Some time ago I began playing around with packing circles and I have some questions that I am hoping someone here can help with. I have linked to three PDF files that should help in understanding my synopsis below. (You will need to click on the blank sheet and then open the PDF’s as I am not sure how to link to them directly – sorry for the inconvenience.) http://cid-64f8d3658ef4ddc3.skydrive.live.com/self.aspx/Public/The%20Whole%20Picture.pdf http://cid-64f8d3658ef4ddc3.skydrive.live.com/self.aspx/Public/The%20Center.pdf http://cid-64f8d3658ef4ddc3.skydrive.live.com/self.aspx/Public/The%20Pattern.pdf In “The Whole Picture” the background is a lattice of hexagonically packed unit circles. The “center” unit circle is arbitrarily selected. From the center circle, circumferential circles are drawn, each 2 times the unit circle diameter greater than the next. By constructing circumferential circles is such a manner, there are a minimum of six unit circles in each successive hexagonal packing that are internally tangential to the circumferential circle associated with that packing. So, from the center circle (packing q=0) the first packing (q=1) contains six circles that are internally tangential to a circumferential circle that has a radius of 3 (unit circle has a radius of one). The second packing (q=2) contains an additional six circles that are internally tangential to a circumferential circle that has a radius of 5. In the “The Center” drawing linked above, you can see that these six unit circles at every packing level are at 60, 120,180, 240, 300, and 360 degrees. Since the centers of these circles form the vertices of the hexagon representative of the associated packing, I have elected not to identify them in any special way. Now, whenever the hexagonal packing level equals a prime number of the form 6n+1, there are 12 additional circles that are internally tangential to the associated concentric circles. In the drawings linked above, these are represented by green shading. Additionally, all multiples of that prime also include 12 additional internally tangential unit circles. These multiple occurrences are represented by magenta shading. For example, at q=7, there are twelve internally tangential circles, shaded green and at q=14, 21, 28… the unit circles are shaded magenta. “The Pattern” has the background of unit and concentric circles turned off. Notice that only when 6n+1 is prime are additional unit circles tangential to the associated concentric circle. So 7, 13, 19, 31… have additional circles but not 25, 49 (except as a multiple of 7), 55, 85… Also, when the packing level equals the multiple of two non identical primes of the form 6n+1, there are 36 additional unit circles that are internally tangential to the associated concentric circle. These are represented in the drawing by cyan shading and are shown at 91 (7*13) and 133 (7*19). The drawings referenced are taken out to only 144 packing level. All that to get to my questions… :) 1) Why do the internally tangential unit circles only appear at primes (and multiples thereof) of the form 6n+1 and not primes of the form 6n-1? 2) Why do the internally tangential unit circles only appear at primes (and multiples thereof) of the form 6n+1 and not all packing levels 6n+1? 3) If you imagine the center of each unit circle of the background as forming a grid, is there a way to determine the angle of the internally tangential circles associated with a packing level, based on whole number steps of that grid? Mahalo!