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December 3rd, 2016, 04:17 PM   #1
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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:


$$πR^2 - (πR^2 - (πR^2)) = πR^2$$


$$πR^2 - (πR^2 - (πr_{1}^{2}))$$


$$πR^2 - (πR^2 - (πr_{1}^2 + 8*πr_{2}^2))$$


$$πR^2 - (πR^2 - (πr_{1}^2 + 8*πr_{2}^2 + 8*πr_{3}^2))$$


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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