Uniformly-Random 12x12 with 2x2 Border

This is a single arbitrary balanced tiling of a 12x12 area surrounded by a border of 2x2 cells. This represents 1 out of the 60,736,978,008,507,298 (60.7 quadtrillion) such balanced tilings that could possibly exist in this area.

It is also part of the 0.042% of tilings which contain at least one 4x4 tile. There are "only" 25,264,337,978,569 (25.3 trillion) such tilings, or approximately 1-in-2403.

This artwork represents the advancement of my paths project from balanced quadtree tilings to arbitrary balanced tilings (which may or may not be valid quadtrees). This allows the program to select from a truly uniform distribution of all possible tilings, instead of favoring only a small subset of outputs like the quadtree

This diagram is a part of my paths art project, and has enabled me to integrate paths that can split and combine to reach different scales. As of last month I have resumed work on this project after a multi-year hiatus, thanks to some unexpected health improvements!

The layout of square tiles in this diagram is subject to a ruleset often known as "balancing" or "restriction". Any given square of size 2^k can only border tiles of size 2^k-1, 2^k, or 2^k+1. In practice, this means that 4x4 tiles cannot touch 1x1 tiles, and 8x8 tiles cannot touch 1x1 or 2x2 tiles, and so on... Additionally, there are rules governing the relative positions of tiles. Specifically, a vertex (corner) of any tile may only touch either [1. other vertices] or [2 the midpoint of a tile's edge]. In practice, this means that tiles of size 2x2 or greater (but not 1x1 tiles) may be placed at a 50% offset, relativel to other tiles of the same size, along one edge axis.

This artwork was minted specifically for Twitter user @NftTrees, who asked me to mint it after I shared it on my Twitter account (https://twitter.com/mathmakesart). Thank you KaiTrees, as always, for your generous offer!