...in all the wild rites and the savage myths, there is at least that twilight which suggests to itself, and by itself, that it might be more enlightened than it is. There is something in the grossest idolatry or the craziest mythology that has a quality of groping and adumbration. There is more in life than we understand; some have told that if we ate a scorpion or worshipped a green monkey we might understand it better. But the evolutionary educator, having never since his birth been in anything but the dark, naturally believes that he is in the daylight. His very notion of daylight is something which is so blank as to be merely blind. There are no depths in it, either of light or darkness. There are no dimensions in it; not only no fourth, but no third, no second, and hardly a first; certainly no dimensions in which the mind can move. Therefore the mind remains fixed, in a posture that is called progressive. It never looks back, even for remembrance; it never looks the other way, even for experiment; it never looks at the other side, even for an adventure; it never winks the other eye. It simply knows all there is; and there does not seem to be much to know. [GKC ILN Aug 6 1932; thanks to Frank Petta and my mother]The first time I went to the Twin Cities to visit Dale Ahlquist, we went to that large shopping town near him - for some reason they call it a "mall". There seemed to be an amusement park inside the mall - which was already so gigantic it was hard to believe we were "inside" - one might have thought we were in a space station halfway between... er, sorry I can't go into that here.
Anyway, we stopped on our way to our noonday meal, and looked at the dozens of poor frightened young children being strapped into some gigantic mechanical thing. Then it started to move, in four or six directions at once, accompanied by screams of terror. I said to Dale how they used to train astronauts in those things, now kids pay money to scream their heads off...
Dale, being a polite Chestertonian, did NOT comment: "Yes, and their parents scream their heads off at how much it costs - once to buy lunch, and once to lose it." But then so much happened that day, perhaps I forgot. (No I did NOT take a ride.)
Now that we've set the appropriate degree of horror:
Please have your ticket ready, fasten your seat belts, tighten your rider-protection equipment. You are going to be displaced into a fractional dimension, courtesy of the American Chesterton Society. Warning: you should have abstained from food and drink for at least the last .03 minutes; in any case, your entrance fee is NOT refundable, and the management assumes no risks to your health or property. You may, however, retain the results of your journey as they will be useful for decorative purposes as we near Winter Tide.
Ahem. (Yes, I do get carried away, it's just such fun to play with these pleasant English word-things after fussing with the brackets, braces, asterisks and semicolons I have to write for work.)
So, here we are, finally ready to follow a very strange path: through the Tollbooth, down the rabbit hole, through the looking glass, by the Straight Road to the Furthest West, over the Mountains of the Dawn, via Platform Nine and Three Quarters... well, actually, on the nearest staircase of your home or office, to that strange step which is just past the first, but not quite the second.
I am going to tell you just a little bit about a strange little branch of mathematics, in which we take the recursion we examined last week and apply it to good old geometry. I can only tell you a little bit about it - as a computer scientist I spend (have spent!) a lot of time wandering along the various halls of the University, and have heard a smidgen here, a drop there... Often these crumbs get wedged into the computer in interesting ways, and they can be fun, and even useful. (I used this one to help a friend design planets... oops, I'm not supposed to talk about that either.) But this one is easy, and it turns out to make a very nice design, providing one remembers the "terminating condition" we talked about last week. Remember, the smallest doll that has no seam? (And what if it didn't?)
This is very much of an audience participation project, and it will be lots of fun to try. You may get tired after a little, and that's OK, because you can always print out our pictures - and if they aren't very nice on your printer, we will look into ways of getting good copies for you, if you ask. But even if all you have is a scrap of paper, a ruler, a pencil, and an eraser - that will be enough. (To do the whole thing, you ought to have a compass, the circle-drawing kind, but that's optional).
OK, ready? Hey, Joe, power up der machine! *CLICK* hummmm...
And when you're ready for the drop, CLICK HERE... (hee hee)
First, I will teach you the "rule". Then we will apply it. This is just a scrap-paper trial, so please play along. It won't hurt at all, and will take just a minute or so. You need a piece of paper, a ruler, a pencil, and an eraser.
The Rule. To do the Rule we are given a line segment.
(Draw a nice handy line maybe about three inches long, from left to right.)
Rule Step 1. Divide the given line segment into thirds.
(Take your pencil and lightly mark the point one third of the distance, and two thirds of the distance. Use a ruler, or just approximate.)
Rule Step 2. Draw an equilateral triangle on the center portion of the line segment.
(Again use your ruler and pencil. Make each side the same length. The picture will now look like a witch's hat - but that's NOT where the magic comes in, hee hee. That's later.)
Rule Step 3. Erase the central portion of the line segment, which is the base of the triangle.
(That's why you need the eraser. It's an important lesson in mystical reality - not every erasing is a mistake! You will now have a kind of V shape with long arms.)
Excellent. That's all the Rule is. (Whew.)
How about a short break for a little Chesterton?
...a hard black outline on a blank sheet of paper, an arbitrary line drawing such as I could make myself with a pen and ink on the paper in front of me - that this thing should come to life was and is a shock to the eye and brain having all the effect of a miracle. That something like a geometrical diagram should take on a personality, should shoot over the page by its own inky vitality, should run races and turn somersaults in its own flat country of two dimensions - this does still startle or stun me like a shot going past my head.He was talking about cartoons, yes indeed. But this is just a curious little pattern.
[GKC ILN Mar 19 1927 CW34:274]
Now let us add the powerful magic of recursion!
Stage TWO. Take your result, and apply the Rule to each of the four NEW line segments you have.
After step 1, applied to all segments:
After step 2, applied to all segments:
After step 3, applied to all segments:
Very nice. You see - now, each of the four new pieces in your first result now looks like that result - just smaller? We've opened our first doll, and found another one inside, just the same.
OH, WOW. you are saying. Now, we do it again...
That is, STAGE THREE, STAGE FOUR, and so on.
Exactly.
But let us be a bit more artistic (if the word be permitted of such bland black-and-white efforts). Let us take a slightly more interesting shape, and apply the Rule in successive stages. Let us, in the name of the Triune God, or the three dimensions if you like, take an equilateral triangle as our start.
You can do this on a nice big sheet of paper if you want, and work carefully, as you will be delighted by the final product - but it will take some work. Just be patient, go all the way around at one level before getting smaller, and stop when things get too small to draw.
Stage One. Here's our starting triangle:
Stage Two. Now apply our Rule to each of the three sides:
Stage Three. And again...
Stage Four. And again...
Stage Five. And again...
Stage Six. And again...
At this point, the changes are too small for the computer to display, so I will quit here.
Now, this is the real-world kind of recursion. We have gone down to the smallest doll, to the pixel-level of the graphics, to the atomic level (Atom in the Greek sense - you cannot cut it any finer!)
But, as we hinted last week - what if there was no terminating condition?
Here again we must pause for a brief comment from a mathematician. We are going to talk (very informally) about a limit. That is, something that is a "final result" of a series of stages, the number of which may increase without bounds. Note that (contrary to the Eagles) we are not "taking it to the limit" by counting to infinity. I really do not have the time or space to explain "limit" now - except that Zeno was wrong. Simply because you can move, you can walk through an infinite number of halfway points from here to there. And the reason is because (as we mnath guys say) the limit of the infinite series is finite. You can add 1/2 and 1/4 and 1/8 and 1/16 and 1/32 and 1/64... and all the infinite fractions which are the reciprocals of the powers of two - and you will get ONE. No more, no less. (the Word, as GKC and St. John say, is One.)
Now, what happens when we apply our Rule along the infinite series of line segments?
Only about 40 years ago, a mathematician named Benoît Mandelbrot was studying the coastline of Britain. Noticing how there seemed to be a similarity of shapes depending on the degree of resolution, he developed the mathematics of such things as we have just considered and found that the result is finite in one sense, though infinite in another... After careful study, he found that somehow the final result is something MORE than a line (which has ONE dimension) but definitely LESS than a planar curve (which has TWO dimensions). He called these things of FRACTionAL dimension fractals.
Remembering that real things do NOT recede to infinity - they stop at some terminating condition, be it pixels, cells, or atoms - it is clear that some things have fractal-like character: tree branches, lightning bolts...
Snowflakes.
Hence, as I said in my title, White Magic. Well, actually it was Father Brown:
When the priest went forth again and set his face homeward, the cold had grown more intense and yet was somehow intoxicating. The trees stood up like silver candelabra of some incredibly cold Candlemas of purification. It was a piercing cold, like that silver sword of pure pain that once pierced the very heart of purity. But it was not a killing cold, save in the sense of seeming to kill all the mortal obstructions to our immortal and immeasurable vitality. The pale green sky of twilight, with one star like the star of Bethlehem, seemed by some strange contradiction to be a cavern of clarity. It was as if there could be a green furnace of cold which wakened all things to lifelike warmth, and that the deeper they went into those cold crystalline colours the more were they light like winged creatures and clear like coloured glass. It tingled with truth and it divided truth from error with a blade like ice; but all that was left had never felt so much alive. It was as if all joy were a jewel in the heart of an iceberg. The priest hardly understood his own mood as he advanced deeper and deeper into the green gloaming, drinking deeper and deeper draughts of that virginal vivacity of the air. Some forgotten muddle and morbidity seemed to be left behind, or wiped out as the snow had painted out the footprints of the man of blood. As he shuffled homewards through the snow, he muttered to himself: "And yet he is right enough about there being a white magic, if he only knows where to look for it."There are lots of other tricks one can play - it is lots easier to do on computers, which don't mind the boring parts and are usually quite neat at inking and erasing and all that. In any case, this concludes our little ride - I hope you aren't queasy - if you have any questions please submit them in writing.
[GKC "The Dagger With Wings" in The Incredulity of Father Brown]
A final note: as disorienting as they may have been, your experiences today CANNOT be used in order to get a ride on the Space Shuttle. You'll have to go to that place in Minnesota for that kind of training - why not do it next June when you come for the Conference?
--Dr. Thursday
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Hint: you can double-click on the "snowflakes" and see them much larger...
ReplyDeletePraise the Lord from the earth, ye dragons, and all ye deeps: Fire, hail, snow, ice, stormy winds, which fulfil his word...
[Ps 148]
But computerized images of fractals are deceiving, aren't they? Such infinite regression coupled with infinite resolution is artificial. Perhaps abstract is a better word for it. In the world of the flesh, such things have their limits.
ReplyDeleteQuite deceiving - yes, Kevin, I tried to emphasize that precise point!
ReplyDeleteAnd most computer monitoring screens have rather poor resolution, sorry - the pixels are quite large.it's like the old challenge to "fold a paper in half eight times"...
There cannot be a "fractal" on a computer, just as there cannot be a "fractal" in the real world, no matter how some whiners pretend. (Haven't these people ever used a microscope?) A true fractal is infinite - in the example I provided, we always have some line-segments left over to do. But in the FRACTAL defined by this rule, no point is adjacent to another at any resolution - it is totally discontinuous. (But the details go out of our range for today.)
Also as you point out, fractals are quite artificial. For all the apparent similarity, Nature isn't "recursive" - at least not in the mathematical sense.
--Dr. Thursday
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ReplyDeleteNow, dear doctor, in reading your three-part posts on three dimensions (and other things), I am left to ponder your point.
ReplyDeleteIs it that this infinite digression, this recursion, this retreat into smaller and smaller rooms in hell, or larger and larger rooms without windows, is hellish because it is counter-dimensional, which is to say, that despite all its apparent infinity -such a journey is flat? That what we need is not the endless loop of self-reference, but the third dimension, the ritual, the drink offered to the god of the vine, to the one who takes the flatness of our fractured fractals and suddenly expands our lives into an undreamed of fullness and depth?
If that is indeed your point, then I commend you, for the eggs you kept hatching seemed to contain only more eggs, the riddles you were posing you seemed to be answering with only more riddles - but the third dimension solves the riddle, and behold! A new thing emerges from the egg!
Kevin, I think you are very close but then I am just an "actor" (hee hee) - I did not invent the truth in the fractals - nor the dimensions or recursion. I am just reciting the lines - or at least the points.
ReplyDeleteThe hint at the center - the final egg with no seam, which needs to be cracked (preferably above a frying pan!) is really the same that GKC stated here:
That circle or disc of the sun set up in the morning of the world by the remote Egyptian has been a mirror and a model for all the philosophers. They have made many things out of it, and sometimes gone mad about it, especially when as in these eastern sages the circle became a wheel going round and round in their heads. But the point about them is that they all think that existence can be represented by a diagram instead of a drawing; and the rude drawings of the childish myth-makers are a sort of crude and spirited protest against that view. They cannot believe that religion is really not a pattern but a picture. Still less can they believe that it is a picture of something that really exists outside our minds. Sometimes the philosopher paints the disc all black and calls himself a pessimist; sometimes he paints it all white and calls himself an optimist; sometimes he divides it exactly into halves of black and white and calls himself a dualist, like those Persian mystics to whom I wish there were space to do justice. None of them could understand a thing that began to draw the proportions just as if they were real proportions, disposed in the living fashion which the mathematical draughtsman would call disproportionate. Like the first artist in the cave, it revealed to incredulous eyes the suggestion of a new purpose in what looked like a wildly crooked pattern; he seemed only to be distorting his diagram, when he began for the first time in all the ages to trace the lines of a form - and of a Face.
[GKC, The Everlasting Man CW2:267-8]
For us, there is something radically more than just a pattern: as much as a flower trancends the most glorious crystal, the slightest animal transcends the most awesome plant, the dullest, weakest, most evil human transcends the whole zoo and hothouse of creatures - there is Something.... no someONE more...
It's really the same play, you know. It's the Surprise, in the most complete sense. Everything hangs on that last sentence - and what a sentence it is. To put it in some more familiar words:
"And the Word was made flesh, and dwelt among us."
The really funny thing is that the Greek verb for "dwelt" happens to have as root the word from which the English word "scene" derives - hence we might paraphrase without violence to:
"And the Word was made flesh, and came on the scene among us."
(God, I hope I don't blow my lines...)
--Dr. Thursday
It occurred to me today that the ending of "The Surprise" is literally a "deus ex machina", a god out of the machine, a much criticized plot device begun by the Greeks in which a god (or in more modern theater an unexpected event) resolves a seemily unresolvable situation. In ancient Greece, an actor portraying a god was literally lowered onto the scene from above, thus saving the day. Wikipedia rightly says, "The first person known to have criticized the device was Aristotle in his Poetics, where he argued that the resolution of a plot must arise internally, following from previous action of the play."
ReplyDeleteThe irony is that while Chesterton was clearly parodying the "deus ex machina" in his resolution of "The Surprise", the parody is itself the true solution - and thus much more than a parody. In "The Surprise" the "deus ex machina" is not a cheap trick in that the resolution does indeed arise from the previous action of the play, and the "god from the machine" is an analogy to the God from the skies, who did indeed descend to save us. This ending functions on several different levels at once.
We have Chesterton 1) parodying a stock stage device, 2) which in his play is not a stock stage device because it arises perfectly from the action that precedes it, and 3) it functions as an allegory for the ultimate resolution - the Incarnation.
Has anybody noticed that this feller could WRITE?