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Hi there,

I'm studying the 3-6 Kisrhombille euclidian tiling thanks to the person behind Fleen.org (http://www.fleen.org/) (see FF post (http://www.fractalforums.com/general-discussion-b77/fractals-made-from-shape-languages-and-fancy-geometry/)).
I'm creating a new topic just to share interestings facts about the tiling itself and not about his awesome fractals.

I'm naming the triangle whose angles are 30,60,90 degrees and sides measures sqrt(3),2,1  (http://mathworld.wolfram.com/30-60-90Triangle.html)"Équerre" because there's no name for it. It means "set square" in french and set squares are either sqrt(2),1,1 (which is called the Isosceles Right Triangle) or this triangle. Plus the word looks like Escher, and I worship him.

The Équerre triangle is a rep-tile (https://en.wikipedia.org/wiki/Rep-tile), meaning it can be divided into three new Équerre sub-triangle (by taking one point in the middle of the hypothenuse and one other point on the other long side at 1/3 from A). If you do this subdivision multiple times you obtain a 3-6 Kisrhombille Euclidian tiling (called Equerre tiling because simpler to write) inside your first triangle.

(http://mathworld.wolfram.com/images/eps-gif/30-60-90Triangle_1000.gif)

Equerre tiling features hexagons (12 connection vertex), equilateral triangles (6 connection vertex) and rhombis (4 connections vertex). I really like this tiling because it's a good mix between classic square tiling, the equilateral triangle tiling and the hexagon tiling. It's like using base 12 instead of base 10, it's more convenient :) .

(http://nocache-nocookies.digitalgott.com/gallery/20/4891_03_07_17_8_33_41.jpeg)

Alternative explanations for the tiling :  (https://github.com/johnalexandergreene/Geom_Kisrhombille/blob/master/README.md)

The dual of this awesome face-transitive tiling is the magnificient vertex-transitive Truncated trihexagonal tiling (https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling) which feature dodecagons, squares and hexagons. You can obtain the dual by taking the barycenter with (1,2,sqrt(3)) coefficients (click on the image to go to the shader) :

(http://nocache-nocookies.digitalgott.com/gallery/20/4891_03_07_17_8_34_53.jpeg)
 (https://www.shadertoy.com/view/XdBfRz)

My shaders studying this tiling : Shadertoy Equerre search (https://www.shadertoy.com/results?query=tag%3Dequerre)
The images are available in this gallery : Equerre Gallery (http://www.fractalforums.com/index.php?action=gallery;su=user;cat=732;u=4891)


Here are the images extracted from the shaders :
(http://nocache-nocookies.digitalgott.com/gallery/20/4891_03_07_17_8_25_53.jpeg)

(http://nocache-nocookies.digitalgott.com/gallery/20/4891_03_07_17_8_37_23.png)

(http://nocache-nocookies.digitalgott.com/gallery/20/4891_03_07_17_8_33_12.jpeg)

I've found something like a Koch curve, but not very pretty yet :

(http://nocache-nocookies.digitalgott.com/gallery/20/4891_03_07_17_8_32_24.jpeg)

I like first, maybe becouse of colours.
The link https://www.shadertoy.com/view/XdBfRz could get you hipnotised ;)
But I hope for complex images like in github site.

I remember simmilar images with octagonal symmetry:
I just found:
http://www.orchidpalms.com/polyhedra/tessellations/type2.htm
http://www.orchidpalms.com/polyhedra/tessellations/type10.htm

(http://www.orchidpalms.com/polyhedra/tessellations/type10cell.gif)
(http://www.orchidpalms.com/polyhedra/tessellations/type2a.gif)

These are national ornaments of some nations. Probably this just means that this type is instrincaly atractive to humans.

There are evolutionary explanations for that. Some human neirons can notice snake by repeating pattern.
(http://www.knowpickens.com/press/snake10-9-13a.jpg)
(http://media.new.mensxp.com/media/content/2017/Apr/can-you-spot-the-snake-in-this-picture-652x400-2-1493218027.jpg)

Added uploaded some octagonal based gif I don't know where I got. in North East Europe and further east these are kind of national ornamentation. The basis seems simmilar (throught I realy don't knwo what is happening).