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About Fractal substitution tiles

Sun Nov 18, 2018, 12:03 PM by teundenouden:iconteundenouden:

Fractal Art Weekend

Suggested prior knowledge:

  • Know how to use Apophysis or Chaotica (fractal editor software)
  • Know some IFS tiling basics (strongly suggested: rep-tiles tutorial ~by tatasz

Some examples of fractal tiles

Just so you know what I will be talking about.
From left to right: Gaucho ~by C-91, Tatlin at Home ~by dark-beam, Coven in circles ~by fardareismai, Low Symmetry ~by pillemaster, Trizaic ~by Senzune.

Twin dragon fractal by teundenouden Butterflies by teundenouden Cubatron Cobalt-01A by teundenouden hypertile-like 12-fold tile by teundenouden
And of course some tilings I did myself.

What is a tiling?

We want to seamlessly cover a surface with a pattern, using a fixed set of shapes (e.g. only parallelograms and triangles). However, it is only allowed to do this using linear transforms, hence it will have to be a substitution tiling. 

There are three types:

1. Rep-tile: A single shape that can be subdivided into smaller copies of itself.Take a square and divide it into 4 smaller copies of itself. This process can be carried out over and over again, causing a pattern as shown on this image:
Untitled Drawing by teundenouden
A single execution of this process is called a “rule”. *

Afbeelding8 by teundenouden

*The formal definition for one loop (input>rule>output) is an iteration.

Here are some examples of rep-tiles:
A selection of rep-tiles by teundenouden
(Try to build some, using the rep-tiles tutorial ~by tatasz)
2. Irrep-tile 
The same as a rep-tile, except for its smaller copies that will be different in size.
Fig218w2 by teundenouden
(from: mathafou)

For more patterns visit
Math Magic ~by Erich Friedman
If you have read the rep-tile tutorial you can also build irrep-tiles. You’ll just have to use multiple scales.
3. Advanced substitution tiles 
These tilings use multiple different shapes and one or multiple scales.
Detached House by teundenouden    Detached house by teundenouden
(substitution tiling) while the image on the right shows the same pattern as the fractal.

What is a substitution tiling?

It is a tiling of which its shapes are being substituted (= replaced) with a pattern. In contrast to rep- and irrep-tiles this pattern will consist of multiple different shapes.

The tiling has to be self-similar in order to make smaller copies fit. Therefore, the shapes should form a bigger shape which its outside border is exactly the same as the shapes it was made with. We didn’t have to worry about this when making rep-tiles, since those shapes were already self-similar.

So, if we want to make a tiling with, e.g. parallelograms and triangles, we need a pattern of shapes that form a bigger triangle and parallelogram. Only with matching outer shapes will make the substitution possible. For example, you can’t replace a circle with a square in a tiling of squares since that will result in leaving gaps or having overlap.
Afbeelding5 by teundenouden

The image below doesn’t show a single substitution tiling. Multiple different shapes are used here (triangle and parallelogram), yet they formed two separate rep-tiles.
Afbeelding3 by teundenouden
This happened because both, the triangle and parallelogram, only consist out of smaller copies of themselves.

This however is not a rep-tile anymore:
Afbeelding 2 by teundenouden
They still have the same outer-shapes, but their inner patterns are different. Every shape has its unique rule that will substitute that same shape with a combination of both shapes. Thus for 2 shapes you’ll need 2 rules.
More in detail: The rule of the triangle will divide every triangle into 3 smaller triangles and 6 smaller parallelograms. After this substitution it is applied again on all new formed triangles. * Apophysis keeps doing this infinitely.
Afbeelding4 by teundenouden
* I didn’t show the rule for the parallelogram, but both rules are applied simultaneously. 

Picking your first substitution tile

Take a look at the encyclopedia of tilings (~by D. Frettlöh & E. Harriss) You won’t be able to make all of them, so here are a few tips on picking your tiling:

  • Pick a tile with a “substitution rule”. This is its blueprint and shows which shapes are used and in what way. 
    • It is even more helpful a patch (=example of the tiling) is given.
  • Look for tiles of which its shapes and patterns have the same outer shape; 
  • Preferably with straight borders.

Naamloos-2 by teundenouden
(modified screenshot from Tilings Encyclopedia nov-2018)

Avoid tilings like these

The shapes on the left side of the arrow don’t match the shapes on the right. However, if you really want to, use crop to make it fit and find a way to fill-up gaps.

Rule by teundenouden    Watanabe by teundenouden    Rule Bow by teundenouden

From left to Right: Watanabe ito-soma 8-fold, Watanabe-ito-soma 12-fold and a bow-tiling.

Make tilings like these

The shapes on the left and right side of the arrow are exactly the same.

Penrose by teundenouden    Rule Equi by teundenouden    Trapezotriangular by teundenouden

From left to Right: Penrose (without rotations), Equithirds, Trapezotriangular.

Get Your hands dirty

Make sure you’ve followed these two tutorials: Advanced tiling:Advanced Linear Tiles Watanabe Ito Soma 12-fold tile: Workflow <<< which is one of the tougher tilings, just to show how complicated it can get. Jawdrop 

And here is a nice list of tiles you should definitely try for yourself:
Just pick one you like and try to work it out :happybounce: 

T2000 (almost the same as equithirds)

Too much math?

Play around and learn from pre-made tiles. Here are some free parameter packs:
Rep-Tile Param Pack by teundenouden 44 lineartiles for JWF (Apophysis compatible) by dark-beam

You can get even more parameters using IFStile. This is a nice piece of software created by Dmitri Mekhontsev.
Download IFStile here and this image should get you started:
IFStile by teundenouden

Can't get enough?

Once you are able to make fancy tiles you can start to spice them up even more.

If the tile has a triangular shape, try to hypertile it: Hypertiling a Tile - Chaotica Tutorial
Make a different tile using modulus instead of linears: Double modulus tutorial

Good luck!

Thanks a lot for reading and if you can't see the tilings for the tiles, you are free to send me a note.


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Teun den Ouden
Artist | Hobbyist | Digital Art


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Orange-Nam Featured By Owner Edited 3 hours ago  New Deviant Student General Artist
Thank you for your llama.Heart 
Vile-Shards Featured By Owner 9 hours ago  New Deviant
Thanks for the llama! I'm only new here so I'm still figuring things out.
But thanks! :happybounce:  
purpledragontreeart Featured By Owner 3 days ago  New Deviant Hobbyist General Artist
thanks for the lama
AsteraRivera Featured By Owner 3 days ago  New Deviant Hobbyist General Artist
Thank you for the llama! Here, have one in return! 3D Llama Badge 
Gurdim Featured By Owner 3 days ago
thanks for the llama :)
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