# gst: overlapping substitutions and tilings.

AA << generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. For such a substitution, the substitution matrix may have non-integer entries. (They) also show that the expansion constant is an algebraic integer under mild conditions. In general, overlapping substitutions may yield a patch with contradictory overlaps of tiles, even if it is locally consistent. >>

AA << give a sufficient condition for an overlapping substitution to be consistent, that is, no such contradiction will emerge. (They) also give many one-dimensional examples of overlapping substitution. (AA) finish by mentioning a construction of overlapping substitutions from Delone multi-sets with inflation symmetry. >>️

Shigeki Akiyama, Yasushi Nagai, Shu-Qin Zhang. Overlapping substitutions and tilings. arXiv: 2407.18666v2 [math.CO]. Aug 30, 2024. 

https://arxiv.org/abs/2407.18666

Also

keyword 'tile', in FonT

https://flashontrack.blogspot.com/search?q=tile 

keyword 'overlap', 'overlapping', in FonT

https://flashontrack.blogspot.com/search?q=overlap 

https://flashontrack.blogspot.com/search?q=overlapping

Keywords: gst, tiles, overlap, overlapping

# gst: riddles of the sphinx tilings, when at low temperatures you could find a chiral order.

<< The sphinx tile, a hexiamond composed of six equilateral triangles in the shape of a sphinx, has interesting and complex tiling behavior due to its chirality and asymmetry. >>️

<< for larger frames (AA) introduce a Monte Carlo (MC) method to create randomized tilings. Key to the latter is the identification of fundamental shapes ("polyads") that have multiple tilings and allow a rejection-free MC move once the polyad is identified. >>

AA << also introduce an Ising-like chiral interaction energy between neighboring sphinx tiles and study how the system behaves as a function of temperature; at low temperatures, chiral ordering is found. >>️

Greg Huber, Craig Knecht, Walter Trump, Robert M. Ziff. Riddles of the sphinx tilings. arXiv: 2304.14388v2 [cond-mat.stat-mech]. May 1, 2023.

https://arxiv.org/abs/2304.14388

Also: 'chiral', 'disorder' in https://www.inkgmr.net/kwrds.html

Keywords: gst, sphinx, sphinx tiles, chirality, asymmetry