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Circling the Square: Cwmbwrla, Coronavirus and Community

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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.

Over 1000 years later, the Old Testament Books of Kings used the simpler approximation π ≈ 3 {\displaystyle \pi \approx 3} . In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve retains the idea of using only restricted geometric methods. The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } . As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared.

James Gregory attempted a proof of the impossibility of squaring the circle in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667.

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.

It was not until 1882 that Ferdinand von Lindemann proved the transcendence of π {\displaystyle \pi } and so showed the impossibility of this construction. The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number π {\displaystyle {\sqrt {\pi }}} , the length of the side of a square whose area equals that of a unit circle. 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. 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 } . After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts.Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. 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.

Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. For example, Dinostratus' theorem uses the quadratrix of Hippias to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it.In Chinese mathematics, in the third century CE, Liu Hui found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century Zu Chongzhi found π ≈ 355 / 113 ≈ 3.

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