Circling the Square: Cwmbwrla, Coronavirus and Community

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Circling the Square: Cwmbwrla, Coronavirus and Community

Circling the Square: Cwmbwrla, Coronavirus and Community

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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. 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. Squaring the circle: the areas of this square and this circle are both equal to π {\displaystyle \pi } .

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. Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to π {\displaystyle \pi } .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. If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number. 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.

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. 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.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. In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. 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.

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