Circling the Square: Cwmbwrla, Coronavirus and Community
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Circling the Square: Cwmbwrla, Coronavirus and Community
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This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental. 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.
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. 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. As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps. 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 } .
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. 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. Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple. 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? 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.
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. 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. 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. 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).
displaystyle \left(9 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. 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. 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. In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } .
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