Thanks Fred. To "get" the 289-omino, it helps to think of the underlying grid as a square grid, and each # as being instead a square that fills the square space available to it. On my Mac I had to copy Fred's depiction to a *text* file, so that it got rid of the vertical line that Mac Mail uses for quoting, and so that each line does not wrap around. —Dan P.S. But what Fraid sed: How did Joerg come to be looking for such a weird tile in the first place? And for that matter: What is known about such tiles in general? Can we assume that the only ways that it tiles the plane are periodic? I seem to recall that a non-periodic connected monotile is knot nown. What about non-periodic monotiles that are *not* connected?
On Nov 7, 2016, at 9:03 AM, James Propp <jamespropp@gmail.com> wrote:
I still don't get it. Can someone make a non-ASCII version?
Thanks,
Jim
On Mon, Nov 7, 2016 at 10:08 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
OK, I get it now. Though what continues to baffle me is how you came to be looking for such an object in the first place!
The version below makes the symmetry more apparent when viewed without proportional spacing (view source, or drag/drop to text file).
WFL
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On 11/6/16, Joerg Arndt <arndt@jjj.de> wrote:
The arrangement in the square grid shown below has four-fold rotational symmetry and none of its squares (apart from the central one) has more than two neighbors.
Of course there are very many such arrangements. This one, however, has a very special property, unique within all such(*) arrangements of 289 squares.
(*) edge-to-edge connected, four fold symmetry, and fewest possible number of neighbors.
Which?
................................................... ................................................... ................................................... ................................................... ..........................#####.................... ..........................#...#.................... ..........................#...#.................... ..........................#...#.................... ..........................#...#.................... ..........................#...#.................... ..........................#...#.................... ..................#########...#.................... ..................#...........########............. ............###...#..................#............. ............#.#...#..........#########............. ............#.#...#..........#..................... ............#.#...#..........#..................... ............#.#...########...#..................... ............#.#..........#...#...#######........... ............#.#..........#...#...#.....#........... ....#########.#..........#...#...#.....#........... ....#.........########...#...#...#.....#........... ....#....................#.......#.....#........... ....#....................#.......#.....#........... ....########.............#.......#.....#........... ...........#.....#################.....#........... ...........#.....#.......#.............########.... ...........#.....#.......#....................#.... ...........#.....#.......#....................#.... ...........#.....#...#...#...########.........#.... ...........#.....#...#...#..........#.#########.... ...........#.....#...#...#..........#.#............ ...........#######...#...#..........#.#............ .....................#...########...#.#............ .....................#..........#...#.#............ .....................#..........#...#.#............ .............#########..........#...#.#............ .............#..................#...###............ .............########...........#.................. ....................#...#########.................. ....................#...#.......................... ....................#...#.......................... ....................#...#.......................... ....................#...#.......................... ....................#...#.......................... ....................#...#.......................... ....................#####.......................... ................................................... ................................................... ................................................... ...................................................
Best regards, jj
Hints below, after some several blank lines.
Hints:
A) 289 = 15^2 + 8^2
B) Consider two copies of the arrangement
C) Consider 289 copies of the arrangement
P.S.: almost-spoiler at http://jjj.de/visual-riddle/visual-riddle.png
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