Warp wrote:

>   That's one thing I like about the Mandelbrot set: At first sight it
> seems that there isn't really all that much variation, but if you just
> keep exploring you'll find some surprises.

Indeed.

Originally I would just sort of explore at random. After a while you get 
to know a few areas such that you can re-find them on que. But that's 
not terribly exciting.

Later I spent some time working out the topology of the M set more 
clearly. (I even have some mathematical rules that describe how the 
periodic cycles work.)

Gradually I came to understand that particular shapes repeat around 
things. Find a 3-way fork, zoom in, and you'll find minibrots decorated 
with 3-way forks. (I quickly discovered that the negative tail has lots 
of nie thin filaments that trace the internal structure of the other items.)

And then, after exploring the M set for years, I discovered something 
completely unexpected: you can find mini Julia sets in there too! Real 
Julia sets have 2-fold symmetry, but these mini copies have in their 
interior shapes with 4-fold symmetry. And then 8-fold, 16-fold, and so 
on, until you find a minibrot at the center.

And then, on exploring further, I discovered that there's a mini Julia 
at every "junction point" inside, not just at the middle. But the ones 
at other junctions have more complicated (and interesting) shapes. For 
example, see

http://www.zazzle.com/MathematicalOrchid/product/228495105959465679

It's a normal "seashell" Julia, but bend into an S-shape. Most unusual.

And then, by choosing mini Julias inside, and going through multiple 
non-central junctions, you can come up with really weird and wild shapes 
[which inevitably end up looking a tad samey after a while].

I'm sure there is still plenty to be discovered in there...

> http://warp.povusers.org/snaps/fract/fract43.jpg

I like this.

> http://warp.povusers.org/snaps/fract/fract16.jpg

I wouldn't call this "unusual", but it is very beautiful.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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