I am thinking of all of these polyominoes as sets of lattice-points, that is, points with integer coordinates. It is these points which I wanted to cover with a disk in the earlier disklike case, and it is these points whose convex hull I am taking, and insisting that no additional lattice points are thereby encompassed. The hexagon case already has a traditional name, *polyhexes*. On Sat, May 7, 2011 at 1:38 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Given a closed & bounded convex figure K:
Does "intersection between any convex figure and the lattice" mean the union of grid squares that intersect K ?
Or the union of grid squares that are each entirely contained in K ?
(Or maybe both definitions yield the same set of polyominoes?)
* * *
Allan's definitions of disklike polyomino, and the one below, suggest another one:
* A polyomino that is topologically a closed disk.
(E.g., this would exclude the ringlike heptomino O_7.)
This would include all n-ominoes through n = 6, but then things start to get interesting.
(A perhaps nicer variant would be to require only that the *open* polyomino be a topological disk. This would rescue O_7.)
* * *
I like hexagonal polyominoes -- say, "heximoes" -- since no two cells can overlap on just a vertex, thus avoiding borderline cases like H_7.
Here the topological-disk ones exclude n-heximoes only for n >= 6.
QUESTION: Is there a simple asymptotic expression H(n) for the number of topological-disk n-heximoes?
(In 3-space one could likewise ask for topological-3-disks tesssellated by truncated octahedral tiles, and an asymptotic formula for their counts, etc.)
--Dan
--------------------------------------------------------------- Allan wrote:
<< On a previous occasion, I talked about enumerating "disklike" polyominoes, polyominoes which occurred as the intersection between a disk and a square lattice. Today I tried counting polyominoes which occurred as the intersection between any convex figure and the lattice. This implies that the convex hull of the polyomino (viewed as a set of lattice points) includes no additional lattice points.
Sometimes the brain has a mind of its own.
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