Geometry on the Beach: Circling the Square

by Jane Morris Pack

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. –G.H. Hardy, A Mathematician’s Apology

Geometry, from the Greek γεωμετρία or “earth measurement,” was used by the Egyptians to reestablish the borders of farmed ground after the Nile had flooded and erased the previous year’s boundary markers.  This marking of the earth developed into a complex system of mathematics which was understood and amplified by the Greeks;  Pythagoras, Euclid and Archimedes are names we associate with geometry. In the Middle Ages mathematicians in medieval Islam contributed most of the further developments until once again the Europeans led the way in the 17th century.

“Squaring the circle” is an attempt to create a square with an area equal to a circle using only a straight edge and a compass.  Ancient geometers worked on this problem and it was only proven impossible in 1882.  Nonetheless the symbolism involved remains potent: that of combining the heavenly circle with the square of earth.  Leonardo explored this image in the famous Vitruvian man drawing which we were privileged to see in Venice this last September.

Steven Kosovac and I gathered with students on a blustery afternoon to draw out our attempt at a related conundrum.  With only pegs and a rope we followed the process from a book and arrived at a large and perfect example of circling the square. The circle we constructed had the same perimeter as a square.