Chesterton and Christmas Talk--A Reader Sends a Report

I was made aware of Chesterton and Christmas: A Serious Affair thanks to your Chesterton blog. Last night I traveled from Philadelphia to New York City to attend this event. In gratitude, I offer you this report.
The event was a collaboration among The Crossroads Cultural Center, the G.K. Chesterton Institute For Faith and Culture at Seton Hall University, and Fordham University's Campus Ministry. Attendance was about two hundred.
Tony Hendra read Chesterton's The House of Christmas, The Nativity, and The Wise Men.
The Choir Of Communion And Liberation, under the direction of Christopher Vath, sang carols from around the world.
Father Ian Boyd and Dr. Dermot Quinn were luminous in Chesterton's Love For Christmas; A Conversation. What stays with me about their presentation is this: that the Incarnation was the foundation of Chesterton's wonder, gratitude, and joy. That God became a man to share in our humanity should be the basis of a lifelong celebration. 
Some GKC zingers:
"Dickens rescued Christmas from the Puritans."
On Shaw's criticism that Christmas is just an excuse for shopkeepers to sell their wares: "That's like saying the purpose of sex is for jewelers to sell rings."
Each Christmas GKC "celebrated the realization that he was not God."
This was a great event and well worth the trip.
John H

Thank you so much for this wonderful report, John. I felt like I got to share a little bit in the joy of this conversation.

And Father Boyd is going to be speaking at ChesterTen in August, 2010.

Wonder and awe at detours

Yesterday, our family was able to witness the launch of Space Shuttle Atlantis and it was an awesome sight. Wow. As I lifted my eyes into the heavens to see those men and women soaring into space, well, I just felt so amazed that we had accomplished such an unbelievable feat. I said a prayer for their safe journey, which again reminded me that there are people on the space station, and we should constantly be praying for them, too.

Almost just as amazing: we travel the highways quite a bit, and the way humans have engineered road work projects is really amazing. No longer do you need to leave the road, but they've figured out how to keep you on the road (maybe you're driving on what used to be the shoulders, but you're still basically on the same path) even while doing huge roadwork projects.

Both are amazing things: roadwork, and shuttle work. Man truly is an amazing being.

How Does One Regain One's Sense of Wonder?

Is it just will-power? Conscious effort? Slowing down, smelling the roses?

Chesterton had the ability to wonder at all of life, and I'm just wondering, how do we regain that child-like, that Chesterton-like sense of wonder?

The World will never want for wonders...

Funny Youtube video of a Chestertonian perspective on the lack of wonder in today's world.

H/T: Rob D.

Dr. Thursday's Post

Introductory: The Bridge to Elfland

Today, the Thursday in the octave of Pentecost, we come down from the "foothills" of Orthodoxy - what GKC calls the "rough review of recent thought" which is madness (the Maniac, chapter 1), as it is centered on self-destruction (the Suicide of Thought, chapter 2).

And lo: we find a bridge. Bridges could easily occupy several bloggs full of writers, whether one approaches from their science, their engineering, their poetry, their art, their symbolic significance....

Or, as GKC might say, bridges can be viewed as the Common Man views them: in the simple, commonplace sense that they provide a way of getting across chasms, rivers, and other such obstacles.

Sometimes the simple and obvious thing defeats us. (It ought to be the other way around: we ought to be using the simple and obvious to defeat our - uh - opposition.) In this next chapter, perhaps one of GKC's greatest and richest writings, we shall see how magic - yes, real, everyday, honest, homely, fairy-tale magic - can be used in this way. (Oh, are you bothered by that "m" word? Lest you misunderstand, I assure you: there is NO danger of demonism here. See my PS at the end.)

But the bridge before us beckons onward, to a wide and lovely land where we shall start our real quest, because, as we heard a week or so ago, "It is time we gave up looking for questions and began looking for answers." [CW1:241]

Wands out, everyone, and let us proceed.
Click wand here; no spell word is required.

The chapter we are entering is called "The Ethics of Elfland". All the lit'ry folk in the audience (you can tell them from the ink stains on their fingers), and the few philosophers who are still with us, will cheer, expecting this will get into some esoteric discussions of truth, fantasy and fiction. And all the scientists moan. (There are SOME scientists out there, I hope; someone has to be turning the crank to keep this network - uh - networking, and your lights shining. I don't count, as I'm on the hike with you! Then again, even the liberal arts folk have web pages now, and use laser printers, how curious.) As I said, the scientists moan, because they think magic and ethics and all that philosophy is boring. How surprised both sides will be! But I am getting ahead of myself.

I have, in the course of my blogging, often mentioned the works of Father Stanley Jaki, a great Chestertonian, a historian of science, and author of several dozen books, including the excellent little tome called Chesterton a Seer of Science. It contains a most important study of this particular chapter, and from it you will learn that about 1/3 of this "elvish" chapter of GKC was reprinted in Great Essays in Science, a title in the Pocket Library, edited by Martin Gardner (a name well-known in science and math circles). As Fr. Jaki revealed,

There was Chesterton in the company of Albert Einstein, Charles Darwin, Henri Fabre, J.R. Oppenheimer, Arthur Stanley Eddington, Alfred North Whitehead, and Bertrand Russell, so many giants in mathematics, physics, and natural history. Chesterton was also in the company of such prominent interpreters of science as John Dewey, Ernest Nagel, and even T. H. and Julian Huxley.
[Jaki, Chesterton: A Seer of Science, 14]

Now, of course, the moans and cheers from the two realms reverse, with the additional effect of a distinct murmur of confusion.

Well, is GKC crazy? (all that talk of Hanwell earlier, hmm...) What is he talking about? Is this science or magic? The real world or the elvish one?

Yes, that's exactly right. (That's the Boolean Yes, if you know what I mean: it's what the kids say when Mom asks if they want ice cream OR cake!) As I said, we are entering into a lovely, beautiful, amazing - and challenging part of our journey.

But I am talking about GKC, not Jaki. I highly recommend Jaki's book, especially for insight into this particular chapter, and the whole intellectual edifice of GKC, but I dare not go too far into it at present. (It's much like fudge, or donuts, or whatever sweet you delight in... you want to keep eating more... I 've got to stop writing these before lunch.) Ahem.

I said there is a bridge here, and I have intentionally provoked all the audience about it, because I, like GKC, am intent on his great engineering project:

"The rebuilding of this bridge between science and human nature is one of the greatest needs of mankind. We have all to show that before we go on to any visions or creations we can be contented with a planet of miracles."
[GKC The Defendant 75]

Incidentally, I first read that GKC quote in the aforementioned book by Jaki! But it is a magic bridge, and dangerous, as all bridges are. The chasm it crosses is of human make, after all, and so it is much worse than any merely natural division.

GKC gives this name, the Ethics of Elfland, because he wants to give us something as one gives to a child. (And now you MUST hear those ancient words: "Amen I say to you, unless you be converted, and become as little children, you shall not enter into the kingdom of heaven." [Mt 18:3]) We need to sit together, yes, the scientists beside the lit'ry folk, and hear Uncle Gilbert tell us a story... "Will there be dragons?" Certainly. "Will there be real trees?" Oh yes. "Will we be there too?" Why, of course. (And it's a good story, I've heard it before...) You will learn as children do, about reality, and about right and wrong... no! it is NOT a sermon! Erase that thought. It is NOT that kind of tale! It is a story, about a marvellous world. (Will anyone recognize it, I wonder...)

Well, I don't want you to be confused here. This chapter is not in the form of a story. GKC keeps to his wandering wonder of words, marching to unheard music... But the music I hear (in the key of "G") is much like that famous "Promenade" from Mussorgsky's "Pictures at an Exhibition", a cumbrous but bold waddle in alternate 5/4 and 6/4 signatures, as GKC stops to look at the pictures - of the real world... and hopes we might waddle along with him and see, and admire...

You may feel, as you enter into this chapter, that it is all verbal fireworks and no fusion. Fusion, you know, is the great power source of the universe: it is what makes the sun light up:

Twinkle twinkle little star:
We know much of what you are!

Atomic fusion makes you shine,
Giving us your light so fine...
Twinkle twinkle little star:
We know much of what you are.

Now to you our eyes we lift,
Thanking God for His great gift,
Twinkle twinkle little star:
We know much of what you are.
[from "Stellar Mechanics for Kids" one of my many unpublished works.]

Ahem. But actually the fireworks are works, even if they are not always firey. As you saw, even during the boring parts of the previous chapters, we are advancing. We shall see more of this very lovely, dangerous, and interesting country, the Elvish world wherein we live... but there is something still greater ahead.

GKC begins his serious work in this nursery "fairy tale" place because he is "now to trace the roots of my personal speculation" [CW1:249] and he finds these roots, not surprisingly, in the fairy tales from his early years. Lovely and thoughtful and rich in ideas, delighting the lit'ry realm... AND! At the same time, he gives, (as Jaki indicates) great, stable, reliable underpinnings to the logical and mechanical and scientific - not by taking away, but by adding...

Not either/or, but both/AND - for such is the Boolean Yes.

For he is a bridge builder. The bridge is splendid, but the other side awaits! Hurry! Let's go!

--Dr. Thursday

PS. I fear I ought to put some kind of explanation about use of "magic", and put it down here so it will be short. (though I will most likely fail in that too!)

The delight I have in telling you about magic is because it is exquisitely relevant to GKC's title. For as I use it, (and perhaps GKC too) "magic" refers to permission, not method. "Magic" is really just another word for "authority". If you are relying on "the proper authority" for your actions (however be the precise method of their enacting), those actions are therefore good. If, however you resort to the wrong "authority" (a pretender to, or a usurper of, the real authority) then those actions are bad. This is all spelled out (no pun intended) in Biringuccio's Pirotechnia... Hence GKC says "Ethics" - for his story is not just for mere delight (which is good too), but primarily for teaching about good - hence about truth.

One more word I must add here, the word "occult", which is from the Latin for hidden, NOT for evil. When the earth shadows the moon during a lunar eclipse, the moon is occulted, or hidden. Many things are occult, especially nowadays. The means by which your computer or your car works... most likely these are hidden from you. (Do YOU know about finite state machines or semiconductors or distributors or carburetors?) In philosophical terms, even a magnet or the substance called AMBER are said to be occult - no, not because we somehow think they are "demonic" - but because the means of their workings are hidden:

...we have to go on using the Greek
name of amber as the only name of electricity because we have no notion what is the real name or nature of
electricity.
[GKC, The Common Man 170]

Yes, the Greek word "Elektron" means "amber"; and "electricity" means no more than "the strange thing amber does". Sure, we know lots about them now, and can use them in marvellous ways, as your reading this demonstrates, but they are still mysterious, and certainly not simple to explain. You need to think about this, and about words, very carefully, or you will FALL OFF THE BRIDGE.

(No we are NOT going into "magic" like you may have read in - uh... well, let that remain occult. Perhaps we'll talk more, but elsewhere, and after you've read the chapter. Not here and now.)

Please, don't get worried here. You have no cause to worry. GKC (and his awkward assistant who is writing this) wants you to receive a good gift, as one gives something safe and beneficial to a child... for "If you then being evil, know how to give good gifts to your children: how much more will your Father who is in heaven, give good things to them that ask him?" [Mt 7:11] But do watch your step as we cross the bridge...

One more thing, most unrelated. I mentioned Martin Gardner... He is quite old, and as yet is not quite convinced about the truth the GKC strived so hard to present in this and other books. Please pray for him.

Thursday's Dr. Thursday Post: Infinity

It may be a stretch of the imagination to connect last Sunday's gospel (the woman at the well) with our discussion of last Thursday - or perhaps not. The woman's "madness" was shattered - as if a spell was broken - by the Voice of Authority who told her "Go get your husband". So deep was her restoration that she was able to bring others to that same fountain... Ah. But for today I shall resist plunging into the deep waters this imagery brings up.

In thinking of insanity, and Lent, I must bring to your attention one of the most unusual and perhaps most insightful views of a gospel event I have ever read. The event is the "Good Thief" hanging in crucifixion next to Jesus - an apologist defending Christ even on Calvary! "We are but suffering as we deserve - but This One has done nothing wrong... Lord, remember me when You come into Your Kingdom."

The insightful view is not mine. It is contained in the rich notes and the amazing play-sequence, "The Man Born To Be King" of Dorothy L. Sayers (DLS), a series of radio plays she wrote on the life of Christ. I don't have the text here to transcribe, so I shall merely give you a hint of her argument. She claims that the Good Thief perhaps took Jesus to be a harmless nut-case - a crazy man - YET - the thief still treats Him kindly, and "plays along" - only to receive a most unexpected reply. The scene DLS only hints at is the one I love to ponder: for behold, later that day, the Lord would tell the thief, "Nope, I wasn't nuts, but it was kind of you to think so. The charity you showed to the harmless lunatic You showed unto Me!" A strange, yet somehow most dramatic view. Read it for yourself.

I had previously thought I would write up a "proof" about GKC's interesting mathematical bit about the circles, but there will be more of this philosophical geometry before you know it, and I don't feel like making such a long detour today. So let us proceed. We have finished GKC's comments on lunacy and madness - which he expresses using the mystery of the circles: infinite in one sense (for it has no end) yet still not so very large (for it is no bigger than it is drawn). We have seen an omnibus labelled "Hanwell" and thought about those unfortunates who believe themselves to be chickens, or glass, or Kings of England, or Jesus. We have heard of the limits of literature, the risks of reason - and been challenged to cut off our own head if it offends us. What is all this? Why are we seriously contemplating insanity? GKC has a reason, and not merely a poetic one.

Click here to continue the adventure.

GKC tells us himself what he is up to:

I have described at length my vision of the maniac for this reason: that just as I am affected by the maniac, so I am affected by most modern thinkers. That unmistakable mood or note that I hear from Hanwell, I hear also from half the chairs of science and seats of learning to-day; and most of the mad doctors are mad doctors in more senses than one. They all have exactly that combination we have noted: the combination of an expansive and exhaustive reason with a contracted common sense. They are universal only in the sense that they take one thin explanation and carry it very far. [CW1:225]

We might take this as the bridge-passage, the musical riff that brings us from Heretics to Orthodoxy. Recall that in Heretics we saw a long line of men - writers, thinkers, philosophers - men whom GKC respects, even admires - some of whom he would readily claim as friends - and yet men with whom he is in bitter and utter disagreement: "a Heretic - that is to say, a man whose philosophy is quite solid, quite coherent, and quite wrong." [Heretics CW1:46]

Those men are the men LIKE the lunatics. Note he does NOT say they ARE lunatics! He is not pulling an ad hominem argument. He is talking about a general idea, dealing with the IDEAS of those men. What does he tell us about them? He says those are the men with the SMALL PATTERNS, even though they are "infinite":

But a pattern can stretch for ever and still be a small pattern. They see a chess-board white on black, and if the universe is paved with it, it is still white on black. Like the lunatic, they cannot alter their standpoint; they cannot make a mental effort and suddenly see it black on white.[CW1:225]

He proceeds to give an example (about materialism) but almost immediately points out that he is NOT making an argument about the detail, but about the generality. He links the flaw in the materialist view of the kosmos back to the flaw in the man in the asylum. It may be true enough. But it is so much smaller a truth than can be found elsewhere.

I hope you are reading along with me - and so you will readily note that it is futile for me to try to skip the example. GKC himself tried to do that. In one of his amazing leaps, he goes from that example to a stark generality of epistemology (the study of knowledge itself): "In one sense, of course, all intelligent ideas are narrow. They cannot be broader than themselves." [CW1:226] It is the paradox of words, the strangeness of a homework assignment like "Define 'infinity' and use it in a sentence." It hints at another mysterious line of GKC's which he put in another mystery: "Alone on earth, the Church affirms that God himself is bound by reason." ["The Blue Cross" in The Innocence of Father Brown]

Whew, let's stop for a bit. Do you feel stuck in a swamp of ideas? You are wrong. It's the brisk fresh air. You are at a peak of a mountain, and seeing a vista. It's at these points where you feel most congested, you are actually most free, and actually presented with a greater wideness of vision than elsewhere. So let us pick this matter apart so we can grasp where we were and better handle where we're going next. I can't do all the epistemology, I didn't bring that in my knapsack today. Let's see if we can deal with it directly. Let's read it again:

"In one sense, of course, all intelligent ideas are narrow. They cannot be broader than themselves."

The point of the paradox is we can handle things far bigger than our hands - because we have words which can reduce infinity to eight letters. (Count them: I, N, F, I, N, I, T, Y.) The strict philosophers will now throw eggs at me, saying I have committed the "Fallacy of Equivocation", confusing the word "infinite" and the idea "infinite". But I catch the eggs, and scramble them to make our lunch. They are not reading along. (Recall "poetry floats on the infinite sea"...) It would be just as adequate for me to cite the Summa of Aquinas (I Q10 A1) to help them out, since they like that kind of citation, it shows I do read those kinds of things. Ahem! But for us, this "paradox" is as simple as this mountain-peak. We're stopped here - and need to choose a path. But we can choose ANY direction - as long as it's down. (We are walking, you know; remember we said last week, "let it be solved by walking".)

The "fallacy of equivocation" is a kind of error in logic, in the use of words. How about an example? Here's one: saying "God is limited because he is only three letters long". But GKC is telling us there is the same kind of error in saying "we cannot hold the idea of 'infinity' since it is INFINITELY BIG". It is a paradox in reason itself, not merely written by GKC, to state without further quibble, that Infinity is narrow, and God is limited. This is not because of the things-in-themselves, but because of our equipment. (We are on a journey, we are NOT going EVERYWHERE AT ONCE. We are walking, and so are SOMEWHERE.)

I will try once more. (This one is great, and will shock any computer scientists in the audience.) Watch carefully, and I will use YOUR computer to represent BIG integers, including for example, the number of electrons required to fill the sphere bounded by the diameter of the most distant galaxies. Or, even bigger: the factorial of that number. Or even bigger: that number raised to its own power... that many times. Big numbers. BIG big numbers. HUGE numbers. (Even more than GKC weighed.) I can even use the computer to deal with transfinite numbers, the mysterious "aleph-one", which is the cardinality of the real system of numbers. And there are even others... Wow, look: before your very eyes, all those things are being communicated by what I have just written! HAVE I NOT COMMUNICATED THEM TO YOU? Of course I have. They are formally represented - as ideas. No, not directly as tick-marks on a sheet of paper or tokens in a box. (Please. Don't be silly. When was the last time you saw 1000 of ANYTHING? We gave that up about 5000 years ago, when the Egyptians began to write Ç to stand for "ten".) You cannot represent such gigantic numbers by that means. It is like asking how much God weighs. It does not have meaning. But you can communicate the idea of such numbers - which means you have communicated the number. The idea of such vast quantities has a meaning, and so we can accomplish the communication of that idea. And if we failed to say it in symbols of mathematics, we would resort to the symbols of poetry: I think of "Tonight" in "West Side Story":

Today the minutes seem like hours,
The hours go so slowly...
And still the sky is light...

That is what GKC is saying. In order to talk about anything, and reason about anything, we use something narrow. We do not have the infinite time or an infinite box of tokens to play around with the real thing, so we use what we can.

OK. Maybe it was futile - I ought to stick to my own toys - so let's resume with GKC, and I will let the high-tech philosophy for others to play with. Perhaps this next sentence will tell you the same thing, which is just GKC's own version of the very important Principle of Contradiction: "Nothing can be, and not be, at the same time."

A Christian is only restricted in the same sense that an atheist is restricted. He cannot think Christianity false and continue to be a Christian; and the atheist cannot think atheism false and continue to be an atheist.
[CW1:226]

Again, please read this carefully. You need to think about the simple sentences, not about some deep quippy insult or brag. You can put in any partisan or sectarian words you like, and it has JUST the SAME meaning and power. YOU ARE ON A PEAK of FREEDOM, my friend, not stuck in a swamp! Try it again. Then we'll proceed.

Now that you have a NEW tool, then, we shall actually approach this example of materialism - and its opponent, spiritualism. (We are using the terms rather generically here; materialism means there is nothing but material: nothing spiritual at all. Whereas spiritualism means there also exists an unseen realm.)

I shall quote at length again, because you are surely tired of reading MY words, and also because the "verbal fireworks" here are SO good:

...there is a very special sense in which materialism has more restrictions than spiritualism. Mr. McCabe thinks me a slave because I am not allowed to believe in determinism. I think Mr. McCabe a slave because he is not allowed to believe in fairies. But if we examine the two vetoes we shall see that his is really much more of a pure veto than mine. The Christian is quite free to believe that there is a considerable amount of settled order and inevitable development in the universe. But the materialist is not allowed to admit into his spotless machine the slightest speck of spiritualism or miracle. Poor Mr. McCabe is not allowed to retain even the tiniest imp, though it might be hiding in a pimpernel. The Christian admits that the universe is manifold and even miscellaneous, just as a sane man knows that he is complex. The sane man knows that he has a touch of the beast, a touch of the devil, a touch of the saint, a touch of the citizen. Nay, the really sane man knows that he has a touch of the madman. But the materialist's world is quite simple and solid, just as the madman is quite sure he is sane. The materialist is sure that history has been simply and solely a chain of causation, just as the interesting person before mentioned is quite sure that he is simply and solely a chicken. Materialists and madmen never have doubts.
[CW1:226-7]

(An aside: if you are wondering who "McCabe" is, you can read his chapter in Heretics CW1:157 et seq. Joseph McCabe (1867-1955) was a Roman Catholic priest turned rationalist.)

And while I greatly doubt that you can possibly be satisfied with my writing, here I must leave you for today. Please try to think a little about these things. Not about the math, or about the epistemology, the knowledge OF meanings of words and ideas - but ABOUT the meanings, and the ideas.

Still lost? When we think about our mother (let us say) we do not think about her picture, but about HER. But when we talk about her, we may show the picture, or use that six-letter word - but everyone knows who it is we are talking about, even if they have never met her or seen her. IN THE SAME WAY: when we think about infinity, we do not think about that splendid and funny little proof of the math dudes about "increasing without bound" or about a bottomless box of tokens - nor simply about that eight-letter word - but we use that word to talk to others about, as I have just done with you.

And this limited limitlessness applies even to the matter of God, which we do not narrow to a mere word of three letters, and Who has even more meaning and even more intimacy to us than our very mothers...

Onward to the next the peak, dudes!

--Dr. Thursday

Dr. Thursday's Thursday Post

Practical White Magic: Climbing to Step One-and-a-Fraction

...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.
[GKC ILN Mar 19 1927 CW34:274]

He was talking about cartoons, yes indeed. But this is just a curious little pattern.

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."
[GKC "The Dagger With Wings" in The Incredulity of Father Brown]

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.

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