Circling the Square: Cwmbwrla, Coronavirus and Community

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If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number.

It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area.The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains regular quadrilaterals, shapes with four equal sides and four equal angles sharper than right angles.

The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to π {\displaystyle \pi } . It takes only elementary geometry to convert any given rational approximation of π {\displaystyle \pi } into a corresponding compass and straightedge construction, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of e {\displaystyle e} , to show that π {\displaystyle \pi } is transcendental and therefore that squaring the circle is impossible. In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } . In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients.

There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Contemporaneously with Antiphon, Bryson of Heraclea argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern intermediate value theorem. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number. Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle.

One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit Adam Adamandy Kochański, producing an approximation diverging from π {\displaystyle \pi } in the 5th decimal place. Over 1000 years later, the Old Testament Books of Kings used the simpler approximation π ≈ 3 {\displaystyle \pi \approx 3} . Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple. This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental.