Polyominoes

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Lewis
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Polyominoes

Post by Lewis » May 14th, 2019, 4:03 pm

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
29x29-squareholes.png (4.11 KiB) Viewed 7667 times

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Moosey
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Re: Polyominoes

Post by Moosey » May 14th, 2019, 4:18 pm

Lewis wrote: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)
Last edited by Moosey on May 14th, 2019, 8:02 pm, edited 1 time in total.
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fluffykitty
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Re: Polyominoes

Post by fluffykitty » May 14th, 2019, 4:29 pm

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!

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Moosey
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Re: Polyominoes

Post by Moosey » May 14th, 2019, 4:59 pm

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!
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Sarp
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Re: Polyominoes

Post by Sarp » May 14th, 2019, 6:43 pm

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.
WADUFI

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Moosey
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Re: Polyominoes

Post by Moosey » May 14th, 2019, 7:09 pm

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?
My CA rules can be found here

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fluffykitty
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Re: Polyominoes

Post by fluffykitty » May 14th, 2019, 8:26 pm

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.

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toroidalet
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Re: Polyominoes

Post by toroidalet » May 14th, 2019, 11:50 pm

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.
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Re: Polyominoes

Post by fluffykitty » May 15th, 2019, 1:47 pm

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!

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Lewis
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Re: Polyominoes

Post by Lewis » May 15th, 2019, 4:41 pm

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
20x38.png (11.94 KiB) Viewed 7593 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.
Moosey wrote: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.

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Re: Polyominoes

Post by Moosey » May 15th, 2019, 4:48 pm

Lewis wrote: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.
Moosey wrote: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.
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Re: Polyominoes

Post by PkmnQ » May 15th, 2019, 11:11 pm

Moosey wrote: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!
Last edited by PkmnQ on May 17th, 2019, 12:17 am, edited 2 times in total.
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Re: Polyominoes

Post by dvgrn » May 16th, 2019, 11:15 am

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
Y-pentomino used to make a larger Y-pentomino
rep12y.gif (4.84 KiB) Viewed 7543 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.

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Re: Polyominoes

Post by Moosey » May 16th, 2019, 11:50 am

dvgrn wrote: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.
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Re: Polyominoes

Post by Freywa » May 17th, 2019, 7:59 am

fluffykitty wrote: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!
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Re: Polyominoes

Post by toroidalet » May 24th, 2019, 3:48 pm

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!
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Re: Polyominoes

Post by dvgrn » May 24th, 2019, 4:09 pm

toroidalet wrote: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.

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Re: Polyominoes

Post by toroidalet » May 24th, 2019, 5:55 pm

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.
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Re: Polyominoes

Post by hkoenig » June 5th, 2019, 3:06 pm

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.

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Re: Polyominoes

Post by Moosey » June 5th, 2019, 4:49 pm

hkoenig wrote: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.
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Re: Polyominoes

Post by hkoenig » June 5th, 2019, 9:08 pm

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

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Re: Polyominoes

Post by Moosey » June 6th, 2019, 8:09 am

hkoenig wrote: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
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Re: Polyominoes

Post by Lewis » May 1st, 2020, 1:35 pm

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:

Image

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

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Re: Polyominoes

Post by hkoenig » May 1st, 2020, 5:58 pm

Impressive! Is that using the Kadon sets?

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