Apollonian Gasket - Construction

Construction

An Apollonian gasket can be constructed as follows. Start with three circles C₁, C₂ and C₃, each one of which is tangent to the other two (in the general construction, these three circles can be any size, as long as they have common tangents). Apollonius discovered that there are two other non-intersecting circles, C₄ and C₅, which have the property that they are tangent to all three of the original circles – these are called Apollonian circles (see Descartes' theorem). Adding the two Apollonian circles to the original three, we now have five circles.

Take one of the two Apollonian circles – say C₄. It is tangent to C₁ and C₂, so the triplet of circles C₄, C₁ and C₂ has its own two Apollonian circles. We already know one of these – it is C₃ – but the other is a new circle C₆.

In a similar way we can construct another new circle C₇ that is tangent to C₄, C₂ and C₃, and another circle C₈ from C₁, C₃ and C₁. This gives us 3 new circles. We can construct another three new circles from C₅, giving six new circles altogether. Together with the circles C₁ to C₅, this gives a total of 11 circles.

Continuing the construction stage by stage in this way, we can add 2·3n new circles at stage n, giving a total of 3n+1 + 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket.

The Apollonian gasket has a Hausdorff dimension of about 1.3057.