Maths question: How do you calculate the fractal length of real things?

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like the coast of britain or a broken eggshell?

I get that you can do

D = log N / log S

(where N=Number of bits in the whole and S=ratio of bits to the whole)

and I can see how this is straight forward for regular maths type fractals like the Koch curve but where do I get N and S from for doing real calculations relating to the real world?

― hmmm (hmmm), Tuesday, 20 April 2004 13:37 (twenty-two years ago)

i think the point is that you get to DEFINE them in order that this question has an answer

(seeing as fractals arose as a way of giving an answer to a question that previously didn't have one - viz what is the length of the coast of britain? - before fractals the answer was "IT DEPENDS!")

― mark s (mark s), Tuesday, 20 April 2004 13:41 (twenty-two years ago)

ie equivalent to the way you decide at which point in the decimal you will start rounding up

― mark s (mark s), Tuesday, 20 April 2004 13:43 (twenty-two years ago)

So how would you go about defining them?

― hmmm (hmmm), Tuesday, 20 April 2004 13:49 (twenty-two years ago)

http://www.property-director.com/images/property-director/tape-measure.jpg

― RJG (RJG), Tuesday, 20 April 2004 13:51 (twenty-two years ago)

deciding at what size the measurement gets too small to count i guess - then translating that into those two terms

― mark s (mark s), Tuesday, 20 April 2004 13:56 (twenty-two years ago)

So I pick what size of bit I want (basically determined by how exactly I am able to measure), estimate N, estimate S and then plug them into the equation?

x-post

Hooray!
Cheers. Interesting, very interesting. I like that it's more of a measurement of inadequacy than of exactness.

― hmmm (hmmm), Tuesday, 20 April 2004 13:59 (twenty-two years ago)

stab it.

― Nellie (nellskies), Tuesday, 20 April 2004 15:01 (twenty-two years ago)

There's no maximum for this value - it's the kind of shape that tends towards infinite length the more you reduce the scale. Except given the way the sea works, I think there is some practical limit to the scale you can consider - I mean, are we talking high or low tide? And even without that, molecules aren't infinitely small. The minimum length I guess would be if you cut out a model in wood or something and wrapped a piece of string around it. This would be a useless value, but there is no clear right scale on which to measure.

― Martin Skidmore (Martin Skidmore), Tuesday, 20 April 2004 16:48 (twenty-two years ago)

Er, yeah, fractals aren't real?

― mei (mei), Tuesday, 20 April 2004 21:20 (twenty-two years ago)

I don't understand what you mean.

― Martin Skidmore (Martin Skidmore), Tuesday, 20 April 2004 21:28 (twenty-two years ago)

ever seen Romanesco Brocolli. Tell me fractals aren't real.

― Ed (dali), Tuesday, 20 April 2004 21:30 (twenty-two years ago)

http://www.midcitynursery.com/seeds/vegpics/56.jpg

― Ed (dali), Tuesday, 20 April 2004 21:31 (twenty-two years ago)

Perhaps the point being made is that in the real world, unlike in fractals, the repeating self-similarity on smaller scales has a limit.

― Martin Skidmore (Martin Skidmore), Tuesday, 20 April 2004 21:32 (twenty-two years ago)

all maths is "not real": numbers don't exist except according to a bunch of "approximation" rules of identity and equivalence which go all squiffy soon as you start looking hard

(eg what does it mean to say there are two of something?: if they're identical there can only be one; viz they are identical except for the property of not being the same as one another = they are different = what does "there's two of them" even mean?)

the problem of reality vs abstraction is common to fractals and to number generally

― mark s (mark s), Tuesday, 20 April 2004 21:34 (twenty-two years ago)

haha i just noticed a giant big unclarity in what i said before: "the measurement gets too small to count" = "count" as in "mean anything in ref yr practical requirements"

as with all measurement, the approximation you make will be reflecting the purpose you are are measuring for

eg do you want to drive a powerboat round the shores of britain and want to know how much petrol to buy; or are you planning to plant a daisy for every inch of coastline and want to know how many seeds to buy or whatever on down

― mark s (mark s), Tuesday, 20 April 2004 21:38 (twenty-two years ago)

hey this here is exactly the issue where that "theory of everything" programme screwed upso badly: it assumed that "infinite divisibility" and "possibility of self-similarity top to bottom" were things that ordinary non-maths trained punters take for granted, but actually they are totally NON-obvious: a requirement of mind it takes mathematicians a lot of practice to instil in themselves (largely bcz it is false in the "real" world, but theoretically convenient in order to learn euclidean/cartesian geometry and newtonian physics) (ie superspecial cases we shd perhaps instead be be FORGETTING??)

― mark s (mark s), Tuesday, 20 April 2004 21:45 (twenty-two years ago)

Wait, what "theory of everything" program? Or wait, you mean TV program(me), don't you.

― Casuistry (Chris P), Tuesday, 20 April 2004 21:50 (twenty-two years ago)

mathematicians: infinity! DO YOU SEE!
non-maths-type punter: no! infinity is silly
mathematicians: haha you dumb-ass!
*later*
mathematicians: actually you were right, infinity IS silly, we were just saying all that stuff for er some very excellent good reason that doesn't really matter any more

― mark s (mark s), Tuesday, 20 April 2004 21:51 (twenty-two years ago)

yes, the TV programme, plus it was called something less melodramatic when it screened in the US also

― mark s (mark s), Tuesday, 20 April 2004 21:53 (twenty-two years ago)

Perhaps the point being made is that in the real world, unlike in fractals, the repeating self-similarity on smaller scales has a limit.
-- Martin Skidmore (lonewolf.cu...), April 20th, 2004.

That's exatly what I mean.

Also, and I can't think of an easy way to put this, but anyway...

It's okay to talk about a maths circle of radius 1 metre and compare it to a real life approximate circle of radius approx 1 metre.

But when you compare a maths fractal to the real life equivalent, you're talking about one infinitecimal thing compared to another, and even two things which are approximately the same size (ie infinitecimal) can be competely different sizes.

I've got this really strong feeling in my mind which I can't put words to. I need to think a bit more.

― mei (mei), Tuesday, 20 April 2004 22:11 (twenty-two years ago)

yes but that's exactly the problem fractals was invented to deal with: it's a way of correctly approximating for shapes or surfaces or whatever where an approximation chosen to operate at the wrong level can potentially introduce errors of catastrophically the wrong-size amount

fractals is a means by which you decide on the approximation which gives you the information you need - ordinary approximation with linear measuring does this too, but with linear measuring the scaling happens correctly pretty much of its own accord (ie when you round up a centimetre it doesn't oops by mistake add five kilometres - whereas if you measured the coast of britain with a five centimetre stick (which is probably roughly rounding up or down by a centimetre) you will perhaps add many unnecessary kilometres to yr speedboat round-trip mileage)

― mark s (mark s), Tuesday, 20 April 2004 23:51 (twenty-two years ago)

(which is probably roughly rounding up or down by a centimetre) = (which probably involves roughly rounding up or down by a centimetre every time you plonk the stick down)

― mark s (mark s), Tuesday, 20 April 2004 23:52 (twenty-two years ago)

Hmm. I obviously need to study fractals a bit.
I thought they just had nice properties which was why they were interesting.

Do you know any good links that would explain what you just said Mark?

― mei (mei), Wednesday, 21 April 2004 05:55 (twenty-two years ago)

They aren't simply pretty things with no meaning: the similarity between the outputs of certain fractals and certain natural things is very striking - there's a great example of an image output by a simple maths formula (not at home so I can't dig it out) which is instantly identifiable by a botanist as a particular fern. Fractals are proving of practical use in some areas too, but again I'll have to dig up some references and examples when I'm at home, if anyone cares.

There's a pretty good basic book on fractals in that copiously illustrated Introducing series, and it is available quite cheaply in many remainder bookstores.

― Martin Skidmore (Martin Skidmore), Wednesday, 21 April 2004 08:22 (twenty-two years ago)

The reason I asked is not, unfortunately, because I want to travel around britain in a boat, but because I want to measure the fractal length of some broken eggshells.

― hmmm (hmmm), Wednesday, 21 April 2004 08:36 (twenty-two years ago)

The simplest self-similar maths object I can think of is probably the sequence 1/2, 1/4, 1/8...1/2^n...

Does that count as a fractal?

I've never heard of the 'introducing series' martin, is there a full name?

― mei (mei), Wednesday, 21 April 2004 18:01 (twenty-two years ago)

They all start with that word: the one I refer to is Introducing Fractal Geometry, by Nigel Lesmoir-Gordon, Will Rood and Ralph Edney. Cost me £2.99.

― Martin Skidmore (Martin Skidmore), Wednesday, 21 April 2004 18:21 (twenty-two years ago)

six years pass...

RIP Benoît Mandelbrot

― Euler, Saturday, 16 October 2010 17:06 (fifteen years ago)

he gave us order out of chaos

― drive this seven inch cheese steak through my philadelphia heart (diamonddave85), Saturday, 16 October 2010 17:39 (fifteen years ago)

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