Circling the Square: Cwmbwrla, Coronavirus and Community
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Circling the Square: Cwmbwrla, Coronavirus and Community
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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.
There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle.
This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental.
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. Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations 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. Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. 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.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. There exist in the hyperbolic plane ( countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. 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.
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. 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. If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number. One of many early historical approximate compassandstraightedge 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.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. 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 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. The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.
Bending the rules by introducing a supplemental tool, allowing an infinite number of compassandstraightedge operations or by performing the operations in certain nonEuclidean geometries makes squaring the circle possible in some sense. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. Over 1000 years later, the Old Testament Books of Kings used the simpler approximation π ≈ 3 {\displaystyle \pi \approx 3} . 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 circlesquaring constructions, largely by amateurs, and by the debunking of these efforts.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. 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. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the method of exhaustion).
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