mathblock - Network

The fourth post in my Turtle tile series is out. Here I tie up some of the loose ends I left earlier in the series to obtain a full proof...

The fourth post in my Turtle tile series is out. Here I tie up some of the loose ends I left earlier in the series to obtain a full proof that the Turtle tile is an aperiodic monotile and that its tilings imply a hierarchical metatile tiling: https://mathblock8128.wordpress.com/2024/08/31/turtle-tiles-part-4-flush-lines-and-deflation-schemes/

The third post in my Turtle tile series is up. Here I show that metatiles(of some sort) always arise in a tiling without using a computer...

The third post in my Turtle tile series is up. Here I show that metatiles(of some sort) always arise in a tiling without using a computer search: https://mathblock8128.wordpress.com/2024/06/08/turtle-tiles-part-3-forced-patches-and-metatiles/

The previous post: https://mathblock8128.wordpress.com/2024/03/20/turtle-tiles-part-1-forcing-non-periodicity/

The previous post: https://mathblock8128.wordpress.com/2024/03/20/turtle-tiles-part-1-forcing-non-periodicity/

I have posted a follow-up to my previous post on the Turtle #aperiodic #monotile. While it is rather dry, the result at the end is...

I have posted a follow-up to my previous post on the Turtle #aperiodic #monotile. While it is rather dry, the result at the end is significant for constraining Turtle tilings. https://mathblock8128.wordpress.com/2024/06/08/turtle-tiles-part-2-recursive-kagome-lattices-and-colour-waves/

The broad strokes of the proof are the same as in the mathstodon post below, but I have explicitly incorporated two powerful...

The broad strokes of the proof are the same as in the mathstodon post below, but I have explicitly incorporated two powerful properties(collation and irrationality) into the proof.

Both properties presuppose the existence of a kagome lattice whose lines come in two colours. The first part of the proof establishes this.

These properties restrict the Turtle tilings much more deeply than I've posted about so far. That will be covered in the next blog post.

https://mathstodon.xyz/@mathBlock/110718945589590196

I've reworked and reposted my proof that the Turtle tile is...

I've reworked and reposted my proof that the Turtle tile is non-periodic:
https://mathblock8128.wordpress.com/2024/03/20/turtle-tiles-part-1-forcing-non-periodicity/

Here are versions of the new metatiles for the hat tiling.

Here are versions of the new metatiles for the hat tiling.

In a private conversation with @pieter he mentioned a technical detail about tilings of the Turtle #aperiodic #monotile. This led me to...

In a private conversation with @pieter he mentioned a technical detail about tilings of the Turtle #aperiodic #monotile. This led me to consider that a particular tile(picture 1) may be interesting if just one were placed in a plane otherwise tiled by Turtle tiles. Since this tile is mirror-symmetric, it cannot decide which reflection of Turtle tiles is predominant. If we choose ourselves and surround this tile with Turtle tiles, then we get something that be assigned metatile edge decorations(picture 2).

We can calculate what the inflation of this new metatile must be, and it can be shown that the metatile forces its own inflation to emerge, which seems to suggest one example of this metatile in a tiling forces the position of all the other metatiles(hence Turtle tiles) in the plane.

The left of picture 3 shows the halo of metatiles around our new metatile, The black dotted outline is the outline of the inflation of our new metatile. However, the right of picture 3 shows a altered halo, suggesting another new metatile.

For clarity the metatile shown in picture 2 will be called the O metatile, and the other the I metatile. To get the I metatile, cut off some bits off of the O metatile(grey outlines parts in top left of picture 4). This also has a metatile representation
(bottom of picture 4), and it seems to also force the plane like the O metatile does.

@zenorogue I was working of something else when I saw something that was reminiscent of the binary tiling. That lead me to come up with the...

@zenorogue I was working of something else when I saw something that was reminiscent of the binary tiling. That lead me to come up with the tile below. Labelled lengths are apparent based on the half-plane representation, if I didn't mess it up.

For the Turtle #aperiodic #monotile, I have found a general rule that prevents the six-tile drainage loop in the left picture from occurring...

For the Turtle #aperiodic #monotile, I have found a general rule that prevents the six-tile drainage loop in the left picture from occurring in any plane tiling, but I have also found a 12-tile loop that that rule cannot exclude.

To explain the rule, first note that one of the corollaries of my non-periodicity proof(https://mathblock8128.wordpress.com/2023/08/15/the-turtle-prototile-is-not-periodic-a-simple-proof-2/) is that one colour of spar line is in the majority(about 72%).

The rule is that for 'adjacent' minority spar lines, there are either two or three majority lines in between. This excludes the first patch outline, but not the second.

The blog post at (https://mathblock8128.wordpress.com/2023/09/16/spar-lines-hexagons-and-tiling-a-finite-boundary/) has more details and...

The blog post at (https://mathblock8128.wordpress.com/2023/09/16/spar-lines-hexagons-and-tiling-a-finite-boundary/) has more details and further discussion. It is strongly suggested to read the previous blog post first. (https://mathblock8128.wordpress.com/2023/08/15/the-turtle-prototile-is-not-periodic-a-simple-proof-2/)

*Note on step 3: It may terminate w/o filling the outline if there exists a ‘loop’ of hexagons and fixed rhombs. Our example outline...

*Note on step 3: It may terminate w/o filling the outline if there exists a ‘loop’ of hexagons and fixed rhombs. Our example outline does not produce those, so it can be completely filled in.

Step 3*: Starting at a hexagon next to only one fixed rhomb, place Turtle tiles until you reach a hexagon next to three uncovered fixed...

Step 3*: Starting at a hexagon next to only one fixed rhomb, place Turtle tiles until you reach a hexagon next to three uncovered fixed rhombs, then find another one-rhomb hex.

If you do this with single Turtle tiles, note that you get one hexagon and one ‘non-free’ (i.e. fixed) rhomb next to each other.

If you do this with single Turtle tiles, note that you get one hexagon and one ‘non-free’ (i.e. fixed) rhomb next to each other.

Step 2: Group positive crossings into threes and draw hexagons over them. Draw rhombs over the negative crossings, decorating by whether the...

Step 2: Group positive crossings into threes and draw hexagons over them. Draw rhombs over the negative crossings, decorating by whether the crossing could be free or not. (Free crossing at bottom left of left picture).

Step 1: Draw the internal lead and lag lines which can be determined from the outline alone.

Step 1: Draw the internal lead and lag lines which can be determined from the outline alone.

A condensed verion of (https://mathblock8128.wordpress.com/2023/09/16/spar-lines-hexagons-and-tiling-a-finite-boundary/): How to (sometimes)...

A condensed verion of (https://mathblock8128.wordpress.com/2023/09/16/spar-lines-hexagons-and-tiling-a-finite-boundary/): How to (sometimes) solve a Turtle #aperiodic #monotile jigsaw puzzle given an outline to fill. Example outline shown below.

Made a follow-up to https://mathblock8128.wordpress.com/2023/08/15/the-turtle-prototile-is-not-periodic-a-simple-proof-2/, linked in reply....

Made a follow-up to https://mathblock8128.wordpress.com/2023/08/15/the-turtle-prototile-is-not-periodic-a-simple-proof-2/, linked in reply. (Hopefully no mistakes)

The original post has been lost due to technical issues. A repost can be found below:...

The original post has been lost due to technical issues. A repost can be found below: https://mathblock8128.wordpress.com/2023/08/15/the-turtle-prototile-is-not-periodic-a-simple-proof-2/

I have made an addendum to my proof that the Turtle prototile is not periodic. This is my attempt to prove that any Turtle tiling is aligned...

I have made an addendum to my proof that the Turtle prototile is not periodic. This is my attempt to prove that any Turtle tiling is aligned to a Laves tiling: https://mathblock8128.wordpress.com/2023/08/09/addendum-turtle-tilings-are-always-aligned-to-a-laves-subtiling/