Given two circles that touch at a point, fill the gap with a chain of touching circles. This is called a Pappus chain. In the image below, I show only two semicircles, and begin the Pappus chain with a circle touching the common diameter.
Now take a circle with center at the point where the two given circles touch, and perpendicular to one of the circles of the Pappus chain we pick out. The inversion at this new circle takes the two given circles to two vertical lines, and the Pappus chain to a chain of circles between these two lines. The picked circle remains fixed. Below the selected circle from the Pappus chain there are precisely as many circles as to the right of the selected circle in the Pappus chain (four in the figure). Thus the height of the selected circle is determined by its diameter and its position in the Pappus chain. That, of course, will only excite the mathematician.
The same construction works in three dimensions. Take an arrangement of spheres between two vertical half planes, and invert them at a half sphere as shown.
The result is an arrangement of spheres between two hemispheres that touch at a point (where the spheres get really small).
I thought this might be an interesting way to fill a dome. Standing in front of the entrance, with reflective spheres and reflective floor, might look like this:
The Inner Frame 