Circling the Square: Cwmbwrla, Coronavirus and Community

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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 } . The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. It was not until 1882 that Ferdinand von Lindemann succeeded in proving more strongly that π is a transcendental number, and by doing so also proved the impossibility of squaring the circle with compass and straightedge.

In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } . 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. Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.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.

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. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of π {\displaystyle \pi } and so showed the impossibility of this construction. 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. color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} .This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental. 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.

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. If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area.displaystyle \left(9 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. 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. 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.

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. 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. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps.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.