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The following is a list of the polyominoes that have been shown to be disks.
I 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.
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