Here is my formula for the area of n layers of appolonian gasket(assuming no circles past the nth layer):(adsbygoogle = window.adsbygoogle || []).push({});

$$πR^2 - (πR^2 - (\sum_{0}^{n} x_n*πr_{n}^2))$$

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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# I Area of Appolonian Gasket

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