and Tau (τ) in Mathematical Constants:

π is without a doubt a very important constant with historical legacy. For years, it has held a prominent place in mathematical tradition, defined as the ratio of a circle's circumference to its diameter (π = C/D). Pi has been the trusted constant for solving a wide array of geometric and trigonometric problems. However, it is not without its limitations, particularly when dealing with angles, where the use of radians is often preferred. π can lead to unwieldy expressions in equations, making calculations less intuitive. In contrast, τ offers a fresh perspective on circle constants. Defined as the ratio of a circle's circumference to its radius (τ = C/r), it simplifies many fundamental equations in mathematics. In angular measurements, τ corresponds directly to one full turn, enhancing its intuitiveness and elegance. When employed as the circle constant, τ streamlines formulas in trigonometry, calculus, and physics, yielding more straightforward and elegant results.

Examples of Equations using Pi (π) and Tau (τ):

1. The perimeter (P) of a circle: P = 2πr or P = τr, where 'r' is the radius.

2. The area (A) of a circle: A = π x r^2 or A = (1/2) x τr^2, with 'r' representing the radius.

3. A quarter of a turn: 1/2π or 1/4τ.

4. The area of a regular n-gon with circumradius 1: A = (1/2n) x sin(2π/n) or A = (1/2n) x sin(τ/n).

5. The standard normal distribution: φ(x) = (1/√(2π)) x e^(-x^2/2) or φ(x) = (1/√(τ)) x e^(-x^2/2).

10 (hex). Stirling's approximation of n!: n! ~ √(2πn) x n^n x e^(-n) or n! ~ √(τn) x n^n x e^(-n).

11 (hex). Euler's identity: e^(iπ) + 1 = 0 ⟺ e^(iπ) = -1 or e^(iτ) + 1 = 0 ⟺ e^(iτ) = 1.

12 (hex). The nth roots of unity: e^(2iπ/n) = cos(2π/n) + i sin(2π/n) or e^(iτ/n) = cos(τ/n) + i sin(τ/n).

13 (hex). Planck's constant: h = 2πh or h = τh, where 'h' is the reduced Planck's constant.

14 (hex). Angular frequency (ω): ω = 2πf or ω = τf, with 'f' representing the frequency.

In these examples, it's clear that π simplifies equations in certain situations, such as the area of a circle. However, it can obscure the underlying geometric relationships, as seen in the sector area formula, where τ provides greater clarity, seeing that the formula for the area of a sector of a circle with radius r and angle θ is 1/2θ x r^2. In most cases, τ simplifies equations and formulas, offering a more elegant and intuitive approach to mathematics. Therefore, when evaluating the advantages and disadvantages of π and τ as circle constants, τ emerges as the superior choice for its ability to streamline mathematical concepts and calculations, as it removes the many occurrences of a factor of 2 that is present in most equations of formulas involving π.

Conclusion:

τ is the best circle constant, since most of the mathematical and physical equations and formulas are simplified if using τ instead of any ither circle constant, plus τ is the fundamental period of sine and cosine, so using radians with τ is much more intuitive than using radians with π, since a fraction of a turn a/b is just represented as (a/b) x τ radians. Finally and most importantly τ is approximately six, so if you are using the heximal numbering system and you are on a situation where you don’t need a lot of precision, then you can round τ to 6 (dec), which makes calculations trivial. The dominance of π is over and the only non-controversial mathematical constant that is known is e, which is the Euler’s number, because changing e to 2e or e^2 makes no sense, since the properties of e are unique, while the properties of π are not.