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Circling the Square: Cwmbwrla, Coronavirus and Community

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In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. For if a parallelogram is found equal to any rectilinear figure, it is worthy of investigation whether one can prove that rectilinear figures are equal to figures bound by circular arcs. It had been known for decades that the construction would be impossible if π {\displaystyle \pi } were transcendental, but that fact was not proven until 1882.

Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. The more general goal of carrying out all geometric constructions using only a compass and straightedge has often been attributed to Oenopides, but the evidence for this is circumstantial. Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple.

Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. 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 the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms.

The problem of finding the area under an arbitrary curve, now known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. If π {\displaystyle {\sqrt {\pi }}} were a constructible number, it would follow from standard compass and straightedge constructions that π {\displaystyle \pi } would also be constructible. 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. Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared.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.

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