h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Mandelbrot Set - A85021616
Author: frleon - U14872593

Here's another entry of mine, this time concerning my favourite subject: Mathematics. Not everybody will profit from this one. You should not be afraid of abstract notation, since there are formulas and ugly letters and indices. Those who fight their way through these will be rewarded with some nice pictures, but for those who aren't interested in mathematics an acid trip will also do.

Hi frleon! Thanks for writing this. We don't appear to have an entry on the Mandelbrot Set so your one is certainly filling a hole in the Guide.

I'll try and give it a detailed read through from the mathematical perspective, since this is something that the average peer reviewer might not know much about. I'll let you know my findings soon.

OK, I've had a quick read, and here are a few things that occur to me:

for an infinite n

I'd prefer "as n tends to infinity", because modern maths shies away from ever stating that a number is infinite.

an=an-12+c -- sorry my version here is hard to read, but the original is also hard to read even with the super and subscripts. I think it would be clearer if you put brackets around the an-1 term:

an=(an-1)2+c

applicating the same formula --> applying the same formula

You have a footnote on "real c" which makes it look like "real c squared". Could you move the footnote somewhere else, so that the 2 of the footnote is not on a mathematical entity?

smaller or equal 0.25 --> less than or equal to 0.25

to justify an own entry --> to justify its own Entry

a pair of buttocks with antennas rotated 90 degrees counter-clockwise -- it's not clear what you mean here. You might be better to take it in stages: "imagine a picture of a pair of buttocks..."

We use British English here, so it should be "anticlockwise" rather than "counter-clockwise".

"the Mandelbrot set is strictly speaking no fractal" -- the Mandelbrot set _is_ fractal. It has fractional dimension. The fact that the smaller copies are not exact copies does not affect this.

You state in a footnote that the Mandelbrot set is the black bit but I think this is important enough to be in the main text. You also should point out that the coloured bits are just ways of representing the speed at which the space around the set diverges.

You link to images on other people's websites. You shouldn't do this. It's OK to link to pages on other people's websites which have those pictures on them, but linking directly to the pictures is like stealing the pictures off their page without giving them any credit.

not even two copies are rotated --> no two copies are rotated

has an infinite circumference --> has a circumference of infinite length

Somewhere I read the set is just appearing to be fractal, but of course you're right.
I'm quite embarrassed because of the infinite number. This is exactly the kind of mistake I love to correct anywhere else.

Bump - is frieon still in the house?

I think that he is busy with school work presently.

I have sufficent mathematics to follow the flow of the entry, but not enough to peer review it, i'll have to leave it to others I'm afraid.

frleon is still around. Any mathematicians in the house?

I wouldn't feel confident completing this one. I'm taking a topology course next year and this covers fractals, but I don't know enough about it now.

My comments as it currently stands are:

* * *



...as opposed to Imaginary numbers which are... er?

Did you intend this to be a joke? It may go over the heads of most readers if you did. If you didn't, then I'd be inclined to describe real numbers as those which you can measure (say).



... justify what's entry into what? This sentence needs to be clearer.



Can you give a link to this particular diagram? (I mean a mathematical diagram - I don't want to see a picture of buttocks with antennae attached)



Please explain this term.

No time to read carefully but it looks more or less right to this ex-mathematician. If you like, you can use my image to illustrate it: http://thesamovar.wordpress.com/2009/03/22/fast-fractals-with-python-and-numpy/

I do keep meaning to read this Entry

I have no idea what this entry is going on about. But it is extremely entertaining.

I am encouraged by the thought that my reaction to Mandelbrot sets is essentially correct.

They are very pretty.

Hello.

I got to the paragraph which states (with subscripts that I cannot replicate in this box:

"This set is named after its discoverer Benoît Mandelbrot who was a French-Polish mathematician living from 1924 until 2010. It is defined as the set of complex numbers c for which an=an-12+c with a0=0 stays convergent as n tends to infinity."


What I ask is whether it might equally be written:

"This set is named after its discoverer Benoît Mandelbrot who was a French-Polish mathematician living from 1924 until 2010. It is defined as the set of complex numbers c for which [with a0=0] an=an-12+c stays convergent as n tends to infinity."?

Otherwise I am stuck.

Yes, that's correct.

I don't know if I'm just suffering from a lack of attention span as I get older and play more and more kneejerk fasttwitch videogames, but I found this entry very hard going.

Fundamentally, I'd like to see it rewritten thus:
- What is an imaginary number?
- What is a complex number?
- How do you multiply complex numbers?
- How can you use a complex number to define a point on a plane?
- What happens if you do the iteration process on every point between (-2,-2) and (2,2), to one decimal place (i.e. (-1.9,-1.9) etc.), iterating 10 times each?
- What happens if you do it to two decimal places? Ten?
- What happens if you iterate 20 times? 50? A million?
- How do you get those colours you always see in the pictures (answer: by contouring - divergence between 0 and 10=black, divergence between 10 and 100=blue, divergence between 100 and 500 = red, etc.)

Fundamentally, the entry should provoke anyone with access to a programmable computer to try to write their own Mandelbrot set generator program. Certainly reading James Gleick's "Chaos" in the mid eighties made me go home and fire up my Spectrum (not the fastest or most graphically capable machine in the world, I'll grant, but even Spectrum basic allowed you to explore the edges of the set...).

Hoovooloo, if you find it hard going, how will the average reader find it. I agree that it needs some rewriting. This will make it longer, but simpler.

I once rote an introduction to complex numbers, so we may not need to start from first principles. (A25746249)

Would it be rude to volunteer to rewrite it?

(This one, not the complex number one...)

It could be re-written, if first we decided that the author had abandoned it, then send it to Flea Market ( this is the protocol for abandoned Entries that we think have merit) and then someone took it under their wing and sorted it out.

The original author would have joint credit, once it was published.