I want to pack x number of pipes into a circle in two different formations; firstly in square formation, secondly in triangular formation.
It seemed obvious to reduce this to a packing problem, i.e. maximizing the amount of shapes that can be fitted onto a circle. With a "regular" formation, i.e. no vertices that coincident between two vertices on an edge or shapes that are rotated in any way:
For the first case, there's analytical estimations used for maximizing the amount of die on a silicon wafer, e.g.: $$DPW=\frac{\pi d^2}{4 S}-\frac{\pi d}{\sqrt{2 S}}$$
Is there a way of accurately approximating the total amount of vertices (my pipes) to the total number of squares? Secondly, can the same be done for a triangular pipe formation?
I've tried looking finding algorithms for MATLAB, but I was not able to find anything useful.