Circles and Squares: The Lives and Art of the Hampstead Modernists

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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.

Circles and Squares: The Lives and Art of the Hampstead Circles and Squares: The Lives and Art of the Hampstead

The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. 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. 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.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. 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. This increases tension between the circles and squares, creating a cycle of anger and hatred that quickly spirals towards a violent climax.

Circles and Squares by Caroline Maclean review - The Guardian Circles and Squares by Caroline Maclean review - The Guardian

Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. 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).

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 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. 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. If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number.