A250001: Circle Arrangements Research
My ongoing investigation into the sequence A250001: "Number of arrangements of n circles in the affine plane."
Approaches
A catalog of distinct methodological approaches I've attempted, their status, and key insights.
- Active Defining relations between arrangements and looking for patterns
- Promising Seeking a characteristic polynomial signature of an arrangement
- Promising Enumeration of well-defined subsets
- Abandoned Brute force enumeration up to some tolerance to get a lower bound
Connections
Related sequences, papers, and mathematical concepts.
- A000081: Planar arrangements of circles, none of which intersect
- A000124: The maximum number of regions produced by N lines in general position
Questions
Specific sub-problems, open questions, and conjectures.
- Only allow N circles to intersect.
- Restrict the centers of the circle to all lay on the same line, or to lay on one of N parallel lines.
- Consider only circles of the same radius.
Insights
Key realizations and pattern observations worth revisiting.
- Containment relations are crucial to this problem. This includes both the simple case of full-circle containment, but also containment of individual intersection points. You also have to pay attention to the case where a union of disks contains a given circle, but neither individual disk from the union contains it fully. All of these different circumstances give rise to distinct arrangements.
- You can arrive at a related figure composed of lines if you draw lines through each pair of circle centers.