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December 5th, 2016, 09:33 AM   #1
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Question Radius unknown

If I have an Apollonian gasket like this one:

Then the radius of the centered circle and the circles right next to that centered circle is easy. radius of centered circle would be a given and since $r + d = R$, once I know the 2 radii in this equation, I can easily divide the diameter that is left by 2 and get the radius. That is all well and good but what about the next layer? Solving for the radius here is much more complicated. Here is the non-trigonometric way of solving for the next radius in either direction:

Make a triangle with sides r+a, r+a, and d

Then set the perimeter equal to x like this:

$$(r+a)+(r+a)+d = x$$

Now we want to find a, the unknown radius.

This is where we need to plug in our values.

So lets set r=2 and d=4

We then get this:

$$(2+a)+(2+a)+4 = x$$

which we can simplify to:

$$8+2a = x$$


$$2a = x-8$$


$$a = \frac{1}{2}x-4$$

Now we can substitute this into our equation with the plugged in values so we get

$$(2+\frac{1}{2}x-4) + (2+\frac{1}{2}x-4) + 4 = x$$

and then solve for x

But we got a problem here. We get x=x which means a is undefined. Uh oh. We need a defined perimeter when we solve the equation so that we can plug it in and find the radius we are looking for. So figuring this out with algebra alone is tough. But it gets even tougher. In the next layer in any direction you get 2 circles of the same radius touching 1 circle and infinitely many circles between those 2 circles. So you have an infinite sum of diameters + 2*unknown radius for 1 of the sides. For this we need to figure out $Θ$ and the arc length respectively and use trig.

So you get more and more unknowns as you get further and further into the Apollonian gasket.

Is there an easier way to find the unknown radius than doing what I just described?
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