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.
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”. *
*The formal definition for one loop (input>rule>output) is an iteration.
Here are some examples of rep-tiles:
The same as a rep-tile, except for its smaller copies that will be different in size.
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.
" (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.
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.
This happened because both, the triangle and parallelogram, only consist out of smaller copies of themselves.
This however is not a rep-tile
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.
* 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.
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.
Make tilings like these
The shapes on the left and right side of the arrow are exactly the same.
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.
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
Too much math?
Play around and learn from pre-made tiles. Here are some free parameter packs:
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:
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
Thanks a lot for reading and if you can't see the tilings for the tiles, you are free to send me a note.