Polyomino tilings
Posted: August 11th, 2017, 1:28 pm
What is known about polyomino tilings in the plane? I've done a bit of preliminary work, finding tilings for every n-omino up to n=5:
But surely someone has investigated all of this already. I just don't know where to look for actual results on such things.
Also, are there any untileable hexominoes? There are definitely heptominoes that cannot be tiled:
What about the following heptomino? Is it tileable?
EDIT: Yes:
Code: Select all
x = 270, y = 25, rule = //5
165.A59.A15.A$.A11.2A10.3A12.2A11.4A10.A12.A12.2A12.2A10.5A9.4A11.4A
11.3A10.3A13.3A13.3A14.3A12.3A14.3A$40.A26.A12.2A12.2A11.2A27.A13.A
14.A12.2A14.A14.2A16.A14.A14.A.A$67.2A11.A113.A4$ABAB6.ABABABAB4.ABAB
ABABA4.2AB2AB2AB2AB4.ABABABAB5.2B2A2C2B5.A3B3AB5.2A2B2A2B6.2A2B2A2B5.
ABABABABA4.2B2A2B2A2B5.4AB4AB7.A11.2A3B2A9.3A2.3D7.3A2B3A2B8.A4.C9.A
24.3C$BABA6.ABABABAB4.ABABABABA4.A2BA2BA2BA2B4.ABABABAB5.ABCABCAB5.2A
BABA2B6.2A2B2A2B5.2A2B2A2B5.ABABABABA4.ABABABABAB6.A4BA4B6.3AB9.3A2B
3A6.3BA.3CD7.2A3B2A3B7.B3A.D3C7.3A2B17.3ACDCD$ABAB6.BABABABA4.ABABABA
BA4.2CA2CA2CA2CA4.ABABABAB5.ABCABCAB5.AB2A2BAB5.2C2D2C2D6.2B2A2B2A5.A
BABABABA4.ABABABABAB7.4AB4A5.CA3BA6.2A3B2A3B5.3ABA3DCD7.2B3A2B3A6.A3B
AC3DC7.BA2BC14.3CABAB3D$BABA6.BABABABA4.BABABABAB4.C2AC2AC2AC2A4.ABAB
ABAB5.2A2C2B2A5.3BAB3A6.2C2D2C2D5.2B2A2B2A5.ABABABABA4.ABABABABAB8.A
4BA7.3CDB3AB4.3A2B3A2B3.3BAB3CDC8.3B2A3B2A5.B3ABD3CD8.2BAB3C2D7.3ACDC
D3B3C$10.ABABABAB4.BABABABAB4.2BC2BC2BC2BC4.BABABABA5.2B2A2C2B5.3ABA
3B5.2A2B2A2B6.2A2B2A2B5.ABABABABA4.2A2B2A2B2A8.B4AB6.AC3DCA3B2.2A3B2A
3B4.3ABA3DCD9.3A2B3A2B5.3BAC3DC7.C2B3ADC2DA7.ABAB3D3ACDCD$10.ABABABAB
4.BABABABAB4.B2CB2CB2CB2C4.BABABABA5.ABCABCAB5.BA2B2ABA6.2A2B2A2B5.2A
2B2A2B5.BABABABAB4.2B2A2B2A2B6.4BA4B5.3ABD3CDB3.3A2B3A2B5.AB3CDC3A7.
2A3B2A3B7.BD3CDA7.3C2DA2DCD3A2B4.3B3CABAB3D$10.BABABABA4.ABABABABA4.
2AB2AB2AB2AB4.BABABABA5.ABCABCAB5.2BABAB2A5.2C2D2C2D6.2B2A2B2A5.BABAB
ABAB4.ABABABABAB6.4AB4A4.CA3BAC3D4.3B2A3B7.A3DCD3BA7.2B3A2B3A7.C3DCB
3A5.DC2DA2D3CBA2B5.3ACDCD3B3C$10.BABABABA4.ABABABABA4.A2BA2BA2BA2B4.B
ABABABA5.2A2C2B2A5.B3A3BA6.2C2D2C2D5.2B2A2B2A5.BABABABAB4.ABABABABAB
7.A4BA6.3CDB3ABD6.2B3A2B6.3CDC3ABA7.3B2A3B2A6.D3CDA3BA5.2DCD3A2BC2BAB
5.ABAB3D3ACDCD$22.ABABABABA4.2CA2CA2CA2CA70.BABABABAB4.ABABABABAB7.B
4AB6.C3DCA3B4.3B2A3B2A5.3DCD3BAB8.3A2B3A2B5.C3DCB3AB4.A2D3CBA2BC2B3A
6.3B3CABAB3D$35.C2AC2AC2AC2A70.BABABABAB4.2A2B2A2B2A5.4BA4B8.D3CDB6.
2B3A2B3A2.3CDC3ABA9.2A3B2A3B5.3CD.3BA5.3A2BC2BAB3C2DA6.3ACDCD3B3C$35.
2BC2BC2BC2BC116.C3D21.CD3.AB27.C4.B6.BA2BC2B3ADC2D8.ABAB3D3ACDCD$35.B
2CB2CB2CB2C119.D21.C4.A40.2BAB3C2DA2DCD9.3B3CABAB3D$233.2B3ADC2DA2D3C
9.3ACDCD3B$237.A2DCD3A2BC9.ABAB3D$237.2D3CBA2B12.3B$241.C2BAB$241.2B
3A$245.A!
Also, are there any untileable hexominoes? There are definitely heptominoes that cannot be tiled:
Code: Select all
x = 3, y = 3, rule = //5
.2A$A.A$3A!
Code: Select all
x = 4, y = 3, rule = //5
A2.A$4A$A!
Code: Select all
x = 36, y = 31, rule = //5
17.B$14.4B$14.BACB.C$12.4A4C.B$12.ABDACD4B$10.4B4DBACB.C$10.BACBDC4A
4C.B$8.4A4CABDACD4B$8.ABDACD4B4DBACB.C$6.4B4DBACBDC4A4C.B$6.BACBDC4A
4CABDACD4B$4.4A4CABDACD4B4DBACB.C$4.ABDACD4B4DBACBDC4A4C.B$2.4B4DBACB
DC4A4CABDACD4B$2.BACBDC4A4CABDACD4B4DBACB.C$4A4CABDACD4B4DBACBDC4A4C$
A.DACD4B4DBACBDC4A4CABDACD$2.4DBACBDC4A4CABDACD4B4D$2.D.4A4CABDACD4B
4DBACBDC$4.A.DACD4B4DBACBDC4A4C$6.4DBACBDC4A4CABDACD$6.D.4A4CABDACD4B
4D$8.A.DACD4B4DBACBDC$10.4DBACBDC4A4C$10.D.4A4CABDACD$12.A.DACD4B4D$
14.4DBACBDC$14.D.4A4C$16.A.DACD$18.4D$18.D!