Turtle tiles part 2: Recursive Kagome lattices and colour waves This is a direct sequel to the last post on the Turtle Tile. Please read that first to make sense of this post. This post has also been heavily edited from a previous version to fit with new insights ...

I also reposted the second post in the series with major edits: mathblock8128.wordpress.com/2024/06/08/t...

Turtle tiles part 3: Forced patches and metatiles This is the third of a series of posts on the Turtle tile. Reading the first post is highly advised. You should be able to skip the second post, but a theorem from that post will be restated without p...

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: mathblock8128.wordpress.com/2024/06/08/t...

Turtle tiles part 1: Forcing non-periodicity This is a rework of a previous post proving that the Turtle tile is non-periodic, to be used as a starting point for further explorations. A edit has been made to change the defined term 'irrational' ...

The previous post: mathblock8128.wordpress.com/2024/03/20/t...

Turtle tiles part 2: Properties of coloured Kagome lattices This is a direct sequel to the last post on the Turtle Tile. Please read that first to make sense of this post. In accounts of the discovery of the Hat aperiodic monotile such as [1], one of the first...

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. mathblock8128.wordpress.com/2024/04/13/t...

Turtle tiles part 1: Forcing non-periodicity This is a rework of a previous post proving that the Turtle tile is non-periodic, to be used as a starting point for further explorations. In March 2023, an aperiodic monotile was revealed to the publ...

Reworked my old proof that the Turtle tile is non-periodic:

This version explicitly uses two powerful properties(collation and irrationality) to carry out the proof.

The consequences of them go farther than I've posted about yet. My next blog post will explore those consequences.

Hello