Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by pierrebai on Tuesday July 11, 2017 @04:50PM
(#14747932)
When they talk about the einstein, I assume they mean a shape that can only tile the plane non-periodically. If the tile is allowed to tile the plane both periodically and non-periodically, the solution would be obvious.
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Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by Terr_ on Tuesday July 11, 2017 @05:27PM
(#14748231)
> I assume they mean a shape that can only tile the plane non-periodically.
Yes, they probably [0] mean aperiodic, or "exclusively non-periodic".
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by GregBuchholz on Tuesday July 11, 2017 @05:14PM
(#14748145)
>If the tile is allowed to tile the plane both periodically and non-periodically, the solution would be obvious.
Maybe you can draw a sketch for those of us who aren't immediately seeing the obvious?
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by kevinwang on Tuesday July 11, 2017 @05:25PM
(#14748216)
Squares, but each row of squares has a random offset
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by irishsultan on Wednesday July 12, 2017 @03:21AM
(#14750744)
That is a periodic tiling (just think about what happens if you follow the tiling along the rows instead of along the columns, you'll periodically see the same pattern).
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by kevinwang on Thursday July 13, 2017 @03:19AM
(#14759366)
No, you wouldn't necessarily
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by irishsultan on Thursday July 13, 2017 @06:40AM
(#14760172)
https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_re...
It's listed as a periodic tiling. If you move the whole plane one square to the left or right you'll find that everything fits into place nicely, making this a periodic tiling.
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by kevinwang on Thursday July 13, 2017 @03:54PM
(#14764608)
Hm, I'm not knowledgeable about this subject but this seems like the offset on each row is NOT random, but it's instead repeating. Imagine instead that each time I added a new row, I chose the offset to be, say, a completely new value.
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by irishsultan on Wednesday July 19, 2017 @06:01AM
(#14803146)
Doesn't matter, because your new row will still be offset with the same random value to existing rows after moving one square to the left or right.
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by Terr_ on Tuesday July 11, 2017 @05:30PM
(#14748253)
Or triangles, with similar "slide along the seam" changes.
Re:Pentagon Tiling Proof Solves Century-Old Math Problem
by nimih on Wednesday July 12, 2017 @12:50PM
(#14754129)
If you allow reflections of the single tile, an easy counterexample is the pinwheel tiling[1], which is a non-periodic tiling composed entirely of isometric triangles.
An (imo less satisfying) example which does not require reflection of the tile would be as follows: Take a rectangle with width equal to twice the height. You can use this tile to create squares which are either split vertically or horizontally. Put a single horizontally split square at the origin, then tile the remainder of the plane with vertically split squares: this tiling is (rather trivially) not periodic.