# Constructing a series of tangent spheres

So I stumbled across this problem a while ago, and have been unable to make any progress with it. Exhaustive searching has found no answers elsewhere.

With four solid balls $A$, $B$, $C$, and $D_1$, all of which are tangent to each other, you construct ball $D_2$ to be tangent to all four of them. Assume that $D_1$ has radius 1. After constructing $D_2$, you create ball $D_3$ tangent to $A$, $B$, $C$, and $D_2$. What is the radius of $D_{2017}$?

Approach

I started by just examining the two dimensional case. In this instance, we would ignore sphere $C$, because you can only have up to 4 mutually tangent spheres in a plane. Remembering that curvature is 1/radius, we So, we let curvatures of $D_1$ equal 1, and call the other two radii a and b respectively. Then, by Descartes's theorem, the curvature _$k_2$ of $D_2$ is defined by $$r_2 = 1+a+b\pm2\sqrt{a+b+ab}$$ So then, the next step would be to iterate this process 2016 times to get $k_2017$. This is where I'm stuck. I can't seem to manipulate the equation into a form that doesn't immediately turn ugly.

After this, the approach for spheres would probably be similar, using the generalization $$\left(\sum _{{i=1}}^{{5}}k_{i}\right)^{2}=3\,\sum _{{i=1}}^{{5}}k_{i}^{2}$$ of Descartes's theorem as applied to spheres. Again, though, I don't see how to manipulate this into a useful form.

• What are the radii of $A,B,C?$ are they 1 as well? – Doug M Jun 28 '17 at 20:24

Project every point on circles $A,B,C$ based on the inverse of the distance from a point on $A$ (marked with the fuzzy dot) i.e. if some point on $B$ is $d$ units from the fuzzy dot, project along the same ray to the point $\frac {1}{d}$ units away.