Aug 122006

The idea that aesthetics has a lot to do with self-similarity occurred to Benoit Mandelbrot thirty or so years before it occured to me, although if you ask Mandelbrot there are vanishingly few ideas that didn’t occur to him first. It’s Newton, Einstein, Archimedes, and Mandelbrot, not necessarily in that order.

So it isn’t original, which doesn’t mean it isn’t true. Mandelbrot’s The Fractal Geometry of Nature is one of the best-selling books on mathematics of all time, mostly because it has a lot of pretty pictures. Fractals are fun to look at. Little copies of the whole structure are buried everywhere within it. Zoom in to 10 angstroms, or out to 10,000 feet: in any part of the picture the same intricate patterns appear.

In self-similarity the part resembles the whole. Symmetry, a different affair, describes a relation between parts. Self-similarity implies some form of symmetry. If A and B are both part of C, and both resemble C, then A will also resemble B, in some sense. But the converse is false. Parts A and B may resemble each other without either resembling the whole, C, in the slightest. One tends to think of self-similarity (and symmetry) as visual, but the principle is structural. The play-within-a-play and the colloquy of the gravediggers in Hamlet are instances of self-similarity. For symmetry, take a lesser play, say Noel Coward’s Nude with Violin, in which a famous modern artist resembling Picasso dies and turns out to have hired a different person to produce the work for each of his “periods.” Each act, in which one of the period painters shows up to claim a share of the estate, resembles the others, but none especially resembles the whole. The play is symmetrical but not self-similar.

Christopher Alexander, in The Nature of Order, gives fifteen principles for “living” architecture, at least six of which can be subsumed under self-similarity. “Local symmetries,” “deep interlock and ambiguity,” “echoes,” “positive space,” “good shape,” and “gradients” are all aspects of what I’m talking about. And a lucky thing too, as one of my principles is that no list of principles should reach double digits.

Alexander also includes “roughness” in his list, which is very much to the point. Fractals fall into two categories: exact and approximate. The first known fractal (arguably), the famous Koch curve, a mathematical monstrosity with unbounded perimeter but finite area, is exact. It is also boring. When you’ve seen one level you’ve seen it all. Everyone likes to look at baby animals. What makes them “cute” is that they are copies of their parents, but inexact ones. They belong to category two.

Self-similarity is a deeply ingrained way for people to see the world. All pre-scientific theories of genesis involve self-similarity. Embryos were first thought to be full-grown animals in miniature, with tiny heads and limbs. It sounds absurd now, but if you knew nothing of cell division and had no microscope handy you might make the same guess. Lamarck precedes Darwin because he assumes that all traits are inherited, that is, that self-similarity applies in full.

If art, as is commonly alleged, speaks to our deepest selves, one would expect it to show self-similarity everywhere, and sure enough it does. Pictures, like Hokusai’s “The Great Wave,” which Mandelbrot reproduces in his book, are the first place to look. Jackson Pollock’s drip paintings are fractal. Music, to my untutored ear, is replete with structures that repeat approximately on different scales, such as leitmotifs taken up first by the winds, then the strings, the brass, and eventually the whole orchestra. I will leave this speculation to those who know, as I do not.

Poetry, however, I do know something about, and perfectly symmetrical verse forms have never gained much traction in English. The sonnet has fourteen lines, and in neither of its standard forms do these lines divide into identical groups. The Petrarchan version has its octet and sextet, and the Elizabethan its three quatrains, along with that last awkward couplet that Shakespeare could never quite figure out what to do with. The villanelle, with its six triplets and 19th line, exhibits the same sort of approximate symmetry. Then there’s the Spenserian stanza, eight lines of pentameter with an alexandrine tacked on to the end. A while back I remarked the stupor produced by extended passages of Pope, in reams of closed, perfectly balanced heroic couplets. Neatness is not all your second-grade teacher cracked it up to be.

Great poems exhibit symmetry and, especially, self-similarity in a high degree, though this may not be obvious to the casual reader. Consider Ben Jonson’s To Heaven:

1 Good and great God, can I not think of Thee,
2 But it must, straight, my melancholy be?
3 Is it interpreted in me disease,
4 That, laden with my sins, I seek for ease?
5 O be Thou witness, that the reins dost know
6 And hearts of all, if I be sad for show;
7 And judge me after: if I dare pretend
8 To aught but grace, or aim at other end.
9 As Thou art all, so be Thou all to me,
10 First, midst, and last, converted One and Three,
11 My faith, my hope, my love; and in this state,
12 My judge, my jury, and my advocate.
13 Where have I been this while exiled from Thee,
14 And whither rapt, now Thou but stoop’st to me?
15 Dwell, dwell here still! O, being everywhere,
16 How can I doubt to find Thee ever here?
17 I know my state, both full of shame and scorn,
18 Conceived in sin, and unto labor born,
19 Standing with fear, and must with horror fall,
20 And destined unto judgment after all.
21 I feel my griefs too, and there scarce is ground
22 Upon my flesh t’inflict another wound.
23 Yet dare I not complain or wish for death,
24 With holy Paul, lest it be thought the breath
25 Of discontent; or that these prayers be
26 For weariness of life, not love of Thee.

Jonson longs for death while rejecting that longing intellectually. To understand this you must enter partly into the Christian experience. If you refuse, then a large portion of great art is partly or wholly closed to you, and the loss is not easily afforded. It may help to remember that you would likely be Christian had you been born in England in the 16th century. Human beings were no different then, you are extremely unlikely to be more intelligent than Ben Jonson was, and the number of atheists in England at the time was an engineering zero.

The poem is a prayer and an apology, and contains, in good fractal style, several prayers and apologies within itself. It is written in heroic couplets, but is 26 lines long, and naturally resists, like most of the classic English verse forms, division into equal parts. There are no enjambments until the very end, in lines 24 and 25. Every full stop ends a line. Yet the movement escapes Pope’s monotony because the argument moves forward and the syntactic unit varies. There are nine sentences (if we count the ejaculation that begins line 15 as part of the question); In line count they run two, two, four, four, two, two, four, two, and four.

The first sixteen lines deal mostly with God, the last ten mostly with man. Both sections are syntactically symmetrical. The first opens with two questions of two lines apiece and closes the same way. The second reverses the procedure, surrounding the short sentence with the two longer ones.

In lines 11 and 12 “faith, hope, and love” are exactly the qualities that a Christian might exhibit toward his “judge, jury, and advocate,” respectively — no other order would do. Jonson’s sleight-of-hand at the end of line 10, where he interposes a reference to the Trinity (“converted One and Three”) to break up his literal trinities, simply beggars praise.

The Elizabethan fondness for wordplay is employed tastefully, for once (cf. “when first your eye I eyed”). “Everywhere” and “ever here,” in lines 15 and 16, are effective; but “disease” and “ease,” in lines 3 and 4, are a masterstroke. “Ease” implies “death,” and sure enough we find “death” at the end of line 23, as distant from the end of the poem as “ease” is from the beginning. “Thee” rhymes with “be” to begin the poem, and “be” with “Thee” to end it. “Me” and “Three” and “Thee” and “me” are rhymed again almost exactly in the middle of the poem, separated by one couplet. “[J]udge me after” in line 7 balances “judgment after all” in line 20. “After all,” a throat-clearing device in most writers, including me, is resurrected here to become living language.

Despite a theme that will not resonate with most modern readers, To Heaven remains one of the most beautiful poems in the language. Can there be any doubt that it is largely the structure — the fractal — that we respond to?

  7 Responses to “The Fractal Geometry of a Poem”

  1. Actually self-similarity still applies pretty well to embryology. Ontogeny doesn’t strictly recapitulate phylogeny, but it does resemble it.

    Fascinating, and not just because Pollock suddenly makes sense to me. I wonder why we find self-similarity so attractive and interesting.

  2. “If art, as is commonly alleged, speaks to our deepest selves, one would expect it to show self-similarity everywhere, and sure enough it does. Pictures, like Hokusai’s “The Great Wave,” which Mandelbrot reproduces in his book, are the first place to look. Jackson Pollock’s drip paintings are fractal.”

    This is a bit confusing. Is it your contention that all art exhibits self-similarity, or only good art? Hokusai and a legion of abstract pollocks don’t count for a large percentage–most “Old Master” painting, or even most Modernism, doesn’t seem very self-similar to me. Music, which almost universally depends on repetition, seems a better bet.

    And what about extended literature? There may be some obvious isolated examples, eg. ‘Finnegans Wake’ or ‘Invisible Cities’ or some Borges story, or a play within a play here, a Chinese box (‘Wuthering Heights’) there–but it’s not the norm, is it?

    Is the upshot of this post, then, only that you like a bit of self-similarity? Structural cohesion?

    Perhaps we can agree on that.

  3. Conrad: perhaps we can agree if I put it a bit differently. Insofar as art lacks mimetic qualities, it relies on structural qualities for its effect, and chief among those are symmetry and self-similarity. Therefore these are found universally in music and abstract or quasi-abstract painting, very commonly in poetry, less commonly in extended literature, and less commonly still in realistic, “Old Master” painting. Your examples go to this point.

  4. In which case, I think the next step would be to ask why those particular artforms invite this type of structural principle. Poetry, being smaller, can more easily accommodate small-scale self-similarities. Abstract painting perhaps needs it for want of other sources of interest–like music, it has no semantic content, and hence relies purely on an inner harmony of form. Poetry aspires to the condition of music (Pater, of course). So: the greater the semantic content of an artform, the less it relies on self-similarity.

    How about that?

  5. you’ll begin to be right about WS when you start thinking about the sonnets rather as a sequence than a collection.

  6. I found this example of fractal poetry (resemblig the Cantor set), in a short but interesting article:


  7. He couldn't deal with all the criticisms of his labyrinthine definition of &##93racism&;39; so he shut down the thread. lol. F*ck him. There are a number of others doing the same thing he does, so who needs him?

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