Squaring The Circle - Modern Approximative Constructions

Modern Approximative Constructions

Though squaring the circle is an impossible problem using only compass and straightedge, approximations to squaring the circle can be given by constructing lengths close to pi. It takes only minimal knowledge of elementary geometry to convert any given rational approximation of pi into a corresponding compass-and-straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly and informally as constructions that are particularly simple among other imaginable constructions that give similar precision.

Among the modern approximate constructions was one by E. W. Hobson in 1913. This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is accurate to 4 decimals (i.e. it differs from pi by about 4.8×10−5).

Indian mathematician Srinivasa Ramanujan in 1913, C. D. Olds in 1963, Martin Gardner in 1966, and Benjamin Bold in 1982 all gave geometric constructions for

which is accurate to six decimal places of pi.

Srinivasa Ramanujan in 1914 gave a ruler-and-compass construction which was equivalent to taking the approximate value for pi to be

giving a remarkable eight decimal places of pi.

In 1991, Robert Dixon gave constructions for

(Kochański's approximation), though these were only accurate to four decimal places of pi.