My Math Forum Apollonian gasket area formula

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 December 3rd, 2016, 05:17 PM #1 Senior Member   Joined: Nov 2013 Posts: 239 Thanks: 2 Apollonian gasket area formula Here is my formula for the area of n layers of appolonian gasket(assuming no circles past the nth layer): $$πR^2 - (πR^2 - (\sum_{0}^{n} x_n*πr_{n}^2))$$ Here R is the radius of the outer circle, r is the radius of an inner circle, x is a function that represents the number of circles in a given layer and n is the number of layers. I know this is right as far as calculating area is concerned but how would I actually represent this if I wanted to show someone else this formula? The reason I only have $πr_{n}^2$ once is because here is what the sum would be like for a successive number of layers. If I assume I have this kind of Apollonian gasket: then the area formula is like this as n increases: n=0 $$πR^2 - (πR^2 - (πR^2)) = πR^2$$ n=1 $$πR^2 - (πR^2 - (πr_{1}^{2}))$$ n=2 $$πR^2 - (πR^2 - (πr_{1}^2 + 8*πr_{2}^2))$$ n=3 $$πR^2 - (πR^2 - (πr_{1}^2 + 8*πr_{2}^2 + 8*πr_{3}^2))$$ etc. But I could easily replace each of those multipliers with $x_1$, $x_2$, $x_3$ etc. So basically every time n increases by 1 is a time when the radius changes in an Apollonian gasket as you get more and more circles inside that 1 outer circle. Would the general formula for any Apollonian gasket I have at the top of this post be the best way to represent this area formula?

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