S. Louise Gould

A silver necklace with a sample of an Apollonian Gasket inspired me to explore these circular constructions. Apollonius of Perga was a student of Euclid’s students in Alexandria. Following the tradition of the ancient Greek geometers, he looked for order in plane geometry. He observed that anytime that you have three circles that are each tangent to the others (but not in the same point) there are usually two more circles tangent to each of the three circles. Using these new circles you can repeat this process adding 2x3n circles at each stage for a total of 3(n+1)+2 circles at each new nth iteration. These constructions became known as Apollonian Gaskets. Descartes observed a pattern in the curvatures $\left (\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4} \right )= $ $2 \left ( \frac{1}{{r_1}^2} + \frac{1}{{r_2}^2} + \frac{1}{{r_3}^2} + \frac{1}{{r_4}^2} \right ) $ (reciprocals of the radii) of the tangent circles that permits you to find the curvature of the tangent circles at each iteration. Historically geometers favored circles with integer curvatures and explored their symmetries. This garment features gaskets that demonstrate no symmetry other than the identity, bilateral symmetry, a set of nested gaskets, a gasket with 120 degree rotational symmetry which does not have integer curvature, and the limit case where two of the circles are represented by parallel lines, that is circles with no curvature.