Apollonian gasket

In mathematics, an Apollonian gasket or Apollonian net is a fractal generated starting from a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga. ( k 1 + k 2 + k 3 + k 4 ) 2 = 2 ( k 1 2 + k 2 2 + k 3 2 + k 4 2 ) . {displaystyle (k_{1}+k_{2}+k_{3}+k_{4})^{2}=2,(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2}).}     (1)Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8)Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)Integral Apollonian circle packing defined by circle curvatures of (−10, 18, 23, 27) In mathematics, an Apollonian gasket or Apollonian net is a fractal generated starting from a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga. 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. 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 C4, 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 sizes of the new circles are determined by Descartes' theorem. Let ki (for i = 1, ..., 4) denote the curvatures of four mutually tangent circles. Then Descartes' Theorem states The Apollonian gasket has a Hausdorff dimension of about 1.3057.