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 } . Therefore, more powerful methods than compass and straightedge constructions, such as neusis construction or mathematical paper folding, can be used to construct solutions to these problems. 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 problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics.

Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries makes squaring the circle possible in some sense.To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. 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. However, they have a different character than squaring the circle, in that their solution involves the root of a cubic equation, rather than being transcendental.

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. Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π {\displaystyle \pi } .After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision. 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. Over 1000 years later, the Old Testament Books of Kings used the simpler approximation π ≈ 3 {\displaystyle \pi \approx 3} . 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. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area.

Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to π {\displaystyle \pi } . 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.

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.

In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. 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. If the areas of the four blue shapes labelled A, B, C and D are one unit each, what is the combined area of all the blue shapes?Squaring the circle: the areas of this square and this circle are both equal to π {\displaystyle \pi } .