I was recently looking over some Islamic geometry patterns, and was struck by the complexity of the constructions needed to create seemingly simple patterns. This got me wondering regarding optimal compass and straightedge constructions. To better define the question:
One is given some initial collection of constructed points, and a new (constructable) point to be constructed.
- Are there known methods for finding the optimal construction (minimum needed number of arcs and lines)?
- If there isn't, are there any methods for reducing the search space, e.g. for computer assisted searches?
- Is there a known method for checking if a given solution is optimal?