Greek keys - Fractal


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A fractal pattern is one where part of the pattern looks like the whole. That involves it being nfinitely complex, which is a bit hard! But one way to approach it is to define the first generation, and then build up more generations. A Greek key is a simple pattern, which you can imagine made of squares. Replace each square with another Greek key, and then do that again, and you start to approach a fractal Greek key!

Four generations of a fractal Greek key

There are two problems. First, a Greek key is not a square but a rectangle. This doesn't seem to matter. Second, a Greek key is notoriously difficult to turn round corners. I wanted to make the fractal pattern to be one continuous line, so I have drawn straight lines (or dots) to get round the corners.


Generation 1: Fractal Greek Key - generation 1

Generation 2: Fractal Greek Key - generation 2

Generation 3: Fractal Greek Key - generation 3

Click here for the fourth generation! If it doesn't appear, try on a different browser - some don't seem to handle large files.



A repeated fractal Greek Key border

While making the previous fractal, I spent much time on the corners, generally failing. But it irritated me, so here is another version. This only has two generations - further generations being files under the technical category called "Life's to short...".

Since this is a more attractive pattern, I've repeated it in case anyone wants a "Greek key of Greek keys".


Generation 1: Fractal Greek Key - generation 1

Greek key of Greek keys
Generation 2: Fractal Greek Key - generation 2



A fractal double Greek Key border

The problems of this niggled at me. I wondered if it would be easier to do with a double key. Short answer - no... At first, while I could get an attractive pattern, it wasn't a continuous line. Rather than waste it, I redefined it as a maze, and put it on my maze website as a Big Spiral Maze.

I continued, but was still having problems with those corners. Here is my final attempt. The Greek keys are packed much closer together, which has caused its own problems. The corners are now zigzags rather than Greek keys. And again, two generations only, I'm afraid!


Generation 1: Double Greek Key Fractal - generation 1

Generation 2: Fractal - generation 2

Here is a line of them, plus colouring to show the four different lines which go to make the pattern.

Greek key of Greek keys

Greek key of Greek keys



A fractal turning Greek Key

This whole project seemed to remind me more and more of unicursal mazes, so I tried a different approach based on Roman mazes. I start with a Roman style maze, but I adapt it to become a continuous line. Then I repeat this pattern four times, but alter it in the centre so the four sections join up. Then I carry on doing that...


Generation 1: Fractal turning Greek Key - generation 1

Generation 2: Fractal turning Greek Key - generation 2

Generation 3: Fractal turning Greek Key - generation 3

Generation 4: Fractal turning Greek Key - generation 4

This all gets a bit Op Art, so here is some colouring to help. The different colours show the various generations embedded in generation 4. I assure you that there is one line throughout (although with different colours in this version). I haven't just repeated the original pattern as that would give lots of different lines.

Fractal turning Greek Key - coloured

Another colouring: Here I coloured in the white bits. I found that there was a continuous area (yellow) and some broken up areas. These seem to form a tiling pattern which I have picked out. There are a few cases where coloured lines cross each other, but I found it so pleasant that I tolerated it.

Fractal turning Greek Key - coloured

The following is a more honest colouring of the previous idea. The discrete background areas are coloured differently (properly). Now this DOES start to look like a fractal!

Fractal turning Greek Key - coloured