Math Embroidery: Archimedes Find Pi

Three circles w/ polygons
Six strand linen and cotton floss on 20 ct. linen

Close up of purple Archimedes transcribed circle
six strand linen and cotton floss on 20 ct. linen

Archimedes, an ancient Greek mathematician, developed a method for finding Pi (necessary if you want to find the area of a circle or cone) that can be used with n-sided polygons. These polygons are inscribed (inside the circle) and circumscribed (outside the circle).  N just represents the biggest number you want to use; Archimedes used a 96-sided polygon and the formula for the area of a right triangle to came up with a value for pi as approximately 22/7.

I didn't attempt a 96 sided polygon because I am sure that on 20 count linen it would look like 3 circles.  Instead, I used 5, 6 and 8-sided shapes.  There is a bit of black work in each polygon.  The yellow/orange figure uses darning stitches but in black work design.  Typically, darning is done with thread the same color as the background fabric because you wouldn't want to see the area of patched fabric.  But darning stitches can be used in black work designs in contrasting colors.