December 3rd, 2016, 04:17 PM  #1 
Senior Member Joined: Nov 2013 Posts: 246 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? 

Tags 
apollonian, apollonian gasket, area, formula, gasket 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Formula to calculate area  DBR  Geometry  5  October 22nd, 2015 02:37 PM 
How to prove formula area of annulus  Happy  Calculus  3  December 15th, 2014 03:06 AM 
Prove the area formula of a circle?  wuzhe  Calculus  4  October 23rd, 2012 05:50 PM 
Problem defining an area formula  Geldon  Algebra  0  October 12th, 2010 09:31 AM 
Differentiation of area of polygon formula  Delos  Calculus  2  August 18th, 2009 12:53 PM 