New maths blog post! The fifth in my series about handling aperiodic tilings combinatorially with string-processing algorithms:

https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-refine/

In this instalment I've automated the process of converting a tiling substitution system into one for which you can build a deterministic transducer, if the starting system didn't already admit one. I discuss the algorithm, its limitations, and a particular success in which it found a more economical substitution system for the hat tiling than any I already knew of.

#TilingTuesday - #Hexagon decomposed into 288 kites.

A design by @DaniLaura on last Tuesday inspired this design :)

These tilings are based on the decomposition of a right kite (with two right angles) whose sides are in the ratio √2. This decomposition can be done with any right kite, but with that ratio one can make nice drawings in the kites which fit neatly with others in the resulting tessellation.
#TilingTuesday #geometry #tiling #mathart

Eye-popping floor tiles, Aswan, Egypt

#TilingTuesday snaking nonagons on a starry sky.

#TilingTuesday

Two different kinds of dodecagons.

(One set is only kinda sorta dodecagons, but you can see what I mean.)

Pavement tiling, Hurghada, Egypt

I have (re)discovered partitions of the rectangles defined by the square root of metallic ratios into similar rectangles. In the case of the golden ratio it was known, as can be seen in the excellent site https://tilings.math.uni-bielefeld.de/substitution/rectangulo-dorado/ .
I think the results are novel for the next ratios, here I present the partitions for the silver and bronze ratios. More complex partitions can be deduced from them.
In the fourth image there is a grid with the eight possible tesselations related to the golden ratio, depending on the orientation of the rectangles produced. All are non-periodic, but some look more regular than other. See continuation post for more.
#TilingTuesday #Mathematics #geometry #tiling

Hexagonal tiling

made with #p5js

Tiled cells done in the now delightfully antiquated Fractal Explorer. TallisV1 Cabs2 #tilingtuesday
One of the first freewares that got me into fractals.

#TilingTuesday
Created in #OneLab

Floor, Kalmar Castle

Pavement tiling, Fes, Morocco

A whole bunch of hex grid tessellation options (incl. more complex & colorful ones) for #TilingTuesday in this old thread...

https://mastodon.thi.ng/@toxi/112498348213405704

Pavement tiling, Ouarzazate, Morocco
#TilingTuesday #geometry #tiling #MathArt #photography #design #TravelPhotography #octagon

Monohedral triangle tiling of the gyroid, which is the dual tessellation of a partial Cayley surface complex of the group:

```
G = ⟨ f₁,t₁ | f₁², t₁⁶, (f₁t₁)⁴, (f₁t₁f₁t₁⁻¹f₁t₁²)² ⟩
```

Ball of radius 21. (1/2)

⋈ Bow ties hexagonal weaving

made with #p5js