Wednesday, 11 January 2017

The Fractal Patch

Take part of a fern, zoom in - it looks like the whole fern. Take a part of that, zoom in again - it still looks like a fern. It displays 'self-similarity' - a part looks like a miniature version of the whole. Welcome to wonderful world of fractals.

As a small child I remember seeing a small twig lying on the ground. I held it up to the sky and squinted a bit.

"I'm a giant", I thought, “I'm holding up a tree.” I'd seen my first example "self-similarity" After that, I didn't give it a lot of thought - until now.

Objects are  self-similar - if a part looks like a  miniature version the whole. It’s something that occurs a lot in nature – ferns, trees, clouds, rocks, crystals, coastlines, rivers, blood vessels.

Copies of the whole repeated at a smaller and smaller scale.

These are a kind of fractal and it's something I’ve been looking out for on the patch (not that I’ve seen a lot of coastlines!).

You could call fractals the geometry of nature.

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line,” as Benoit Mandlebrot, the inventor of fractal geometry said.

He realised that nature is ‘rough’, ‘messy’, ‘jagged’ so it can’t be described with the perfect ‘smooth’ shapes of traditional geometry.

Fractals  abound in  the plant kingdom – trees are a good example. When you look at a tree you a see what appears to be very complicated, messy shape. It's generated, however by very simple rules.

1)    Grow a bit
2)    Branch in two
3)    Grow some more (but a bit less than before)
4)    Branch again
5)    Grow some more (but a bit less than last time)
etc, etc.

The place where branching occurs – the node – is the start of what could be seen as two small-scale trees. In this way  so you generate self-similarity – you get mini-trees, in turn comprised of mini trees.  Complexity emerges from simple rules.

So the process of repeating smaller and smaller copies gives rise to fractal shapes. In nature this repetetion doesn’t continue indefinitely – 2 or 3 times in the case of a fern - more in the case of a tree.

Mathemetic fractals, such as the famous Mandlebrot Set do repeat infinitely. So the amount of detail is infinite.

Recent Patch Sightings
1/1 - Willow tit - in the garden
5/1 - Pink-footed goose - skein over
5/1 - Winter moth
19/1 - Mistle Thrush, Song thrush, Great Tit, Coal tit in song
Angelica - a globe of globes
- it's fractal
Ice crystals are fractal
A twig lying on the ground...
...looks like a small tree
when stuck in the ground
...because trees are fractal

Following simple rules gives rise to fractal shapes.

Change the rule that says 'branch at 45 degrees' (as above) to branch at 10 degrees and you get a completely different kind of tree (although I'm not quite sure what kind tree this is - maybe an acacia -

The previous 'trees' look too  symmetrical to be very convincing. In the real world there would be more randomness - caused by wind and sun direction, branches breaking, disease, and the rules not being followed exactly. Add a bit more randomness and it starts to look 'messy' enough - more like a real tree.

Leaf veins have fractal branching patterns - like rivers or lightning.

It's usually easy to tell where 'man-made' ends and 'nature' begins. Man-made objects are usually 'smooth', objects in nature are usually 'rough'.  Man-made object usually conform to the straight lines, rectangles, circles of traditional geometry - objects in nature don't.

Clouds are fractal.

Friday, 9 December 2016

Textures of Autumn

Autumnal Leaves, seeds, fruits, fungi, on the patch - arranged by colour (with maybe a couple of Autumn insects in there) - mouseover the picture for description, click for larger version.