Polyominoes

[Lewis]

Anyone else on here have an interest in polyominoes or other polyforms (outside of a cellular-automata context, I suppose)?
I've been playing on with various sets a fair bit recently, pentominoes and hexominoes mainly, just constructing (or attempting to construct) rectangles and other shapes with them. The most interesting/challenging thing I've managed to make so far is the construction in the picture below, all 108 heptominoes into a 29x29 square with a symmetrical hole.

29x29-squareholes.png (4.11 KiB) Viewed 11161 times

[Moosey]

Anyone else on here have an interest in polyominoes or other polyforms (outside of a cellular-automata context, I suppose)?
I've been playing on with various sets a fair bit recently, pentominoes and hexominoes mainly, just constructing (or attempting to construct) rectangles and other shapes with them. The most interesting/challenging thing I've managed to make so far is the construction in the picture below, all 108 heptominoes into a 29x29 square with a symmetrical hole.

[a thing]

I find this really fascinating. I might try to see if something similar is possible with different size polyominoes.
(By hand)

[fluffykitty]

What's the smallest set of polyominoes that tiles the plane aperiodically (with rotations/reflections), measured by total square count? Translating the Robinson tiles gives 142 cells: (Wikipedia image order if you're wondering)

Code: Select all

x = 23, y = 15, rule = B/S012345678
10b2o$2bobo5b3o7bo$b5o4b3o4b5o$2b5o3b3o3b5o$b5o3b5o3b5o$2bobo7bo7bo3$
2b2o4b4ob2o$2b3o3b7o3bobo$b5o3b5o3b5o$7ob7o3b5o$b6o2b6o3b5o$2b3o3b7o3b
obo$3bo4b2obob2o!

Code: Select all

x = 37, y = 37, rule = Codd
2D.D.2D.B.2D.D.2D.B.2D.D.2D.B.2D.D.2D$7D3B7D3B7D3B7D$.6D3B6D5B6D3B6D$
7D3B7D3B7D3B7D$.5D5B5D5B5D5B5D$7DBCB7DBCB7DBCB7D$4DB2D3C2DB4D3C4DB2D
3C2DB4D$.5B6C5B3C5B6C5B$5B7C5B3C5B7C5B$.5B5C5B5C5B5C5B$4DB2D3C2DB4DCB
C4DB2D3C2DB4D$7D2CB7D3B7DB2C7D$.5D5B5D5B5D5B5D$7D3B7D3B7D3B7D$.6D3B6D
5B6D3B6D$7D3B7DBCB7D3B7D$2DBDB2D2BC2DBDB2D3C2DBDB2DC2B2DBDB2D$.5B5C5B
6C5B5C5B$5B5C5B7C5B5C5B$.5B5C5B5C5B5C5B$2DBDB2D2BC2DBDB2D3C2DBDB2DC2B
2DBDB2D$7D3B7D2CB7D3B7D$.6D3B6D5B6D3B6D$7D3B7D3B7D3B7D$.5D5B5D5B5D5B
5D$7D2CB7D3B7DB2C7D$4DB2D3C2DB4D2BC4DB2D3C2DB4D$.5B5C5B5C5B5C5B$5B7C
5B3C5B7C5B$.5B6C5B3C5B6C5B$4DB2D3C2DB4D3C4DB2D3C2DB4D$7DBCB7D2CB7DBCB
7D$.5D5B5D5B5D5B5D$7D3B7D3B7D3B7D$.6D3B6D5B6D3B6D$7D3B7D3B7D3B7D$2D.D
.2D.B.2D.D.2D2B.2D.D.2D.B.2D.D.2D!

[Moosey]

Here’s an attempt at using all hexominoes for something.
Can this be somehow completed to make something interesting?

Code: Select all

x = 83, y = 63, rule = Codd
6A7.5A8.5A8.5A8.4A9.4A9.4A$17.A11.A11.A13.2A11.A11.2A$68.A$5.C11.C64.
C$2.C.C9.C.C51.C10.C.C$3.C11.C49.C.C12.C$66.C4$4A9.4A9.A13.A13.A13.A
9.4A$.A.A9.A2.A9.4A9.4A9.4A9.4A11.A$29.A12.A12.A12.A11.A$16.C$13.C.C$
14.C5$4A10.A13.A10.3A10.3A10.3A10.3A$.2A10.4A9.4A11.3A10.2A11.2A10.3A
$15.A12.A26.A11.A8$3A10.A13.A11.3A10.3A10.3A10.3A$A.2A9.3A10.3A12.A
12.A11.2A11.2A$15.2A11.2A11.2A10.2A12.A11.A$3.C$C.C13.C38.C$.C11.C.C
36.C.C$14.C38.C4$3A10.A13.A11.2A11.A13.A11.2A$3A10.3A10.3A11.3A9.3A
10.3A11.2A$14.2A11.2A11.A11.A.A10.A.A12.2A3$16.C12.C25.C12.C$13.C.C
10.C.C23.C.C10.C.C$14.C12.C25.C12.C8$26.A$20.2A3B3A$17.B3ACBA2B2A$17.
BAB3C3A2C$17.3BCBCABA2C$17.4A4B2C$17.A4C5A$17.A2C6BA!

[Sarp]

I am. I've recently printed out a set of heptiamonds and had some fun with them. I also find polyform compatibility a fascinating subject.

[Moosey]

Here’s a rectangle which contains at least one of every hexomino

Code: Select all

x = 24, y = 18, rule = Codd
2B5C3B3AD2C2A2C2D2C$4B2AC3B3AD2C2A2C2DCB$A4D4AC3E2DC2A2CD2CB$2A2C2D2B
2C2BEDACD3BDACB$3AC3B2C2B2ED2A2DBD3AB$3B3CBC2B3DB2A2DBDACAB$3B6A3CD2B
ADCBD3CB$3C3B4DCBDCB4CDC2AE$2CA2B2A2D2CBDCB2AC2DC2AE$C3AB2A4EBACB3AB
2ACAE$2A3C2A2EA2BAC2DA2BA2CAE$3C2A3B3ABACD3B2ABC2E$B3ACBA2B2ADACD4CAB
CDA$BAB3C3A2CDA2D2CD3BCDA$3BCBCABA2CDA2C4DCBADA$4A4B2C2DBC2AD4CADA$A
4C5AD2B2C2A3BCADA$A2C6BA3BC2A3B3ADA!

Someone can 4-color mapping theorem it if they like.


Are there smaller rectangles?
Obviously it must have at least 210 cells since that’s 35*6


Challenge:
Find a box with at least one of every hexomino, with at most 5 colors, where no two copies of the same hexomino share a color.

Every hexomino must have a distinct color from every other one it touches.

EDIT:
Remember how much easier smaller polyominoes are

Code: Select all

x = 3, y = 3, rule = Codd
B2C$2BC$3A!

EDIT:
Also, a way to show that you need at least four colors to color any map

Code: Select all

x = 6, y = 5, rule = Codd
6D$DA2CBD$DA2CBD$D2A2BD$6D!

Are there any proofs like this that show the other part of the 4-color mapping theorem, that on a plane or sphere you need at most 4?

[fluffykitty]

Well, you can do it with tetrominoes by just removing the left, right, and bottom columns of that. I'm not sure if there are any other particularly interesting ones though.

[toroidalet]

I believe that this set (123 cells) is aperiodic:

Code: Select all

x = 30, y = 11, rule = //5
19.C$17.5C$17.5C$16.7C$17.5C$9.B7.5C$2A.2A3.5B4.5C3.3D.D$5A3.5B4.5C4.
4D$.3A3.7B2.7C2.5D$5A3.5B4.5C3.4D$2A.2A3.2B.2B4.C.3C3.D.3D!

Sample partial tiling:

Code: Select all

x = 97, y = 97, rule = //5
17.B9.D9.B9.B9.B9.D9.B$.2A.2A2B.2B2A.2A5B2A.2A5D2A.2A5B2A.2A5B2A.2A5B
2A.2A5D2A.2A5B2A.2A2B.2B2A.2A$.5A5B5A5B5A5D5A5B5A5B5A5B5A5D5A5B5A5B5A
$2.3A7B3A7B3A7D3A7B3A7B3A7B3A7D3A7B3A7B3A$.5A5B5A5B5A5D5A5B5A5B5A5B5A
5D5A5B5A5B5A$.2AB2A5B2AB2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A2BC2B2AB2A2BC
2B2AC2A2DB2D2AC2A2BC2B2AB2A5B2AB2A$.5B3DBD5B10C5B10C5B5C5B10C5B10C5B
3DBD5B$.6B4D5B10C5B10C5B5C6B9C6B9C6B4D5B$2.4B5D4B11C4B11C4B7C4B11C4B
11C4B5D4B$.5B4D6B9C6B9C6B5C5B10C5B10C5B4D6B$.5BDB3D5B10C5B10C5B5C5B
10C5B10C5BDB3D5B$.2AB2A5B2AB2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A5C2AB2A2B
C2B2AC2A2DB2D2AC2A2BC2B2AB2A5B2AB2A$.5A5B5A5B5A5D5A5B5A5C5A5B5A5D5A5B
5A5B5A$2.3A7B3A7B3A7D3A7B3A7C3A7B3A7D3A7B3A7B3A$.5A5B5A5B5A5D5A5B5A5C
5A5B5A5D5A5B5A5B5A$.2AB2A2BC2B2AB2A5B2AB2A5D2AB2A5B2AC2ACB3C2AC2A5B2A
B2A5D2AB2A5B2AB2A2BC2B2AB2A$.5B5C5B3DBD5B3CDC5B3DBD5C5B5C3DBD5B3CDC5B
3DBD5B5C5B$.5B5C6B4D5B5C6B4D5C5B6C4D5B5C6B4D5B5C6B$.4B7C4B5D4B7C4B5D
4C7B4C5D4B7C4B5D4B7C4B$6B5C5B4D6B5C5B4D6C5B5C4D6B5C5B4D6B5C5B$.5B5C5B
DB3D5B5C5BDB3D5C2BC2B5CDB3D5B5C5BDB3D5B5C5B$.2AB2A5C2AB2A5B2AB2A5C2AB
2A5B2AC2A5C2AC2A5B2AB2A5C2AB2A5B2AB2A5C2AB2A$.5A5C5A5B5A5C5A5B5A5C5A
5B5A5C5A5B5A5C5A$2.3A7C3A7B3A7C3A7B3A7C3A7B3A7C3A7B3A7C3A$.5A5C5A5B5A
5C5A5B5A5C5A5B5A5C5A5B5A5C5A$.2AD2ACB3C2AD2A2BC2B2AC2A5C2AC2A2BC2B2AB
2A5C2AB2A2BC2B2AC2A5C2AC2A2BC2B2AD2ACB3C2AD2A$.5D5B5D10C2AC2A10C5B5C
5B10C2AC2A10C5D5B5D$.5D5B6D9C5A10C5B5C6B9C5A10C5D5B6D$.4D7B4D11C3A11C
4B7C4B11C3A11C4D7B4D$6D5B5D10C5A9C6B5C5B10C5A9C6D5B5D$.5D2BC2B5D10C2A
C2A10C5BCB3C5B10C2AC2A10C5D2BC2B5D$.2AD2A5C2AD2A2BC2B2AC2A5C2AC2A2BC
2B2AB2A5B2AB2A2BC2B2AC2A5C2AC2A2BC2B2AD2A5C2AD2A$.5A5C5A5B5A5C5A5B5A
5B5A5B5A5C5A5B5A5C5A$2.3A7C3A7B3A7C3A7B3A7B3A7B3A7C3A7B3A7C3A$.5A5C5A
5B5A5C5A5B5A5B5A5B5A5C5A5B5A5C5A$.2AB2A5C2AB2A5B2AB2A5C2AB2A5B2AB2A2B
C2B2AB2A5B2AB2A5C2AB2A5B2AB2A5C2AB2A$.5B5C5B3DBD5B5C5B3DBD5B5C5B3DBD
5B5C5B3DBD5B5C5B$.5B5C6B4D5B5C6B4D5B5C6B4D5B5C6B4D5B5C6B$.4B7C4B5D4B
7C4B5D4B7C4B5D4B7C4B5D4B7C4B$6B5C5B4D6B5C5B4D6B5C5B4D6B5C5B4D6B5C5B$.
5BCB3C5BDB3D5BCD3C5BDB3D5B5C5BDB3D5BCD3C5BDB3D5BCB3C5B$.2AB2A5B2AB2A
5B2AB2A5D2AB2A5B2AB2A5C2AB2A5B2AB2A5D2AB2A5B2AB2A5B2AB2A$.5A5B5A5B5A
5D5A5B5A5C5A5B5A5D5A5B5A5B5A$2.3A7B3A7B3A7D3A7B3A7C3A7B3A7D3A7B3A7B3A
$.5A5B5A5B5A5D5A5B5A5C5A5B5A5D5A5B5A5B5A$.2AC2A2BC2B2AB2A2BC2B2AC2A2D
B2D2AC2A2BC2B2AD2ACB3C2AD2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A2BC2B2AC2A$.
10C5B10C5B10C5D5B5D10C5B10C5B10C$.10C5B10C5B10C5D5B5D10C6B9C6B9C$11C
4B11C4B11C4D7B4D11C4B11C4B11C$.9C6B9C6B9C6D5B6D9C5B10C5B10C$.10C5B10C
5B10C5D2BC2B5D10C5B10C5B10C$.2AC2A2BC2B2AB2A2BC2B2AC2A2DB2D2AC2A2BC2B
2AD2A5C2AD2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A2BC2B2AC2A$.5A5B5A5B5A5D5A
5B5A5C5A5B5A5D5A5B5A5B5A$2.3A7B3A7B3A7D3A7B3A7C3A7B3A7D3A7B3A7B3A$.5A
5B5A5B5A5D5A5B5A5C5A5B5A5D5A5B5A5B5A$.2AB2A5B2AB2A5B2AB2A5D2AB2A5B2AB
2A5C2AB2A5B2AB2A5D2AB2A5B2AB2A5B2AB2A$.5B3CBC5B3DBD5B3CDC5B3DBD5B5C5B
3DBD5B3CDC5B3DBD5B3CBC5B$.5B5C6B4D5B5C6B4D5B5C6B4D5B5C6B4D5B5C6B$.4B
7C4B5D4B7C4B5D4B7C4B5D4B7C4B5D4B7C4B$6B5C5B4D6B5C5B4D6B5C5B4D6B5C5B4D
6B5C5B$.5B5C5BDB3D5B5C5BDB3D5B3CBC5BDB3D5B5C5BDB3D5B5C5B$.2AB2A5C2AB
2A5B2AB2A5C2AB2A5B2AB2A5B2AB2A5B2AB2A5C2AB2A5B2AB2A5C2AB2A$.5A5C5A5B
5A5C5A5B5A5B5A5B5A5C5A5B5A5C5A$2.3A7C3A7B3A7C3A7B3A7B3A7B3A7C3A7B3A7C
3A$.5A5C5A5B5A5C5A5B5A5B5A5B5A5C5A5B5A5C5A$.2AD2A5C2AD2A2BC2B2AC2A5C
2AC2A2BC2B2AB2A2BC2B2AB2A2BC2B2AC2A5C2AC2A2BC2B2AD2A5C2AD2A$.5D2BC2B
5D10C2AC2A10C5B5C5B10C2AC2A10C5D2BC2B5D$.5D5B6D9C5A10C5B5C6B9C5A10C5D
5B6D$.4D7B4D11C3A11C4B7C4B11C3A11C4D7B4D$6D5B5D10C5A9C6B5C5B10C5A9C6D
5B5D$.5D5B5D10C2AC2A10C5B5C5B10C2AC2A10C5D5B5D$.2AD2A3CBC2AD2A2BC2B2A
C2A5C2AC2A2BC2B2AB2A5C2AB2A2BC2B2AC2A5C2AC2A2BC2B2AD2A3CBC2AD2A$.5A5C
5A5B5A5C5A5B5A5C5A5B5A5C5A5B5A5C5A$2.3A7C3A7B3A7C3A7B3A7C3A7B3A7C3A7B
3A7C3A$.5A5C5A5B5A5C5A5B5A5C5A5B5A5C5A5B5A5C5A$.2AB2A5C2AB2A5B2AB2A5C
2AB2A5B2AC2ACB3C2AC2A5B2AB2A5C2AB2A5B2AB2A5C2AB2A$.5B5C5B3DBD5B5C5B3D
BD5C5B5C3DBD5B5C5B3DBD5B5C5B$.5B5C6B4D5B5C6B4D5C5B6C4D5B5C6B4D5B5C6B$
.4B7C4B5D4B7C4B5D4C7B4C5D4B7C4B5D4B7C4B$6B5C5B4D6B5C5B4D6C5B5C4D6B5C
5B4D6B5C5B$.5B5C5BDB3D5BCD3C5BDB3D5C2BC2B5CDB3D5BCD3C5BDB3D5B5C5B$.2A
B2A2BC2B2AB2A5B2AB2A5D2AB2A5B2AC2A5C2AC2A5B2AB2A5D2AB2A5B2AB2A2BC2B2A
B2A$.5A5B5A5B5A5D5A5B5A5C5A5B5A5D5A5B5A5B5A$2.3A7B3A7B3A7D3A7B3A7C3A
7B3A7D3A7B3A7B3A$.5A5B5A5B5A5D5A5B5A5C5A5B5A5D5A5B5A5B5A$.2AB2A5B2AB
2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A5C2AB2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A
5B2AB2A$.5B3DBD5B10C5B10C5B5C5B10C5B10C5B3DBD5B$.6B4D5B10C5B10C5B5C6B
9C6B9C6B4D5B$2.4B5D4B11C4B11C4B7C4B11C4B11C4B5D4B$.5B4D6B9C6B9C6B5C5B
10C5B10C5B4D6B$.5BDB3D5B10C5B10C5BCB3C5B10C5B10C5BDB3D5B$.2AB2A5B2AB
2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A5B2AB2A2BC2B2AC2A2DB2D2AC2A2BC2B2AB2A
5B2AB2A$.5A5B5A5B5A5D5A5B5A5B5A5B5A5D5A5B5A5B5A$2.3A7B3A7B3A7D3A7B3A
7B3A7B3A7D3A7B3A7B3A$.5A5B5A5B5A5D5A5B5A5B5A5B5A5D5A5B5A5B5A$.2A.2A2B
.2B2A.2A5B2A.2A5D2A.2A5B2A.2A2B.2B2A.2A5B2A.2A5D2A.2A5B2A.2A2B.2B2A.
2A$19.B9.D9.B19.B9.D9.B!

Note that although tiles 2, 3, and 4 have asymmetric edge markings, they effectively tile as though those markings are symmetric.

[fluffykitty]

Improved to 117:

Code: Select all

x = 35, y = 8, rule = //6
3.2A6.B$.5A3.5B2.2C.2C2.D2.2D2.2E.2E$6A3.4B3.2C.C3.4D3.5E$7A3.3B4.3C
4.3D4.3E$.6A3.4B3.4C3.4D2.5E$.5A3.5B2.2C.2C2.5D2.2E.2E$2.2A6.2B13.D$
25.D!

[Lewis]

I built another rectangle I haven't seen anywhere before, again using one of each of the heptominoes: a 20x38 with 4 holes (placed symmetrically).

20x38.png (11.94 KiB) Viewed 11087 times

This ended up taking an entire evening of playing around with my little homemade heptomino set. I had attempted this rectangle ages ago, and had completed it but with the all holes shifted off by one, only realising afterwards that it was slightly wonky.

Are there smaller rectangles?
Obviously it must have at least 210 cells since that’s 35*6

Challenge:
Find a box with at least one of every hexomino, with at most 5 colors, where no two copies of the same hexomino share a color.

Every hexomino must have a distinct color from every other one it touches.

I have a feeling a rectangle like this should be possible with 36 hexominoes, i.e. a full set with one extra duplicate. I'll have to have a try at this at some point, the colouring constraint is probably going to make it really tricky though.

[Moosey]

I built another rectangle I haven't seen anywhere before, again using one of each of the heptominoes: a 20x38 with 4 holes (placed symmetrically).

20x38.png

This ended up taking an entire evening of playing around with my little homemade heptomino set. I had attempted this rectangle ages ago, and had completed it but with the all holes shifted off by one, only realising afterwards that it was slightly wonky.

Are there smaller rectangles?
Obviously it must have at least 210 cells since that’s 35*6

Challenge:
Find a box with at least one of every hexomino, with at most 5 colors, where no two copies of the same hexomino share a color.

Every hexomino must have a distinct color from every other one it touches.

I have a feeling a rectangle like this should be possible with 36 hexominoes, i.e. a full set with one extra duplicate. I'll have to have a try at this at some point, the colouring constraint is probably going to make it really tricky though.

The 4-color mapping theorem guarantees you can use at most 4 colors to do the coloring, so that’s not a problem unless you’re doing the challenge.

[PkmnQ]

Here’s a rectangle which contains at least one of every hexomino

Code: Select all

x = 24, y = 18, rule = Codd
2B5C3B3AD2C2A2C2D2C$4B2AC3B3AD2C2A2C2DCB$A4D4AC3E2DC2A2CD2CB$2A2C2D2B
2C2BEDACD3BDACB$3AC3B2C2B2ED2A2DBD3AB$3B3CBC2B3DB2A2DBDACAB$3B6A3CD2B
ADCBD3CB$3C3B4DCBDCB4CDC2AE$2CA2B2A2D2CBDCB2AC2DC2AE$C3AB2A4EBACB3AB
2ACAE$2A3C2A2EA2BAC2DA2BA2CAE$3C2A3B3ABACD3B2ABC2E$B3ACBA2B2ADACD4CAB
CDA$BAB3C3A2CDA2D2CD3BCDA$3BCBCABA2CDA2C4DCBADA$4A4B2C2DBC2AD4CADA$A
4C5AD2B2C2A3BCADA$A2C6BA3BC2A3B3ADA!

Someone can 4-color mapping theorem it if they like.

I'm working on it. I only have one problematic hexomino to solve.

EDIT:
Done!

Code: Select all

x = 24, y = 18, rule = Codd
2B5C3B3CD2C2A2C2D2C$4B2AC3B3CD2C2A2C2DCD$A4D4AC3A2DC2A2CD2CD$2A2C2D2B
2C2BADACD3BDACD$3AC3B2C2B2AD2A2DBD3AD$3B3CBC2B3DB2A2DBDACAD$3B6A3CD2B
ADCBD3CD$3D3C4BCBDCB4CDC2AB$2DA2C2A2B2CBDCB2AC2DC2AB$D3AC2A4DBACB3AB
2ACAB$2A3D2A2DA2BAC2DA2BA2CAB$3D2A3B3ABACD3B2ABC2B$B3ACBA2B2ADACD4CAB
CDA$BAB3C3A2CDA2D2CD3BCDA$3BCBCABA2CDA2C4DCBADA$4A4B2C2DBC2AD4CADA$A
4C5AD2B2C2A3BCADA$A2C6BA3BC2A3B3ADA!

[Naszvadi]

My 2cents:

https://github.com/PetterS/glpk/blob/ma ... tomino.mod
https://github.com/PetterS/glpk/blob/ma ... tiling.mod

See more at:
http://www.gnu.org/software/glpk/glpk.html

[dvgrn]

My two cents: anyone interested in polyomino problems might want to page through the archives on www.mathpuzzle.com. There's something omino-related in almost every annual / bi-monthly / monthly archive, like this from 29 March 2005:

Y-pentomino used to make a larger Y-pentomino

rep12y.gif (4.84 KiB) Viewed 11042 times

That was apparently an unusually tricky problem to solve... See also the tiling problems at the end of 2012, and the huge multi-omino bulls-eye in 2011, and the links from those entries to other polyomino websites, and the "Polyominoes" page in the right sidebar which is also full of interesting links.

[Moosey]

My two cents: anyone interested in polyomino problems might want to page through the archives on http://www.mathpuzzle.com. There's something omino-related in almost every annual / bi-monthly / monthly archive, like this from 29 March 2005:

Code: Select all

For some reason, a gif

That was apparently an unusually tricky problem to solve... See also the tiling problems at the end of 2012, and the huge multi-omino bulls-eye in 2011, and the links from those entries to other polyomino websites, and the "Polyominoes" page in the right sidebar which is also full of interesting links.

There’s a Moscow puzzles puzzle involving that.

[Freywa]

What's the smallest set of polyominoes that tiles the plane aperiodically (with rotations/reflections), measured by total square count?

The MathWorld page on polyomino tilings gives a 60-square aperiodic by Matthew Cook:

Code: Select all

x = 15, y = 12, rule = 345/2/4
11.C2.C$11.4C$.7A4.2C$2.A.A7.2C$A.A8.4C$4A7.C2.C3$.4B2.7B$.B.B4.2B$3.
B.2B.3B$2.7B.B!

[toroidalet]

I believe these tiles should get 41:

Code: Select all

x = 14, y = 64, rule = //5
7.3A.3A$.A5.7A$4A3.7A$.A7.3A$.A5.4A$7.4A$7.3A10$7.3B.3B$.B5.7B$4B3.7B
$.B7.3B$.B5.4B$7.4B$7.3B13$7.3C.3C$.C5.7C$4C3.7C$.C7.3C$.C5.4C$7.4C$
7.3C16$7.3D.3D$.D5.7D$4D3.7D$.D7.3D$.D5.4D$7.4D$7.3D!

[dvgrn]

I believe these tiles should get 41:

Code: Select all

x = 14, y = 64, rule = //5
7.3A.3A$.A5.7A$4A3.7A$.A7.3A$.A5.4A$7.4A$7.3A10$7.3B.3B$.B5.7B$4B3.7B
$.B7.3B$.B5.4B$7.4B$7.3B13$7.3C.3C$.C5.7C$4C3.7C$.C7.3C$.C5.4C$7.4C$
7.3C16$7.3D.3D$.D5.7D$4D3.7D$.D7.3D$.D5.4D$7.4D$7.3D!

I don't think so. Part of the requirement is that the tile set can't allow any periodic tilings of the plane. That doesn't seem to be true of your two tiles:

Code: Select all

x = 108, y = 74, rule = //5
41.D9.3D$41.D8.4DB$39.4D7.4DB$38.3CD3C4.3D4B$38.7CA.7DB3A$37.A7CA.7D
4AC$23.D9.3D4A3C4A3DC3D4AC$23.D8.4DBA4C3BA3B4C3A4C$21.4D7.4DBA4C7BDC
7AC3D$20.3CD3C4.3D4B3CD7BDC7A4DB$20.7CA.7DB3A4D3B4D3AB3A4DB$19.A7CA.
7D4ACD4B3CD3C4B3D4B$5.D9.3D4A3C4A3DC3D4ACD4B7CAB7DB3A$5.D8.4DBA4C3BA
3B4C3A4C3BA7CAB7D4AC$3.4D7.4DBA4C7BDC7AC3D4A3C4A3DC3D4AC$2.3CD3C4.3D
4B3CD7BDC7A4DBA4C3BA3B4C3A4C$2.7CA.7DB3A4D3B4D3AB3A4DBA4C7BDC7AC3D$.A
7CA.7D4ACD4B3CD3C4B3D4B3CD7BDC7A4DB$4A3C4A3DC3D4ACD4B7CAB7DB3A4D3B4D
3AB3A4DB$.A4C3BA3B4C3A4C3BA7CAB7D4ACD4B3CD3C4B3D4B$.A4C7BDC7AC3D4A3C
4A3DC3D4ACD4B7CAB7DB3A$2.3CD7BDC7A4DBA4C3BA3B4C3A4C3BA7CAB7D4AC$4.4D
3B4D3AB3A4DBA4C7BDC7AC3D4A3C4A3DC3D4AC$5.D4B3CD3C4B3D4B3CD7BDC7A4DBA
4C3BA3B4C3A4C$5.D4B7CAB7DB3A4D3B4D3AB3A4DBA4C7BDC7AC3D$6.3BA7CAB7D4AC
D4B3CD3C4B3D4B3CD7BDC7A4DB$8.4A3C4A3DC3D4ACD4B7CAB7DB3A4D3B4D3AB3A4DB
$9.A4C3BA3B4C3A4C3BA7CAB7D4ACD4B3CD3C4B3D4B$9.A4C7BDC7AC3D4A3C4A3DC3D
4ACD4B7CAB7DB3A$10.3CD7BDC7A4DBA4C3BA3B4C3A4C3BA7CAB7D4AC$12.4D3B4D3A
B3A4DBA4C7BDC7AC3D4A3C4A3DC3D4AC$13.D4B3CD3C4B3D4B3CD7BDC7A4DBA4C3BA
3B4C3A4C$13.D4B7CAB7DB3A4D3B4D3AB3A4DBA4C7BDC7AC3D$14.3BA7CAB7D4ACD4B
3CD3C4B3D4B3CD7BDC7A4DB$16.4A3C4A3DC3D4ACD4B7CAB7DB3A4D3B4D3AB3A4DB$
17.A4C3BA3B4C3A4C3BA7CAB7D4ACD4B3CD3C4B3D4B$17.A4C7BDC7AC3D4A3C4A3DC
3D4ACD4B7CAB7DB3A$18.3CD7BDC7A4DBA4C3BA3B4C3A4C3BA7CAB7D4AC$20.4D3B4D
3AB3A4DBA4C7BDC7AC3D4A3C4A3DC3D4AC$21.D4B3CD3C4B3D4B3CD7BDC7A4DBA4C3B
A3B4C3A4C$21.D4B7CAB7DB3A4D3B4D3AB3A4DBA4C7BDC7AC3D$22.3BA7CAB7D4ACD
4B3CD3C4B3D4B3CD7BDC7A4DB$24.4A3C4A3DC3D4ACD4B7CAB7DB3A4D3B4D3AB3A4DB
$25.A4C3BA3B4C3A4C3BA7CAB7D4ACD4B3CD3C4B3D4B$25.A4C7BDC7AC3D4A3C4A3DC
3D4ACD4B7CAB7DB3A$26.3CD7BDC7A4DBA4C3BA3B4C3A4C3BA7CAB7D4AC$28.4D3B4D
3AB3A4DBA4C7BDC7AC3D4A3C4A3DC3D4AC$29.D4B3CD3C4B3D4B3CD7BDC7A4DBA4C3B
A3B4C3A4C$29.D4B7CAB7DB3A4D3B4D3AB3A4DBA4C7BDC7AC3D$30.3BA7CAB7D4ACD
4B3CD3C4B3D4B3CD7BDC7A4DB$32.4A3C4A3DC3D4ACD4B7CAB7DB3A4D3B4D3AB3A4DB
$33.A4C3BA3B4C3A4C3BA7CAB7D4ACD4B3CD3C4B3D4B$33.A4C7BDC7AC3D4A3C4A3DC
3D4ACD4B7CAB7DB3A$34.3CD7BDC7A4DBA4C3BA3B4C3A4C3BA7CAB7D4AC$36.4D3B4D
3AB3A4DBA4C7BDC7AC3D4A3C4A3DC3D4AC$37.D4B3CD3C4B3D4B3CD7BDC7A4DBA4C3B
A3B4C3A4C$37.D4B7CAB7DB3A4D3B4D3AB3A4DBA4C7B.C7AC$38.3BA7CAB7D4ACD4B
3CD3C4B3D4B3CD7B.C7A$40.4A3C4A3DC3D4ACD4B7CAB7DB3A4D3B4.3AB3A$41.A4C
3BA3B4C3A4C3BA7CAB7D4ACD4B7.4B$41.A4C7BDC7AC3D4A3C4A3DC3D4ACD4B8.B$
42.3CD7BDC7A4DBA4C3BA3B4C3A4C3B9.B$44.4D3B4D3AB3A4DBA4C7B.C7AC$45.D4B
3CD3C4B3D4B3CD7B.C7A$45.D4B7CAB7DB3A4D3B4.3AB3A$46.3BA7CAB7D4ACD4B7.
4B$48.4A3C4A3DC3D4ACD4B8.B$49.A4C3BA3B4C3A4C3B9.B$49.A4C7B.C7AC$50.3C
D7B.C7A$52.4D3B4.3AB3A$53.D4B7.4B$53.D4B8.B$54.3B9.B!
#C [[ THUMBNAIL THUMBSIZE 2 ]]

Even the single t-shaped tile by itself can tile the plane periodically.

Or am I missing something obvious here? The aperiodic-only requirement isn't clearly stated in the question, I guess, but if that constraint is removed, a single domino tile can tile the plane aperiodically.

[toroidalet]

It appears that I didn't check thoroughly enough. This pair of polyominoes (36) might work, though:

Code: Select all

x = 16, y = 6, rule = //5
10.2A2.2A$2.A7.2A.3A$.4A5.5A$4A7.5A$2.2A8.A.2A$3.A10.2A!

(Based on the matching-free variant of the Trilobite and Cross tiles (Figure 9, right before the references section))
I'm still testing, so I haven't confirmed they are aperiodic yet.

[hkoenig]

If you'd like to get nice sets of physical polyominoes, see Kaidon Enterprises--

http://www.gamepuzzles.com/polycube.htm

I've bought several sets of polyominoes, polyiamond, polycubes and Penrose tiles from them.

[Moosey]

If you'd like to get nice sets of physical polyominoes, see Kaidon Enterprises--

http://www.gamepuzzles.com/polycube.htm

I've bought several sets of polyominoes, polyiamond, polycubes and Penrose tiles from them.

I’m a little irritated how they call the R-pentomino an F-pentomino. Also S->N, O->I, Q->L, but the R would be the one that would get on my nerves.

[hkoenig]

The original names in Golomb's 1965 book were T--Z, and F, L, I, P, N.

[Moosey]

The original names in Golomb's 1965 book were T--Z, and F, L, I, P, N.

I guess they make more sense that way

[Lewis]

Bumping this truly ancient thread because I built another nice thing with a full set of pentominoes, hexominoes, heptominoes and octominoes a little while ago:



It's not as impressive as the big 'bulls-eye' solution linked to further up the thread but I think it's still kinda cool. All found by hand; here's another picture of it, hogging some serious table real estate:
https://1.bp.blogspot.com/-gR5crtyJwbQ/ ... ntric1.jpg

[hkoenig]

Impressive! Is that using the Kadon sets?