The Last Archimedean

Geometry Level 1

Ludolph van Ceulen used Archimedean polygons to calculate the value of pi out to 35 decimal places.

True or false: Using this approximation for pi and an exact equatorial diameter for Earth, the calculation of the length of Earth's equatorial circumference would be off by less than the diameter of a proton.

False True

Denton Young by Denton Young , 47, USA

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2 solutions

Michael Mendrin
Aug 7, 2015

Archimedes originally used an hexagon, doubled four times to get a 96-gon, in which to compute the approximation 22/7. Ludolph van Ceulen started with a square, and doubled it 60 times to get a polygon with a number of sides 19 digits long. See this short paper on this subject.

Updating Archimedes

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Denton Young
Aug 7, 2015

The diameter of a proton is about 1 0 − 15 10^{-15} meters.

The margin of error introduced is 1 0 − 35 10^{-35} * the diameter of the Earth in meters. This diameter is about 1.3 ∗ 1 0 7 1.3 * 10^{7} m, so the margin of error is about 1.3 × 1 0 − 28 1.3 \times 10^{-28} m, far less than the diameter of a proton.

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