muzik wrote:I don't know how this one works out:
Code: Select all
x = 38, y = 11, rule = //5
3.C$2.2CB3A12.B2.C2.B2.C2.B2.C$.3C2B3AD9.3B3C3B3C3B3C$2.CA3BA2DC9.3B
3C3B3C3B3C$2.3ABC3D2C8.3A3D3A3D3A3D$2.2AB3CDA3C6.3A3D3A3D3A3D$2.A3B2C
3AC8.A2.D2.A2.D2.A2.D$2.3BCDC2AB$.3D2C2DA3B$3D3C3D2B$.D2.C2.D.B!
Code: Select all
x = 52, y = 41, rule = //5
20.A$19.3AC$18.B2A3CA$17.2BAD2C3AC$16.3B2DCB2A3CA$16.AB3D2BAD2C3AC$
15.3ACD3B2DCB2A3CA$14.B2A3CAB3D2BAD2C3AC$13.2BAD2C3ACD3B2DCB2A3CA$12.
3B2DCB2A3CAB3D2BAD2C3AC$12.AB3D2BAD2C3ACD3B2DCB2A3CA$11.3ACD3B2DCB2A
3CAB3D2BAD2C3AC$10.B2A3CAB3D2BAD2C3ACD3B2DCB2A3CA$9.2BAD2C3ACD3B2DCB
2A3CAB3D2BAD2C3AC$8.3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3CA$8.AB3D2BAD2C
3ACD3B2DCB2A3CAB3D2BAD2C3AC$7.3ACD3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3C$
6.B2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C$5.2BAD2C3ACD3B2DCB2A3CAB3D
2BAD2C3ACD3B2DC$4.3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D$4.AB3D2BAD2C
3ACD3B2DCB2A3CAB3D2BAD2C3ACD$3.3ACD3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3C
$2.B2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C$.2BAD2C3ACD3B2DCB2A3CAB3D
2BAD2C3ACD3B2DC$3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D$.B3D2BAD2C3ACD
3B2DCB2A3CAB3D2BAD2C3ACD$3.D3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3C$5.B3D
2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C$7.D3B2DCB2A3CAB3D2BAD2C3ACD3B2DC$9.B3D
2BAD2C3ACD3B2DCB2A3CAB3D$11.D3B2DCB2A3CAB3D2BAD2C3ACD$13.B3D2BAD2C3AC
D3B2DCB2A3C$15.D3B2DCB2A3CAB3D2BAD2C$17.B3D2BAD2C3ACD3B2DC$19.D3B2DCB
2A3CAB3D$21.B3D2BAD2C3ACD$23.D3B2DCB2A3C$25.B3D2BAD2C$27.D3B2DC$29.B
3D$31.D!
EDIT: An octomino with what I believe to be not only a unique tiling, but also a unique (and also anisotropic) 3-coloring of such tiling:
Code: Select all
x = 117, y = 118, rule = //4
60.A2.A$59.6A$59.6C$60.CABC$56.C2.C2A2BA2.A$55.6CAB6A$55.6BAB6C$56.BC
AB2A2BCABC$52.B2.B2C2ACABC2A2BA2.A$51.6BCA6CAB6A$51.6ACA6BAB6C$52.ABC
A2C2ABCAB2A2BCABC$48.A2.A2B2CBCAB2C2ACABC2A2BA2.A$47.6ABC6BCA6CAB6A$
47.6CBC6ACA6BAB6C$48.CABC2B2CABCA2C2ABCAB2A2BCABC$44.C2.C2A2BABCA2B2C
BCAB2C2ACABC2A2BA2.A$43.6CAB6ABC6BCA6CAB6A$43.6BAB6CBC6ACA6BAB6C$44.B
CAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$40.B2.B2C2ACABC2A2BABCA2B2CBCAB2C
2ACABC2A2BA2.A$39.6BCA6CAB6ABC6BCA6CAB6A$39.6ACA6BAB6CBC6ACA6BAB6C$
40.ABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$36.A2.A2B2CBCAB2C2ACA
BC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$35.6ABC6BCA6CAB6ABC6BCA6CAB6A$35.
6CBC6ACA6BAB6CBC6ACA6BAB6C$36.CABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C
2ABCAB2A2BCABC$32.C2.C2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACAB
C2A2BA2.A$31.6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$31.6BAB6CBC6ACA6BAB6CBC6A
CA6BAB6C$32.BCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BC
ABC$28.B2.B2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A
2BA2.A$27.6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$27.6ACA6BAB6CBC6ACA6BAB
6CBC6ACA6BAB6C$28.ABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABC
A2C2ABCAB2A2BCABC$24.A2.A2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2B
ABCA2B2CBCAB2C2ACABC2A2BA2.A$23.6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB
6A$23.6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$24.CABC2B2CABCA2C2ABCAB
2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$20.C2.C2A2BA
BCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A
2BA2.A$19.6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$19.6BAB6CBC6ACA
6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$20.BCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCAB
C2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$16.B2.B2C2ACABC2A2B
ABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC
2A2BA2.A$15.6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$15.6ACA6BA
B6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$16.ABCA2C2ABCAB2A2BCABC2B2CAB
CA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$
12.A2.A2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACA
BC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$11.6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BC
A6CAB6ABC6BCA6CAB6A$11.6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6B
AB6C$12.CABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA
2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$8.C2.C2A2BABCA2B2CBCAB2C2ACA
BC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C
2ACABC2A2BA2.A$7.6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB
6A$7.6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$8.BCAB2A
2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCA
B2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$4.B2.B2C2ACABC2A2BABCA2B2CBCAB2C2AC
ABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C
2ACABC2A2B$3.6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB
$3.6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB$4.ABCA2C
2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABC
A2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2B$4.2B2CBCAB2C2ACABC2A2BABCA2B2CB
CAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B
2CBCAB2C2ACABC$5.BC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6B
CA6C$5.BC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA$.AB.2B
2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCAB
C2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2A$2A2BABCA2B2CBCAB2C2ACABC2A2BABC
A2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2B
ABCA2B2CBCAB$.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC
6B$.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC$2A2BCABC2B
2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCAB
C2B2CABCA2C2ABCAB2A2BCABC2B2C$.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B
2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABC
A$5.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$5.AB6CBC6ACA
6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB$4.2A2BCABC2B2CABCA2C2ABCAB2A
2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCA
B2A2B$5.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B
2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC$9.AB6ABC6BCA6CAB6ABC6BCA6CAB6A
BC6BCA6CAB6ABC6BCA6C$9.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA
$8.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C
2ABCAB2A2BCABC2B2CABCA2C2A$9.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CB
CAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB$13.AB6ABC6BCA6CAB
6ABC6BCA6CAB6ABC6BCA6CAB6ABC6B$13.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA
6BAB6CBC$12.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC
2B2CABCA2C2ABCAB2A2BCABC2B2C$13.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B
2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA$17.AB6ABC6BCA6CAB6ABC
6BCA6CAB6ABC6BCA6CAB6A$17.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB$16.
2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2AB
CAB2A2B$17.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABC
A2B2CBCAB2C2ACABC$21.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6C$21.AB6CBC6A
CA6BAB6CBC6ACA6BAB6CBC6ACA$20.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABC
A2C2ABCAB2A2BCABC2B2CABCA2C2A$21.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA
2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB$25.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6B$
25.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC$24.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC
2B2CABCA2C2ABCAB2A2BCABC2B2C$25.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B
2CBCAB2C2ACABC2A2BABCA$29.AB6ABC6BCA6CAB6ABC6BCA6CAB6A$29.AB6CBC6ACA
6BAB6CBC6ACA6BAB$28.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB
2A2B$29.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC$33.AB6ABC
6BCA6CAB6ABC6BCA6C$33.AB6CBC6ACA6BAB6CBC6ACA$32.2A2BCABC2B2CABCA2C2AB
CAB2A2BCABC2B2CABCA2C2A$33.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCA
B$37.AB6ABC6BCA6CAB6ABC6B$37.AB6CBC6ACA6BAB6CBC$36.2A2BCABC2B2CABCA2C
2ABCAB2A2BCABC2B2C$37.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA$41.AB6ABC6B
CA6CAB6A$41.AB6CBC6ACA6BAB$40.2A2BCABC2B2CABCA2C2ABCAB2A2B$41.AB.2A2B
ABCA2B2CBCAB2C2ACABC$45.AB6ABC6BCA6C$45.AB6CBC6ACA$44.2A2BCABC2B2CABC
A2C2A$45.AB.2A2BABCA2B2CBCAB$49.AB6ABC6B$49.AB6CBC$48.2A2BC2.C2B2C$
49.AB5.ABCA$55.6A!
EDIT 2: A proof that the pre-pulsar heptomino is untileable:
Starting with a single heptomino:
The three-cell "hollow" within the heptomino must be filled with a one-cell corner and a two-cell edge (three one-cell corners is obviously impossible):
Code: Select all
x = 5, y = 2, rule = //5
AC2BA$5A!
This can take on either of the following two forms:
Code: Select all
x = 18, y = 6, rule = //5
5.2B8.2B$6.B8.B$6.B8.B$5CDB3.5CB$C2.AC2BA2.C2.AC2BA$3.5A5.5A!
In the first form, the marked cell cannot belong to any heptomino, as that would necessitate overlap. Therefore, the second form is the only one possible.
Applying this form to the darker-orange heptomino gives this, and then this (ignoring the other heptomino as it is unnecessary for the proof):
Code: Select all
x = 18, y = 8, rule = //5
2.2B8.2B$2.B9.B$2.B5C4.B5C$2.BC3.C4.BC2D.C$A.2BA5.A.2BAD$5A5.5AD$15.D
$14.2D!
This shows that any heptomino in the tiling must be a part of one of these larger C4_4-symmetric 28-ominoes. In addition, any heptomino must be a part of
exactly one 28-omino to prevent overlap. So one can treat the heptomino tiling as simply a tiling of such 28-ominoes.
Starting from a single 28-omino, the marked cell can only belong to another 28-omino in the following way:
Code: Select all
x = 26, y = 9, rule = //5
2.2A16.2B$2.A11.2A4.B$2.6A6.A5.6B$2.4ABA6.6A4B.B$A.4A8.4ABA4B$6A6.A.
4A6B$5.A6.6A5.B$4.2A11.A4.2B$16.2A!
Extending this gives:
Code: Select all
x = 15, y = 15, rule = //5
8.2B$2.2A4.B$2.A5.6B$2.6A4B.B$2.4ABA4B$A.4A6B$6A3.2DB$3.2CA3.D2B$3.C
2A3.6D$3.6C4D.D$3.4CDC4D$.C.4C6D$.6C5.D$6.C4.2D$5.2C!
It is clearly impossible to fill the central 9-cell region using 28-ominos. Therefore, it is impossible to tile the plane using the original heptomino. QED, am I correct?
EDIT 3: Simpler Herschel tiling:
Code: Select all
x = 32, y = 34, rule = //5
15.3A$13.3BA$13.B3A3C$12.3B3DC$12.3AD3C3A$10.3BA3D3BA$10.B3A3CB3A3C$
9.3B3DC3B3DC$9.3AD3C3AD3C3A$7.3BA3D3BA3D3BA$7.B3A3CB3A3CB3A3C$6.3B3DC
3B3DC3B3DC$6.3AD3C3AD3C3AD3C3A$4.3BA3D3BA3D3BA3D3BA$4.B3A3CB3A3CB3A3C
B3A3C$3.3B3DC3B3DC3B3DC3B3DC$3.3AD3C3AD3C3AD3C3AD3C$.3BA3D3BA3D3BA3D
3BA3D$.B3A3CB3A3CB3A3CB3A3C$3B3DC3B3DC3B3DC3B3DC$3.D3C3AD3C3AD3C3AD3C
$2.3D3BA3D3BA3D3BA3D$5.B3A3CB3A3CB3A3C$4.3B3DC3B3DC3B3DC$7.D3C3AD3C3A
D3C$6.3D3BA3D3BA3D$9.B3A3CB3A3C$8.3B3DC3B3DC$11.D3C3AD3C$10.3D3BA3D$
13.B3A3C$12.3B3DC$15.D3C$14.3D!