# Möbius Tripping

“In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)” — Stanislaw Ulam (1909-84)

# Square Routes Re-Re-Re-Re-Re-Revisited

For a good example of how more can be less, try the chaos game. You trace a point jumping repeatedly 1/n of the way towards a randomly chosen vertex of a regular polygon. When the polygon is a triangle and 1/n = 1/2, this is what happens:

Chaos triangle #1

Chaos triangle #2

Chaos triangle #3

Chaos triangle #4

Chaos triangle #5

Chaos triangle #6

Chaos triangle #7

As you can see, this simple chaos game creates a fractal known as the Sierpiński triangle (or Sierpiński sieve). Now try more and discover that it’s less. When you play the chaos game with a square, this is what happens:

Chaos square #1

Chaos square #2

Chaos square #3

Chaos square #4

Chaos square #5

Chaos square #6

Chaos square #7

As you can see, more is less: the interior of the square simply fills with points and no attractive fractal appears. And because that was more is less, let’s see how less is more. What happens if you restrict the way in which the point inside the square can jump? Suppose it can’t jump twice towards the same vertex (i.e., the vertex v+0 is banned). This fractal appears:

Ban on choosing vertex [v+0]

And if the point can’t jump towards the vertex one place anti-clockwise of the currently chosen vertex, this fractal appears:

Ban on vertex [v+1] (or [v-1], depending on how you number the vertices)

And if the point can’t jump towards two places clockwise or anti-clockwise of the currently chosen vertex, this fractal appears:

Ban on vertex [v+2], i.e. the diagonally opposite vertex

At least, that is one possible route to those three particular fractals. You see another route, start with this simple fractal, where dividing and discarding parts of a square creates a Sierpiński triangle:

Square to Sierpiński triangle #1

Square to Sierpiński triangle #2

Square to Sierpiński triangle #3

Square to Sierpiński triangle #4

[…]

Square to Sierpiński triangle #10

Square to Sierpiński triangle (animated)

By taking four of these square-to-Sierpiński-triangle fractals and rotating them in the right way, you can re-create the three chaos-game fractals shown above. Here’s the [v+0]-ban fractal:

[v+0]-ban fractal #1

[v+0]-ban #2

[v+0]-ban #3

[v+0]-ban #4

[v+0]-ban #5

[v+0]-ban #6

[v+0]-ban #7

[v+0]-ban #8

[v+0]-ban #9

[v+0]-ban (animated)

And here’s the [v+1]-ban fractal:

[v+1]-ban fractal #1

[v+1]-ban #2

[v+1]-ban #3

[v+1]-ban #4

[v+1]-ban #5

[v+1]-ban #6

[v+1]-ban #7

[v+1]-ban #8

[v+1]-ban #9

[v+1]-ban (animated)

And here’s the [v+2]-ban fractal:

[v+2]-ban fractal #1

[v+2]-ban #2

[v+2]-ban #3

[v+2]-ban #4

[v+2]-ban #5

[v+2]-ban #6

[v+2]-ban #7

[v+2]-ban #8

[v+2]-ban #9

[v+2]-ban (animated)

And taking a different route means that you can find more fractals — as I will demonstrate.

# Tolk of the Devil

I’ve said it before and I’ll say it again: I wish someone would translate Lord of the Rings (1954-5) into English. By that I mean (of course) that I wish someone would translate LOTR into good English. I’ve looked at Tolkien’s bad English in “Noise Annoys” and “Science and Sorcery”. Here’s another example:

Pippin declared that Frodo was looking twice the hobbit that he had been.

“Very odd,” said Frodo, tightening his belt, “considering that there is actually a good deal less of me. I hope the thinning process will not go on indefinitely, or I shall become a wraith.”

“Do not speak of such things!” said Strider quickly, and with surprising earnestness. – The Fellowship of the Ring (1954), Chapter 11, “A Knife in the Dark”

Strider should have added: “Or in such a way!” In the second paragraph, Frodo suddenly talks like a Guardian-reader. Why on earth did Tolkien use “thinning process”, “indefinitely” and “actually” amid otherwise good, simple English? Thinning is obviously a “process”, so there’s no need to say it is, and “indefinitely” and “actually” are badly out of a place in a fantasy novel, let alone in dialogue there. “Considering” is less bad, but it should go too. I would improve the paragraph like this:

“Very odd,” said Frodo, tightening his belt, “seeing that there is now a good deal less of me. I hope the thinning will not go on much longer, or I shall become a wraith.”

Now there’s nothing incongruous: the only un-English word is “very”, but that doesn’t seem un-English on the tongue or to the eye. The Guardianese is gone, but it should never have been there in the first place. Tolkien should not have written like that in Lord of the Rings. And not just as a professional scholar of language: simply as a literate Englishman. H.W. Fowler’s Modern English Usage (1926) had been in print for twenty-eight years when The Fellowship of the Ring was first published. It’s hard to believe that Tolkien wasn’t familiar with it.

If he wasn’t, that’s a great pity. If he was, the bad prose in LOTR becomes even more inexplicable and unforgiveable. Alas for what might have been! Imagine if, per impossibile, Tolkien’s masterwork had been edited by the second-greatest Catholic writer of the twentieth-century, namely, Evelyn Waugh.

When bad prose appears in something by Waugh, it’s deliberate:

I had a fine haul – eleven paintings and fifty odd drawings – and when eventually I exhibited them in London, the art critics, many of whom hitherto had been patronizing in tone as my success invited, acclaimed a new and richer note in my work.

Mr. Ryder [the most respected of them wrote] rises like a fresh young trout to the hypodermic injection of a new culture and discloses a powerful facet in the vista of his potentialities … By focusing the frankly traditional battery of his elegance and erudition on the maelstrom of barbarism, Mr. Ryder has at last found himself.Brideshead Revisited (1945), Book II, “A Twitch Upon the Thread”, ch. 1

Waugh was deliberately mocking the mixed-metaphor-strewn prose and pretensions of modern critics. Waugh paid great attention to language and compared writing to carpentry. It was a craft and good craftsmen do not work carelessly or use bad materials. Nothing in Brideshead is careless or casual, as we can see when the narrator, Charles Ryder, first meets the “devilish” æsthete Anthony Blanche, who has “studied Black Art at Cefalù” with Aleister Crowley and is “a byword of iniquity from Cherwell Edge to Somerville”. Blanche has a stutter and Waugh uses the stutter to underline his iniquity. Or so I would claim. Here is Blanche engaging in some papyrocentric performativity:

After luncheon he stood on the balcony with a megaphone which had appeared surprisingly among the bric-à-brac of Sebastian’s room, and in languishing, sobbing tones recited passages from The Waste Land to the sweatered and muffled throng that was on its way to the river.

“’I, Tiresias, have foresuffered all,’” he sobbed to them from the Venetian arches –
“Enacted on this same d-divan or b-bed,
I who have sat by Thebes below the wall
And walked among the l-l-lowest of the dead….”

And then, stepping lightly into the room, “How I have surprised them! All b-boatmen are Grace Darlings to me.” Brideshead Revisited, Book I, “Et in Arcadia Ego”, ch. 1

Talking about the Greek sage Tiresias, who experienced life as both a man and a woman, Anthony Blanche, a man whose surname is the feminine form of the French adjective blanc, meaning “white”, stumbles over the initial consonants of three words: “divan”, “bed” and “lowest”. Is it a coincidence that the same consonants, in the same order, appear in the Greek diabolos, meaning “devil”?

I don’t think so. If Blanche had stuttered on “surprised” too, I would be even more certain. But the –s isn’t essential. After all, it was lost as diabolos journeyed from Greek to Latin, from Latin to French, and from French to English, where it appears as “Devil”. And what does Charles Ryder later call Anthony Blanche after Blanche has spent an evening tête-à-tête trying to turn Ryder against Ryder’s great friend Sebastian Flyte? You can find out here, as Ryder discusses the evening with Sebastian:

“I just wanted to find out how much truth there was in what Anthony said last night.”

“I shouldn’t think a word. That’s his great charm.”

“You may think it charming. I think it’s devilish. Do you know he spent the whole of yesterday evening trying to turn me against you, and almost succeeded?”

“Did he? How silly. Aloysius wouldn’t approve of that at all, would you, you pompous old bear?” – Brideshead Revisited, Book I, “Et in Arcadia Ego”, ch. 2

Blanche is “devilish” and his reputation for “iniquity” is well-deserved. That’s why I think the three words over which Blanche stutters were carefully chosen by Waugh from The Waste Land. Waugh was a logophile and that is exactly the kind of linguistic game that logophiles like to play.

# Thalassic Classic

The Silver Moon Nocturne by Montague Dawson (1890-1973)

(Click for larger)

# Lesz is More

• Matematyka jest najpotężniejszym intelektualnym wehikułem, jaki kiedykolwiek został skonstruowany, za pomocą którego uciekamy przed czasem, lecz nie ma powodu przypuszczać, że mogłaby kiedyś umożliwić tego rodzaju ucieczkę, jaką ucieleśnia pogoń za Absolutem. — Leszek Kołakowski

• Mathematics is the most powerful intellectual vehicle that has ever been constructed, by means of which we flee ahead of time, but there is no reason to suppose that it could someday enable the kind of escape embodied by the pursuit of the Absolute. — Leszek Kołakowski

# Koch Rock

The Koch snowflake, named after the Swedish mathematician Helge von Koch, is a famous fractal that encloses a finite area within an infinitely long boundary. To make a ’flake, you start with an equilateral triangle:

Koch snowflake stage #1 (with room for manœuvre)

Next, you divide each side in three and erect a smaller equilateral triangle on the middle third, like this:

Koch snowflake #2

Each original straight side of the triangle is now 1/3 longer, so the full perimeter has also increased by 1/3. In other words, perimeter = perimeter * 1⅓. If the perimeter of the equilateral triangle was 3, the perimeter of the nascent Koch snowflake is 4 = 3 * 1⅓. The area of the original triangle also increases by 1/3, because each new equalitarian triangle is 1/9 the size of the original and there are three of them: 1/9 * 3 = 1/3.

Now here’s stage 3 of the snowflake:

Koch snowflake #3, perimeter = 4 * 1⅓ = 5⅓

Again, each straight line on the perimeter has been divided in three and capped with a smaller equilateral triangle. This increases the length of each line by 1/3 and so increases the full perimeter by a third. 4 * 1⅓ = 5⅓. However, the area does not increase by 1/3. There are twelve straight lines in the new perimeter, so twelve new equilateral triangles are erected. However, because their sides are 1/9 as long as the original side of the triangle, they have 1/(9^2) = 1/81 the area of the original triangle. 1/81 * 12 = 4/27 = 0.148…

Koch snowflake #4, perimeter = 7.11

Koch snowflake #5, p = 9.48

Koch snowflake #6, p = 12.64

Koch snowflake #7, p = 16.85

Koch snowflake (animated)

The perimeter of the triangle increases by 1⅓ each time, while the area reaches a fixed limit. And that’s how the Koch snowflake contains a finite area within an infinite boundary. But the Koch snowflake isn’t confined to itself, as it were. In “Dissecting the Diamond”, I described how dissecting and discarding parts of a certain kind of diamond could generate one side of a Koch snowflake. But now I realize that Koch snowflakes are everywhere in the diamond — it’s a Koch rock. To see how, let’s start with the full diamond. It can be divided, or dissected, into five smaller versions of itself:

Dissectable diamond

When the diamond is dissected and three of the sub-diamonds are discarded, two sub-diamonds remain. Let’s call them sub-diamonds 1 and 2. When this dissection-and-discarding is repeated again and again, a familiar shape begins to appear:

Koch rock stage 1

Koch rock #2

Koch rock #3

Koch rock #4

Koch rock #5

Koch rock #6

Koch rock #7

Koch rock #8

Koch rock #9

Koch rock #10

Koch rock #11

Koch rock #12

Koch rock #13

Koch rock (animated)

Dissecting and discarding the diamond creates one side of a Koch triangle. Now see what happens when discarding is delayed and sub-diamonds 1 and 2 are allowed to appear in other parts of the diamond. Here again is the dissectable diamond:

Dia-flake stage 1

If no sub-diamonds are discarded after dissection, the full diamond looks like this when each sub-diamond is dissected in its turn:

Dia-flake #2

Dia-flake #3

And now discard everything but sub-diamonds 1 and 2:

Dia-flake #4

Dia-flake #5

Dia-flake #6

Dia-flake #7

Dia-flake #8

Dia-flake #9

Dia-flake #10

Now full Koch snowflakes have appeared inside the diamond — count ’em! I see seven full ’flakes:

Dia-flake #11

Dia-flake (animated)

But that isn’t the limit. In fact, an infinite number of full ’flakes appear inside the diamond — it truly is a Koch rock. Here are examples of how to find more full ’flakes:

Dia-flake 2 (static)

Dia-flake 2 (animated)

Dia-flake 3 (static)

Dia-flake 3 (animated)

Previously pre-posted:

Dissecting the Diamond — other fractals in the dissectable diamond