ConwayLife.com - A community for Conway's Game of Life and related cellular automata
Home  •  LifeWiki  •  Forums  •  Download Golly

Polyomino tilings

A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.

Polyomino tilings

Postby A for awesome » 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:
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!

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:
x = 3, y = 3, rule = //5
.2A$A.A$3A!

What about the following heptomino? Is it tileable?
x = 4, y = 3, rule = //5
A2.A$4A$A!

EDIT: Yes:
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!
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce
User avatar
A for awesome
 
Posts: 1730
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

Re: Polyomino tilings

Postby bprentice » August 11th, 2017, 2:15 pm

bprentice
 
Posts: 534
Joined: September 10th, 2009, 6:20 pm
Location: Coos Bay, Oregon

Re: Polyomino tilings

Postby muzik » August 11th, 2017, 2:22 pm

An octomino:
x = 30, y = 16, rule = //5
13.3B$13.BAB2A$9.3D2BABA3B$9.DCD2C3ABAB2A$5.3B2DCDC3D2BABA3B$5.BAB2A
3CDCD2C3ABAB2A$.3D2BABA3B2DCDC3D2BABA3B$.DCD2C3ABAB2A3CDCD2C3ABAB2A$
2DCDC3D2BABA3B2DCDC3D2BABA$2.3CDCD2C3ABAB2A3CDCD2C3A$4.2DCDC3D2BABA3B
2DCDC$6.3CDCD2C3ABAB2A3C$8.2DCDC3D2BABA$10.3CDCD2C3A$12.2DCDC$14.3C!


B:
x = 21, y = 14, rule = //5
A.2A3.A.2A3.A.2A$3A2C.C3A2C.C3A2C.C$BA2B3CBA2B3CBA2B3C$3B2DCD3B2DCD3B
2DCD$AB2A3DAB2A3DAB2A3D$3A2CDC3A2CDC3A2CDC$BA2B3CBA2B3CBA2B3C$3B2DCD
3B2DCD3B2DCD$AB2A3DAB2A3DAB2A3D$3A2CDC3A2CDC3A2CDC$BA2B3CBA2B3CBA2B3C
$3B2DCD3B2DCD3B2DCD$.B2.3D.B2.3D.B2.3D$5.D6.D6.D!


H:
x = 29, y = 21, rule = //5
.A$.3A4.A$.ABA3C.3A4.A$3BAC3DABA3C.3A4.A$BAB3CD3BAC3DABA3C.3A$B3A3DBA
B3CD3BAC3DABA3C$.ABA3CB3A3DBAB3CD3BAC3D$3BAC3DABA3CB3A3DBAB3CD$BAB3CD
3BAC3DABA3CB3A3D$B3A3DBAB3CD3BAC3DABA3C$.ABA3CB3A3DBAB3CD3BAC3D$3BAC
3DABA3CB3A3DBAB3CD$BAB3CD3BAC3DABA3CB3A3D$B3A3DBAB3CD3BAC3DABA3C$.ABA
3CB3A3DBAB3CD3BAC3D$3BAC3DABA3CB3A3DBAB3CD$B.B3CD3BAC3DABA3CB3A3D$B3.
3DB.B3CD3BAC3DABA3C$7.B3.3DB.B3CD3BAC3D$14.B3.3DB.B3CD$21.B3.3D!
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik
 
Posts: 3248
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

Re: Polyomino tilings

Postby muzik » August 11th, 2017, 2:36 pm

I don't know how this one works out:
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!


Some hexes:
x = 12, y = 26, rule = //4
2A.2A.2A.2A$A2.A2.A2.A$A2BA2BA2BA2B$2AB2AB2AB2AB$2CB2CB2CB2CB$C2BC2BC
2BC2B$C2AC2AC2AC2A$2CA2CA2CA2CA$2BA2BA2BA2BA$B2AB2AB2AB2A$B2CB2CB2CB
2C$2BC2BC2BC2BC$2AC2AC2AC2AC$A2CA2CA2CA2C$A2BA2BA2BA2B$2AB2AB2AB2AB$
2CB2CB2CB2CB$C2BC2BC2BC2B$C2AC2AC2AC2A$2CA2CA2CA2CA$2BA2BA2BA2BA$B2AB
2AB2AB2A$B2CB2CB2CB2C$2BC2BC2BC2BC$2.C2.C2.C2.C$.2C.2C.2C.2C!

x = 16, y = 24, rule = //3
3AB3AB3AB3AB$ABABABABABABABAB$A3BA3BA3BA3B$B3AB3AB3AB3A$BABABABABABAB
ABA$3BA3BA3BA3BA$3AB3AB3AB3AB$ABABABABABABABAB$A3BA3BA3BA3B$B3AB3AB3A
B3A$BABABABABABABABA$3BA3BA3BA3BA$3AB3AB3AB3AB$ABABABABABABABAB$A3BA
3BA3BA3B$B3AB3AB3AB3A$BABABABABABABABA$3BA3BA3BA3BA$3AB3AB3AB3AB$ABAB
ABABABABABAB$A3BA3BA3BA3B$B3AB3AB3AB3A$BABABABABABABABA$3BA3BA3BA3BA!
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik
 
Posts: 3248
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

Re: Polyomino tilings

Postby A for awesome » August 11th, 2017, 3:49 pm

muzik wrote:I don't know how this one works out:
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!

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:
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:
x = 5, y = 2, rule = //5
A3.A$5A!

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):
x = 5, y = 2, rule = //5
AC2BA$5A!

This can take on either of the following two forms:
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):
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:
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:
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:
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!
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce
User avatar
A for awesome
 
Posts: 1730
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

Re: Polyomino tilings

Postby calcyman » August 12th, 2017, 10:43 am

My friend Joseph Myers conducted arguably the most comprehensive study on tilings of polyominos, polyiamonds, and polyhexes:

https://www.polyomino.org.uk/mathematic ... rm-tiling/

This was part of an attempt to find an aperiodic monotile; whilst unsuccessful in this regard, he succeeded in breaking various records (such as finding a 16-hex where any fundamental region must have at least 10 tiles).
What do you do with ill crystallographers? Take them to the mono-clinic!
User avatar
calcyman
 
Posts: 1948
Joined: June 1st, 2009, 4:32 pm

Re: Polyomino tilings

Postby A for awesome » August 12th, 2017, 11:39 am

calcyman wrote:My friend Joseph Myers conducted arguably the most comprehensive study on tilings of polyominos, polyiamonds, and polyhexes:

https://www.polyomino.org.uk/mathematic ... rm-tiling/

This was part of an attempt to find an aperiodic monotile; whilst unsuccessful in this regard, he succeeded in breaking various records (such as finding a 16-hex where any fundamental region must have at least 10 tiles).

Wow. Okay, forget I ever posted this topic...
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce
User avatar
A for awesome
 
Posts: 1730
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

Re: Polyomino tilings

Postby Rhombic » August 12th, 2017, 2:56 pm

For every n, there is a tileable n-polyomino (certainly, since a one-cell line is one). One of these n-cell polyominoes has the greatest number m of tiling collocations.
Define M(n) as the sequence with the maximum m for every n-omino.
How does M(n) grow, or tend to grow? (linear? log? sqrt?)

PS stacking lines with offsets per line obviously counts as one (1) single disposition, because it's 1D ordering with a flat surface; think that for any 1D ordered pattern with two flat surfaces there should only be ONE valid disposition containing that given 1D ordering. Otherwise M(n)=n
SoL : FreeElectronics : DeadlyEnemies : 6a-ite : Rule X3VI
what is “sesame oil”?
User avatar
Rhombic
 
Posts: 1056
Joined: June 1st, 2013, 5:41 pm

Re: Polyomino tilings

Postby wwei23 » August 13th, 2017, 10:39 am

muzik wrote:Some hexes:

Any jinxes or curses?
User avatar
wwei23
 
Posts: 935
Joined: May 22nd, 2017, 6:14 pm
Location: The (Life?) Universe

Re: Polyomino tilings

Postby muzik » August 13th, 2017, 10:50 am

wwei23 wrote:
muzik wrote:Some hexes:

Any jinxes or curses?

The latter goes against rule 2b, unfortunately.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik
 
Posts: 3248
Joined: January 28th, 2016, 2:47 pm
Location: Scotland


Return to The Sandbox

Who is online

Users browsing this forum: No registered users and 2 guests