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Monday, 18 May 2009

Better than sex?

I've heard it said... honest! You may or may not agree, I could not possibly bring myself to comment. What I will say is that Seamus Heaney in Stepping Stones, his biography by interview, talks about the great harvest of inspiration that he derived from P.V. Glob's book, The Bog People. He says of it that it has never left him. The Beauty of Fractals by H. -O.Peitgen and P.H.Richter had the same effect on me when, like Heaney and his Bog People, I bought it for myself as a Christmas present, he in 1969, me around 1986 or the year after, since when fractals have been an abiding interest of mine.

A quote from the book:

Where the world ceases to be the stage for personal hopes and desires, where we, as free beings, behold it in wonder, to question and contemplate, there we enter the realm of art and science. If we trace out what we behold and experience through the language of logic, we are doing science; if we show it in forms whose interrelationships are not accessible to our conscious thought but are intuitively recognized as meaningful, we are doing art. Common to both is the devotion to something beyond the personal, removed from the arbitrary. A Einstein

A fractal is an irregular or fragmented shape, like a cloud or a

coastline, into which you can zoom, almost without limit, dividing and subdividing it into smaller and smaller parts, each of which is a clone of the original whole.

My post concerns fractal plots, of which the black beetle (Mandelbrot) shown above is probably the most iconic. By zooming in on the first section of the antenna protruding from its head you are rewarded with

the plot following it (above left). Zoom in on that and the next appears... and so on ad infinitum. These, then are mathematical plots, obtained from an equation by repeatedly applying a process (a

function or formula)to that equation, often hundreds, even thousands of times. The colour is applied to show how many times the function had to be applied for the plot to arrive at

that particular point. Because of the vast numbers of calculations required these plots could not be realised before the advent of computers. (Mathematicians knew the plots were there. What they had not realised was how beautiful they would prove to be.) Now anyone can produce them. Those above I produced using Chaospro, one of the many fractal generators available free on the web.

The images below are from the Julia sets. These are found by exploring the periphery of the Mandelbrot.

Thoughts on a Fractal Plot

There's something in the way infinity
evolves from repetition and goes on
to cosy up to what is local and well-loved.

It has to do with how a line repeats itself
and yet surprises you, how little gasps
of disbelief greet what was sure to be.

It has to do with laws of nature laid
in stone, and flexible as oil, though hid
in folds away from human thought and prying eyes.

We've seen it the way that men like Koch
and Mandelbrot set out to map their world,
who gave geometry its extra twist, who said:

there's something in the complex plane - a void,
a vacancy, a darkness, an abyss -
whereby infinity bleeds back into the world.

And where to us it seems most whimsical,
in sudden exit or return, it tells in truth
of discipline and order writing home.

It had to do with clouds, their borders
and their surfaces, when Gaston Julia had yet
to intuit how beauty slept

with storm and chaos in a geometric set,
how Adam's single fruit upon its single tree
would bifurcate to great complexity.

It had to do with how the new (non-linear)
would wipe the lines that Euclid drew, and how
a simple plot could crown such turbulence as queen.

There's something in oneself that echoes this,
first turns it like a wooden bowl upon a lathe,
then fills it with earth's most delicious fruits.

D King


Shadow said...

that einstein quote? darn well said that. everyone should read it and take it in...

Dedene said...

Thank you for explaining fractals! I would have never had the patience to read the book. Now I can sound intelligent when I talk to people. LOL!
A lovely post, really. Thanks for the Einstein quote as well!

The Weaver of Grass said...

I love that last verse Dave - superb metaphor. Like also "flexible as oil" another brilliant image. As for the Better than sex thing - I coulnd't possibly comment.
The images are so beautiful that I shall now go to Chaospro and investigate. Honestly - if we carry on like this we shall all be sitting at the computer all day! Thanks for a lovely post.

Rachel Fox said...

Better than sex? Depends on the sex surely...

Carl said...

Wonderful Dave. I enjoy creating Fractals and have had a little break from it. I'll have to start up again.


Highton-Ridley said...

Aah, fractals... a journey I made in the late 80s early 90s.

Did u realise that at the very moment of the big bang, fractals were present in the very first bit of the expansion, at the very moment when the singularity first stopped being singular?

I guess this isn't surprising if you think that information has to come from somewhere.

In a very real sense, fractals are the DNA of our universe - they are everywhere. And just like we inherent DNA from our parents, the fractals that appear everywhere have to come from somewhere.

So not surprising that they should be present in the first moments of the creation of our universe.

So says me, anyway :)


Tabor said...

"It has to do with laws of nature laid
in stone, and flexible as oil, though hid
in folds away from human thought and prying eyes." such a magnetic but also scary image.

John Hayes said...

Really fantastic post, & a really good poem to top it off. The last stanza is quite amazing.

readingsully2 said...


Highton-Ridley said...

Here's a related little poem I remember:

Big Fleas have little fleas,
Upon their backs to bite 'em,
And little fleas have lesser fleas,
And so ad infinitum!


Comedy Goddess said...

You could probably explain it to me a thousand times, and I would still not get it.

Oh well, the pictures are worth a thousand words to me. They raise my consciousness just looking at them.

Lady Glamis said...

This is an amazing concept, really. I start to think of writing and the amazing ways I can apply this concept to the way I think and plan my stories and characters. It's a beautiful, amazing vision. Wow. I'm definitely going to have to to do a post on this... Thanks!

jinksy said...

At this point, I feel my brain has fractured with all the talk of fractals you've bombarded it with! Did enjoy you poem without ill effects, though!

Starlene said...

These fractals made me think of holograms - in which the blueprint for the whole is present in every part, ad infinitum.

And your thoughts on fractals are as unexpectedly beautiful as the fractals themselves. : )

Pearl said...

How I admire your intellect, Dave. Your blog has produced, by far, the most interesting thoughts in my head today...
Thank you!

Meri Arnett-Kremian said...

Fascinating post. But I have to admit that some of the fractal images look like tie-die patterns.

Aniket said...

I knew fractals before. But this was a wonderful take on them.

The quest to infinity is indeed most fascinating.

hope said...

This may be the only time math is appealing to me...just beautiful! Thanks for sharing.

The Things We Carried said...

I was stunned by Einstein being the author of that quote! Wow. Your blog is always fascinating!

Karen said...

Dave, I think this may be the best piece you've written (or that I've read here on your blog). It is intelligent, perceptive, thought-provoking, informative, and poetic. It turns from the subject to the self - the object of all poetry, I think. Excellent poetry.

Cloudia said...

"There's something in the way infinity
evolves from repetition and goes on
to cosy up to what is local and well-loved."

I always emerge taller and smarter after a "stroll" with you, Dave!

So very well done. Instructional and affecting. Not an easy marriage.

Dave King said...

ShadowYer, good ole Einstein, eh?

DedeneYou've read my book, How to sound intelligent without really trying! Good for you!

The Weaver of Grass My wife thinks I'm spending too long. Guess it's a danger! Thanks for the comment.

Rachel That it does, that it surely does!

Carl Great minds...

Highton-Ridley Welcome to my blog. I didn't know they were there at that time, but it does make sense. Thanks for that.
And thanks for the P.S. Very apt.

Tabor Interesting you should pick out that: for the second time running someone has alighted on the last line or so to be incorporated.

John Many thanks for the response.

Readingsully2 Welcome and many thanks for stopping by.

Comedy Goddess I am not a mathematician, by any means. With a fractal explorer you could investigate them by trial and error, if you are interested.

Lady Glamis And thanks for that. Yes, I'm sure it could prove a fertile way forward.

Jinksy The poem came first - the rest was originally going to be just an introduction!

Starlene Thank you very much for that. yes, I see the similarity between the two.

Pearl Welcome to you and many thanks for the compliment.

Meri See what you mean, but really the fascination is in the concept and the process as much as the result.

Aniket They swim into my consciousness from time to time.

Hope Know what you mean!

The Things We Carried I guess he was a stunning sort of guy.

Karen Really, your comment could only make me feel humble. Thanks for it.

Cloudia Thanks for that. I, too, enjoy the walk. Nice having you along.

claude said...

Hello Dave !
It is very difficult for me because my english is very poor.
Your post reminds me when I was little girl I try to make some fractals. On a paper I put a few inkstains on a white paper I folded. I remember I was very glad when the fractal looks like a butterfly.

Dave King said...

Your observation is probably a comment on the extent to which the exploration of fractals has influenced designers.

Dave King said...

ClaudeWelcome. Your English seems very good to me. Thank you for sharing that with us.

jinksy said...

How I wish I had an email address that I could get to work so's I could talk to you. You've just left a comment on my today's blog, and I can't reply any other way but this... The 'minute', with it's required sixty syllables, is not necessarily as easy as one might think. I found it a real trial to get the twelve lines, and I think your doctor's surgery effort needs a few more to pass the 60 mark on the syllables, an all! Rinkly Rimes seems to be on a roll though, as she's done another one on her post today! Brave lady...

Just Jules said...

seriously neglecting my bloggy visits - Spring, far too busy for us

Tumblewords: said...

A simple yes, probably. :) Your poem is lovely and speaks well to fractals and the layered delving of each. I illustrate much of my poetry blog and books with fractals and often times find the creation of a series can consume an entire block of time. Yes, probably!

Dick said...

Another fascinating post that reaches the parts that others miss. And a fine, clear, elegant poem to celebrate the theme.

Ronda Laveen said...

The fractals are magnificent. Loved your "Thoughts On.." and the Einstein quote. Guess I bloody well like the entire post.

Art Durkee said...

For those who are interested in creating their own fractal art, I highly recommend Tierazon, a cheap Win program that is definitely a lot of bang for the buck:


Do a Google search and you'll find lots of galleries and tutorials. Here's one sample gallery:


If you want to go deeper, I recommend you look into what some call the Buddhabrot, an amazing variant on the Mandelbrot set:



Personally, these are some of my favorites, although I haven't yet figured out how to get my fractal art software to generate them easily. Quite different, very spiritual.

Poetic Artist said...

This was so interesting and You always make me want to learn more and more..And to buy more books. LOL
Thank you for always putting comments on my blog. I really appreciate the time you take.

Adrian LaRoque said...

Great post Dave, congratulations!

Rinkly Rimes said...

I don't know a great deal about fractals but what I do know fascinates me. I've written two poems on the subject. You might like to read them.



A Cuban In London said...

Oh, dave, I loved the boundless freedom of this post. From the initial sentence (I can see you are enjoying that Seamus Heany, maybe I will have to get my hands on it soon, too) to your thoughts on fractal plot, this was a good ride. Many thanks.

Greetings from London.

Dave King said...

Jinksy How about, I email you and you can reply? I must admit I'd overlooked the minute requirement - I was just filling in time and didn't have the blog with me! You are quite right.

Just Jules All a matter of priorities. Understood. Thanks for visiting.

Tumblewords I can well see how that could be. Thanks.

DickMany thanks for both kind comments.

Ronda Much thanks. Very generous comments.

Art I will definitely have a look at your addresses myself. Much appreciate the suggestions, as I am sure others will, also.

Poetic Artist Anything I can do to encourage more book buying is okay with me... I always enjoy going to your blog.

Adrian Thanks muchly.

Rinkly Rimes Welcome and thanks a lot. I will surely go read your poems.

A Cuban in London Very much appreciate the comments - as I am sure you will the book!

Art Durkee said...

Note, there are several other books now available on the topic, some equally or more beautiful.

For example:

"The Colours of Infinity: The Beauty and Power of Fractals." This is an anthology with several contributors. Most notably it comes with a DVD featuring Arthur C. Clarke as host and commentator.

Eliot Porter and James Gleick, "Nature's Chaos." A book of photographs by Porter than Gleick uses to illustrate how fractal math describes nature's true forms more accurately than other geometric system we know. This is a stunningly beautiful book, with no math in it to challenge the unwary.

Robin Chapman and Julien Clinton Sprott. "Images of a Complex World: The Art and Poetry of Chaos." Gorgeous fractal art combined with poetry inspired by the art.

John Briggs: "Fractals: The Patterns of Chaos: Discovering a new aesthetic of art, science, and nature." A very good book, that does what the title claims. Full of illustrations and commentary.

And for those who get interested enough to pursue fractals to their source, i.e. their discoverer:

Benoit Mandelbrot: "The Fractal Geometry of Nature." This is actually a very readable, very beautiful book. It contains Mandelbrot's own history of how he stumbled across fractals, and why. There's a lot of good art in here, and a little bit of math, but it's very easy to read.

One thing I love about fractals is how simple the math is. It's mostly recursive equations, that cycle the output data points back into the input, and recycle them. That's how the complexity is formed: from very simple initial conditions.

Another really good book on all this is the only comprehensive overview history of chaos science and math, written for the general reader:

James Gleick: "Chaos."

Yeah, I've been involved with this stuff a long time. It's nice to see that it still inspires.

David Cranmer said...

Thanks Dave. A very interesting and informative post.

Sarah Laurence said...

Beautiful indeed – images and words. Fractals rock!

Jeanne said...

I was introduced to Mandelbrot via a sci-fi murder mystery a few years back (I'm thinking the author was Kate Wilhelm, but wouldn't swear to it.)

Loved your poem -- "laid in stone, flexible as oil" AND you used the word "bifurcate."

How cool is that?

Art Durkee said...

Jeanne, if it's the book I'm thinking of, it was indeed by Kate Wilhelm. It was titled "Death Qualified." It revolved around a scientist's discovery that a retraining of human potential could be done using a video program he invented that was heavily involved with fractals. Quite a suspenseful novel, as I recall.

Dave King said...

Art Thanks for all the useful info'. I should have mentionmed, maybe, that a handy "layman's" introduction , I found, was Chaos by James Gleick. It is perhaps a bit dated now (understatement - my paperback copy was published in 1987), but still gives the bare bones, not just of fractals, but an overview of the theory.

David Welcome to the blog and many thanks for stopping by to comment.

Sarah Well put. They certainly do!

Jeanne Yes, fractals pop up everywhere, I find, even in the most unlikely places. Thanks for the kind comment.

willow said...

That last line is just earthy and wonderful, Dave. Fascinating Einstein quote, too.

Dave King said...

Willow Thanks for that.

Mad Aunt Bernard said...

I love your post - fractals are fascinating (I had the Ultrafractal software for a while and could spend HOURS creating them). It was nice to find them here, as they remind me to look at the world with fresh eyes.