Oliver Steele

Tilings

Here's a link for context https://www.ias.ac.in/public/Volumes/reso/011/06/0042-0050.pdf

Smith's paper is paywalled, https://muse.jhu.edu/article/600574/summary

Jacob Rus

This one is good: http://archive.bridgesmathart.org/2020/bridges2020-191.pdf

A few more links:  
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.297.4115&rep=rep1&type=pdf  
http://arearugscarpet.blogspot.com/2014/04/the-curse-of-truchets-tiles.html  
http://mypages.iit.edu/~krawczyk/rjkisama11.pdf  
http://jacques-andre.fr/faqtypo/truchet/

Thank you! I have put some of these here, some of these in my other notebook, and some both places. The Cameron Brown article is particularly useful – I originally got into this in order to try halftoning effects, that he demonstrates.

Actually what most caught my eye in this notebook was the relation of these tiles to http://tamivox.org/dave/train/index60.html

With physical realization http://www.tinkeringtechie.com/projects/woodworking/toys/wooden-train

A generalized algorithm for polygons having 8 and 12 sides would make it possible to use truchet tiles in conjunction with archimedean tilings - see https://observablehq.com/@esperanc/archimedean-tilings Just itching to do it!

Claudio Esperança

There's still a tiny bug in the all-corners case, it seems:

https://i.imgur.com/qMpCr8P.png

(slightly above the middle)

Job van der Zwan

That's just the result of attaching wings to a half-plane tile where the plane takes up the majority of the tile, def looks a bit odd though. Probably should just replace that tile with a completely black one.

Sylvan Wu

I'm not following. All other tiles result in continuous connections though, what makes this one special?

There’s nothing special about it, its strange appearance is more coincidence that it being unique. The collection that this notebook is in has another notebook that explains how winged tiles are made. 

Basically when adding wings, the small black circle thats added on the edge that doesn’t touch the half-plane lines up near perfectly with the half-plane, which gives it its odd look.

I’m sincerely sorry for my terrible English.

Your English is great, don't worry about it! I'll re-read the notebooks and see if I can make sense of why this is the logical consequence of the construction

Thank you both for the discussion. I believe Sylvan Wu is correct, and I agree with his suggestion to replace the tile by an all-black one.

Here is an illustration that may make it clearer:

https://imgur.com/a/1zgclb7

In the upper left corner, edges 0 and 2 are connected; the upper right corner adds wings. In the lower left corner, edges 3 and 5 are connected; the lower right corner adds wings to this tile. The issue with the lower right corner is that very little of the arc that connects edges 3 and 5 remains visible; the white dots over the vertex obscure most of the arc, and make it appear broken.

I actually tried out removing large arcs; there is some commented-out code that does this. This made the tiles less interesting. The Truchet-Carlson square tiles added another tile type that the code in this notebook does not generate, which is an all-black tile. I suspect that removing the problematic large-arc tiles and adding all-black tiles will be more interesting, without the broken-looking tiles. I hope to try that the next time I have a chance to work on this.