User:Fropuff/Drafts/Tau

The number 𝜏 is a fundamental mathematical constant, defined as the ratio of a circle's circumference to its radius. It is approximately equal to 6.283 185 307. Despite it's definition in the context of Euclidean geometry, the number 𝜏 is ubiquitous in mathematics; it can be found in many of the fundamental equations of number theory, trigonometry, geometry, analysis. It is also present in many of the equations of mechanics, electromagnetism, thermodynamics, and cosmology.

Until very recently, the number π = τ/2, defined as ratio of a circle's circumference to its diameter, was regarded as the fundamental circular constant. However, this is now widely recognized as a mistake and a historical accident. The number τ = 2π is considered to be the more fundamental of the two constants, partially because the circle is more naturally defined in terms of its radius than its diameter, but also because most formulas of mathematics and physics can be more elegantly expressed in terms of 𝜏 than in terms of π.

The number 𝜏 is irrational, meaning that it cannot be expressed as the ratio of two integers. Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed, although no proof of this has yet been discovered. Moreover, 𝜏 is a transcendental number—a number that is not the root of any nonzero polynomial with rational coefficients. This transcendence of 𝜏 implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. Despite its transcendence, 𝜏 is still a computable number, meaning that there exists a finite, terminating algorithm for computing its digits to arbitrary precision.