teundenouden's avatar


Teun den Ouden
247 Deviations

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:
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.

Join the community to add your comment. Already a deviant? Log In
Currently bezo97 is teaching me to program with processing and it is a lot of fun. I am trying to get hold of the IFS fundamentals. Hopefully this will make me able to understand and explain how fractals are formed. So far we've succesfully made a little program that plots a sierpinski gasket. In a previous version I added color but in this one we wanted to display the log-density like it does in apophysis. A very neat feature is that the sierpinski changes shape when moving the mouse! My homework for now would be to add weights to the transforms.

Im having sooo much fun :D
Sierpinski Log by teundenouden
Join the community to add your comment. Already a deviant? Log In


1 min read
[Update 6-6-2018]: Succesfully re-rendered the image, however the spiral seems to be dissapearing when using different size. I fixed this using some curve adjustments.


Currently im re-rendering my deviation "Shattered Liquorice".
Both resolution (width in px) and density will be 10 000. Since this fractal is super glitchy I've tried many different ways to achieve this, but none found to be succesful.
So I've chosen to render it regulary as an unsliced image, with the 32-bit version of apophysis using a single core. So far everything seems to go fine and my pc has been rendering for 50 hours and 58 minutes. The remaining rendertime is estimated to be somewhere around 7 hours more.
Hopefully there will be no gnomes visiting in my sleep and pull out the powercord.
Join the community to add your comment. Already a deviant? Log In
Dear Ladies, Gentlemen and Genderfluids,

Since I am following some accelerated math courses, I will be inactive for probably one or two weeks.
I have been lazy and need a kick in the butt. So I will disconnect my computer in order to eliminate distractions of any kind.
Hopefully I will pass my tests, but for now I must go all out.

I apologize for the inconvenience, but I hope you understand.

Yours Sincerely,
Join the community to add your comment. Already a deviant? Log In
Cycle Of Life by teundenouden
Won 3th price with this at a gallery in Delft (the Netherlands) :)

Jesus by teundenouden
Join the community to add your comment. Already a deviant? Log In

About Fractal substitution tiles by teundenouden, journal

Learning processing by teundenouden, journal

Waiting... by teundenouden, journal

Absent for a while by teundenouden, journal

Just some more :) by teundenouden, journal