Here's a reference I can't consult right away, which seems at least partly germane: Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008. On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.