Return path of the fractal?
Saturday, October 13th, 2007 at 7:00 amThe sky this morning reminds me of a dusty blue-and-white china jug: splotchy, dim blue clouds on a brownish-cream background. The buildings all look very blue and the distance is misty. I have been getting up early to write. Writing Horn is scary because I have very little idea about what’s going to happen. I can see ahead to a certain point, then it all gets vague, rather like this morning’s view. I’m completely incapable of saying, “Ok, this will happen.” I’m not always so utterly unable to plan, or at least make contingency plans, but this book feels like a foreign territory that I’m exploring without a map. And there are odd things in it that I can’t explain yet. There’s a tower room with windows that look out onto an orderly land of trees and lawn where it’s always (I’m fairly sure, though not positive) a spring day. One of the characters disappeared into it before the story begins. I know what the land signifies symbolically but not how and why it exists physically. Forage says (I think) that it is the “return path of the fractal”. Now, Forage in the book certainly doesn’t know what a fractal is, but the Forage who stands over my shoulder is obviously more au fait with mathematics. I unfortunately lost my ability to understand maths somewhere in high school. It got terribly hard one day, it seemed, and that was that. My math grades went from easy As to struggling Cs and Ds. Part of my problem is that I have a very concrete way of thinking. I can’t understand abstract explanations very well. No matter what the subject, I always need explanation in the form of concrete examples, or at least decent metaphors. So I understand the aspect of a fractal that shows up in pictures, namely that it’s “a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole”. But then I read that a fractal’s Hausdorff Dimension is greater than its topological dimension, and I find that I understand the explanation of a Hausdorff dimension exactly as much as I expected, which is to say, not at all. Sadly there’s no such thing as Advanced Math for Dummies.
While writing I’ve been thinking about geometric patterns in nature–spirals, in particular, since spirals are found at the largest and the smallest scales in nature–and of course, Forage’s horn is one. But a spiral doesn’t have a return path, unless you turn around and go back again, and I have no idea whether a fractal can have a return path, or what such a path would mean, mathematically or visually. Googling didn’t bring any enlightenment, though it did lead me to this interesting page, which I at least half understand - though if the whole Meaning of Life really is matter’s search for antimatter, I’m glad we conscious beings can embroider that backcloth with fancier meanings of our own–that we can dally, as opposed to quest, which is something I wanted to address (or indulge in) in this book.
At any rate, does “the return path of the fractal” make sense to anyone reading this? I should be very grateful for any help!
October 13th, 2007 at 10:18 am
…Oh, dear. Just this morning, for no reason I could ascertain, I suddenly had this concept of myself walking along a spiral, circling around my original path over and over. I wondered whether I was heading toward the edge of the spiral or the center. Then the answer came to me that either way I was walking, it would still lead back to the center.
I think we did it again…
October 13th, 2007 at 7:15 pm
Ah, fractals. Be still my beating heart. Have you ever read James Gleick’s book “Chaos” ?
In the context of the images in your head and words on the page, however, it is more around supporting the understanding of what one is actually looking at out of the windows of the tower, and what is beyond them
The concept of an “orderly line of trees and lawn where it is always…a spring day…” is about as far from chaotic, on the outset, as one would expect. It is a metaphor for a constant, a “carrier signal” to use a data comms term, against which all other things resonate? To explain another way…does the consistent, unnatural representation of an ideal world represent the “zero state” or “normalcy” or “test pattern” of alternative worlds/universes, towhich the aforementioned character has journeyed before the book began?
To drill down into fractal geometry is to try and measure the length of a coast line of an island…you can break out your Tape Measure of Extraordinary Length and say “100km around”…but then if you start “measuring around” the nooks, crannies, inlets and rivers…you find you are measuring it at 200km around…and if you drill down further and take every rock, plant, ripple of sand, etc… all of a sudden you are up to 500km, and so on and so on. Just as if you drill down on the perfect spring day trees-and-lawns place, the frame of reference shifts and shifts…tree trunks are uneven, blades of grass grown tall and short, dandelions sprout and spread…bugs crawl, and things crawl after the bugs. The deeper you look, the more universes open up, until you are looking between the molecules. As you pull back and pull back, like a fast zoom out of Google Earth, the field and trees reveal themselves. You are returning backwards along the path of the fractal geometry you have travelled down along…and possibly ‘backwards’ in time to where you began.
The thing you will find with the spiral patterns in nature - the description of a Fibonacci sequence…you have talked about it earlier - i.e. the additional of sequential numbers and their sum (1+2=3 +3=6 +4=10 +5=15 etc.) is well documented. What happens when fractal paths and a naturally-occuring Fibonacci sequence combine? Something purely chaotic meeting something mathematically ‘real’ and so well ordered? Well, you end up with your path between worlds, and your anchor to take you back out again. What if the unicorn horn was the key? Do the spirals on the horn curve up and describe a Fibonacci sequence? Seems only natural.
Some other things to tie together. Octotrope. Are there eight windows in the tower in Castle Orpheus? Are some boarded up, for reasons yet to be explained? As some fractured, burnt or abandoned? Is the octotrope the gothic mathematical-mechanical device, which has the power to adjust, tune and refine what the watch tower is able to “do”, i.e. tap into energy of the world, calculate the fractal path to be described, and harness it to open up views and links to other worlds…like the orrery in Iain M. Bank’s Feersum Endjinn…but which Rootmold broke by meddling with it, and was turned into a slug so that he/she would no longer have hands to meddle?
(ps. Putting aside that rant for a sec…isn’t / wasn’t your data a math’s teacher??? There are not enough artist-mathematicians in the world, and arguably far too many warrier-clerics, and elven-thieves…)
October 14th, 2007 at 8:17 am
Laurie - we did!! And then I went looking for some tax receipts, and in the bag where they were I found a book I’d bought and forgot about, Gaston Bachelard’s “The Poetics of Space”. I opened it casually, at random, and found that I was looking at a page dealing with the spiral geometry of shells and shells as toys for the imagination.
Al - I think I’ve read some of that book, or a similar book. I’ve read a book with pictures of fractals on the front…
“does the consistent, unnatural representation of an ideal world represent the “zero state” or “normalcy” or “test pattern” of alternative worlds/universes, to which the aforementioned character has journeyed before the book began?” - it quite possibly does. I’m going to keep that possibility in mind. Forage and Reason don’t actually go into the tree world (which they call the Arboretum). Reason is tempted; Forage is not. But - it might be the path by which they eventually return to Castle Orpheus.
I hadn’t thought of the “zoom out” as the path of return. That’s also possible. I’m not sure what it would mean psychologically, unless it might mean a move from an inward, solipsistic focus to an outward one. Forage and Reason are going to meet a very solipsistic king (who either is, or thinks he is, Rudolf II of Bavaria) who is obsessed with the creation of a miniature ideal world in a Mannerist garden. So perhaps the Arboretum somehow links to that. I had also thought of it as a new world growing inside the current one - a world where time hasn’t started and everything is ideal because nothing has happened yet - a world entirely without the anxieties (or the pleasures) of narrative.
In fact… it is timeless; Castle Orpheus is half-timeless; Mackerel Town is perhaps a quarter timeless; and Forage and Reason are going to be thrust into a world where time marches on, thought Rudolph would have it otherwise. So the Arboretum might be an original point through which one can eventually loop - though not to enter a state of pre-Freudian bliss, because you can’t return to that, but to enter something more like hard-won enlightenment. So that when you enter the Arboretum from the windows of Castle Orpheus, it’s like going back to the womb; but if you enter it from its other side in the wider world, it’s a different place - though still similar, like a fractal.
Maybe. But I like that idea.
I don’t think animal horns are Fibonacci spirals - they’re too long and slender. Although they might be Fibinacci spirals in three dimensions, with the vertical distance from coil to coil making up for the lack of increasing width. I’ll have to research it and see if there are common spiral sequences in nature other than Fibonacci ones.
Haha - yes, the tower does have eight windows! I hadn’t thought of that. They are actually the only windows in perfect repair in the whole castle (I think). From the outside, in the Arboretum, the castle doesn’t exist; there’s only the tower. I’ve got it in mind that if the ruined tower were rebuilt, it might complete a circuit and start Time off in the Arboretum; but I’ll have to see if the story needs or wants that.
Yep, Dad’s a (retired) maths lecturer. And good at drawing pictures. Now that I think of it, I’m fairly sure he thinks Space/Time is finite and funnel-shaped, and that when you get to the widest point of the funnel you end up back at the beginning. If it were a spiralling funnel, that would be cool. I shall have to ask him. There are even more arguably not enough warrior-mathematicians or elven-Sumo wrestlers. What’s a fat elf to do?
Thanks very much for the long and thoughtful comment - you’ve helped me along, I think!
October 14th, 2007 at 8:41 am
Solipsism is a wonderful thing to explore…again, my fave author Iain M. Banks created a cute little group of mercenary solipistic characters in “Against a Dark Background”, who thought they were ALL God, and that the others around them were merely their apparences.
[ Indeed, on Facebook, I put my religion down as “Metaphysical Solipsism” and no one has challeged it…leading me to think that I Must Be Right…
The new-world in the current-one is very Mandlebrot (the pretty spiral pictures you have seen in the centre-exhibits of Chaoas); it could stretch in both directions. What if the current-one is a world within a macro-world? The Arboretum could be the singularity - the connection point, the collapse point. According to Hawking, Feynmann, Davies and, I think, Einstein, the idea of time, space and singularities (known by the cutesy name of black holes) is that the closer you get to the singularity, the more things back along your path appear to “speed up”, as you are experiencing time moving more slowly relative to things further ‘out’ that you…, as time has trouble “escaping” and “moving” away, just like light does (hence why they look dark).
So…therefore…stand outside the Arboretum/singularity and it looks frozen in time. Stand within the Arbortetum and look ‘out’ and everything appears to still be moving forward. The further away it is, the faster it appears to move. This then links to what you are saying about the perception of time movement in Castle Orpheus (1/2 frozen), Mackerel Town (1/4 frozen), etc. There may be a physical/dimensional distance if you were to draw an arbitary straight line between these points, which is why the movement of time speeds up the more you move away from it.
Maybe the zoom in/out along the path impacts the perception, and perhaps the behaviour, of the inhabitants…
Elven-sumo. Mmm. Surely there have to be some 1/2 elves who cop the too-human end of the genetic stick in that regard? I think you are tapping into a hidden underbelly of elven ethnic purity there…
October 17th, 2007 at 9:42 am
I like it that the world growing inside a world is a Mandelbrot thing! Time movement in those places in the story probably isn’t what science would demand, but this isn’t hard sci fi, after all, so an analogy seems fine. For instance, the Arboretum isn’t perfectly frozen in time; breezes blow, clouds move across the sky; it’s as if Time there is a gently rippling pond rather than a river.
If there’s a line between the points, it’s a spiral, going west from Reason’s tower to Forage’s, then north to the town, then east to the outside world. But a spiral is fine, I guess!
I like very much the idea that time has trouble escaping - in fact, I might use that exact phrase. Forage himself exists half in linear time and half out of it; he has multiple pasts, and periodically he “remembers” another one, though they all result in him living in Castle Orpheus.
October 17th, 2007 at 1:52 pm
Glad to be of service…and never let science get in the way of a good fantasy world creation!