Here is my formula for the area of $n$ layers of appolonian gasket(assuming no circles past the $n$th layer):
$$πR^2 - \left(πR^2 - \left(\sum_0^n x_n\cdot\pi r_n^2\right)\right)$$
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:
$n=0$
$$πR^2 - (πR^2 - (πR^2)) = πR^2$$
$n=1$
$$πR^2 - (πR^2 - (πr_1^2))$$
$n=2$
$$πR^2 - (πR^2 - (πr_{1}^2 + 8 πr_2^2))$$
$n=3$
$$πR^2 - (πR^2 - (πr_{1}^2 + 8πr_2^2 + 8πr_3^2))$$
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?
I mean there is just 1 part of it telling me "Your wrong" and that is the way I formatted this sum:
$$\sum_0^n x_n πr_n^2$$
I mean x = some function is unusual but it isn't wrong. In fact it would be even more unusual to say something like this:
$$x = f(x)$$
This implies a line with a slope of 1 or as I sometimes call it, a 1:1 slope
or
$$X(x)$$
This is just odd having a function with the same letter as the variable
Another thing getting me about the sum is the sigma. It isn't normal for a sum to not have a variable equaling something but in this case I think it is required to not have that variable since I am summing from 0 to n and everything is in terms of n(multipliers, area of circle formulas).
