Re: Chaos Theory - Yes!

Ryan Heise (rheise@nospam.socs.uts.edu.au)
Thu, 18 Jul 1996 15:56:16 +1000 (EST)

On Thu, 18 Jul 1996, Scott Hopwood wrote:

> On Thu, 18 Jul 1996, Ryan Heise wrote:
>
> I think you are giving fractal a bit of a loose definition. A fractal
> isn't any pattern, but a pattern that is self-similar at different levels
> of scale.

I agree with that definition.

> Chaos theory deals with finding order in *apparent* randomness.

True.

> True randomness will not compress, and fractal compression will only work
> on data that repeats at different scales. Unfortunately this is not a
> feature of many data sets.

And the question is: "Is there such thing as true randomness?"

> investigating *nature*. Few elements of data we deal with are natural,
> with the exception of natural images, sounds (not music, it seems that
> music designed along fractal concepts general ends up in elevators), and

I find fractal music amazing myself.

> stock exchange prices (think about it a second, and it is obvious they
> recursive, a little longer and you can see they are fractals. Spend some
> money to predict them, and you'll soon go broke).

It's those butterflies again...

Chaos theory leaves room for an ultimate formula which could predict the
stock exchange perfectly, however, the probability of creating something
perfect is 'zero'.

> Documents, while
> displaying patterns, are not fractal. Basically, gzip will always do a
> better job on the written word them then any fractal algorithm.

You will not find every possible fractal algorithm published - are you saying
that there is no room for development? I dissagree that gzip will _always_ do
a better job than fractal compression.

> > It is generally impossible to find one formula which will best compress a
> > whole file. The file must be divided into sections each with their own
> > fractal properties. By separating parts of the file, you can increase the
> > level of compression because different parts of the file may have fractal
> > properties very different from other parts of the file.
>
> As you split up the file, you have to store more information to put the
> pieces back together again. Because of this you start getting diminishing
> returns on each split. The first is a big gain, but the twentieth may use
> up more space then number nineteen.

That is correct.

> > > Sorry, this wouldn't work (IMHO). There are theoretical limits to how far
> > > you can push lossless compression (which is what you want for data, not
> > > images). An encyclopedia isn't fractal in nature (self similar at
> > > different scales), hence would not compress using this technique.
> >
> > It's amazing isn't it. No-one can pinpoint the exact limit but we are always
> > getting closer to it. Different data would have different limits ofcourse - a
> > file full of 0's would have an obvious limit - or not so obvious! Think about
> > it.
>
> No, actually the limit was pin-pointed quite some time ago. I don't know
> the details, and I'm a little to lazy to wonder down to the library for
> the sake of an argument but it does depend on the data set, and it depends
> on prior knowledge of the type of information being communicated (hence
> the not so obvious limit on the file full of '0's).
>
>
>>You are probably interested about how a fractal could compress an
>>encyclopedia. You are probably thinking about how fractals would not produce
>>the exact results and would cause misspelt words etc. This is not a problem
>>however - it is possible to find a level of compression that will produce
>>_exact_ results. For example, real numbers produced by the algorithm might be
>>rounded up or down to an absolute value like an ascii character. If the lower
>>character is wanted, the compressor would aim to produce a value less than 0.5
>>higher than the wanted character.
>
> Sorry, this is a case of the wrong algorithm for the job. If you look back
> to what fractal compression is, you will see that it just won't work on
> prose (unless artificially constructed). It relies on identifying patterns
> and encoding them, not just any pattern, but *fractal* patterns. If the
> patterns don't appear in the data - and that does not mean the data is
> random - you won't get any compression, even if there are other patterns
> that could be exploited.
>
> Part of the magic of the fractal compression is that you can display your
> image at some (nominal) resolution (of information, not image), then zoom
> in as far as you like to see more detail. Think about what this would mean
> to a word or sentence compressed in this manner. How can you print a
> word at half it's resolution? You can't. Words and sentences are all or
> nothing things, they have no similarity on different scale.

I agree that words are all or nothing which means that your whole paragraph
doesn't make complete sense. You say that a characteristic of fractals is
that you can zoom in - but why would you want to zoom 'past' the detail of a
word. Given a word 'Windows95' or maybe something like 'sucks', what more
information could you possibly get out of these words (hehehe).
Alternatively, why would you want to read a text file that has only been
half-decompressed ie. 80% of the target iterations. The iteration
level would be saved in the compressed file and it would not to go past the
minimum iteration level that accurately defines the word.

This is from research that a friend of mine did 2 years ago, so I can't give
you all the details.

>>Like I said before, you have to also forget about what people usually think of
>>as fractals - pretty patterns. You are the first person I have heard who has
>>said that an encyclopedia isn't fractal in nature. I am one of many who
>>believe (like to beleive) that everything is fractal in nature and everything
>>in nature is fractal and all of these fractals are derrived from super
>>fractals... and there is one super-duper-fractal. It's hard to believe this
>>becuase... well... it's hard to believe. The more you read about chaos
>>theory, though, the closer that statement seems to be to the truth.
>
> [I'm wincing as I say this] You're Wrong! [ouch! they hurt, I promise not
> to say it again].
>
> Fractals do not nature make,
> nature does the fractals make.
>

I don't mind if we have different opinions.

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"Don't anthropomorphize computers. They hate that."
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Ryan Heise rheise@nospam.progsoc.uts.edu.au