Fractals: It’s What’s for Dinner

Romanesco broccoli from the Mt. Pleasant farmers' market, Washington, D.C., 10 October 2009

Ever since I read that romanesco broccoli was a fractal I’ve been on the lookout for it. It finally turned up along with all the varieties of cauliflower at the Mt. Pleasant farmer’s market, so I snatched it up and tonight I broke that fractal down into-a little-a tiny cubes and fried it in olive oil, salt and pepper and white wine.

(My picture is nowhere as cool as this New York Times picture of the day from 7 October 2009)

Information Phenotype

The fractal tree at the corner of 18th and Lamont, Washington, D.C., 29 April 2009

I have previously commented on my love of the movie π (“The Supernovae in Your Coffee Cup,” 2 November 2008). It left me with two enduring images: mixing coffee and cream as an example of turbulence, previously discussed; and the branching of tree limbs as an example of fractal symmetry. I love winter for its exposure of this fabulous phenomena, innocuously right over our heads. I am always a little sad for the arrival of spring and the enshrouding of all these thought-provoking fractals in greenery.

The picture above is of my favorite tree in the neighborhood where I live. The degree to which the pattern of major arc over two-thirds of growth length followed by sharp break and lesser arc over remainder of growth length is repeated trunk to twig is amazing. Notice the arc of the trunk: unlike many trees which follow one rule for trunk and a separate rule for the branches, this tree follows a single rule throughout.

We think of a fractal as a recursive algorithm, a mathematical formula. But there’s no math in that tree. The recipe for that fractal is coded somewhere in the tree’s DNA. But the DNA contains no fractal. The DNA is a bunch of nucleotides that are transcribed by messenger RNA that code amino acids that assemble into proteins that form the structures of cells. The cells then split and differentiate in response to a complex of internal chemical signals and environmental stimuli to grow in a pattern that is the fractal.

One might say that there is a fractal somewhere in that tree, but there are so many transformation rules between nucleotide sequence and fractal growth pattern, that it is only in a manner of speaking. I am reminded of Wittgenstein’s discussion of what constitutes following a rule and going against it (Philosophical Investigations, trans. G. E. M. Anscombe [Malden, Mass.: Blackwell, 1953]):

198. “But how can a rule shew me what I have to do at this point? Whatever I do is, on some interpretation, in accord with the rule.” — That is not what we ought to say, but rather: any interpretation still hangs in the air along with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning.

“Then can whatever I do be brought into accord with the rule?” — Let me ask this: what has the expression of a rule — say a sign-post — got to do with my actions? What sort of connection is there here? — Well perhaps this one: I have been trained to react to this sign in a particular way, and now I so react to it.

But that is only to give the causal connection; to tell how it has come about that we now go by the sign-post; not what this going-by-the-sign really consists in. On the contrary; I have further indicated that a person goes by a sign-post only in so far as there exists a regular use of sign posts, a custom.

199. Is what we call “obeying a rule” something that it would be possible for only one man to do, and to do only once in his life? — This is of course a note on the grammar of the expression “to obey a rule.”

It is not possible that there should have been only one occasion on which someone obeyed a rule. It is not possible that there should have been only one occasion on which a report was made, an order given or understood; and so on. — To obey a rule, to make a report, to give an order, to play a game of chess, are customs (uses, institutions).

To understand a sentence means to understand a language. To understand a language means to be master of a technique.

200. It is, of course, imaginable that two people belonging to a tribe unacquainted with games should sit at a chess-board and go through the moves of a game of chess; and even with all the appropriate mental accompaniments. And if we were to see it we should say they were playing chess. But now imagine a game of chess translated according to certain rules into a series of actions which we do not ordinarily associate with a game — say into yells and stamping of feet. And now suppose those two people to yell and stamp instead of playing the form of chess that we are used to; and this in such a way that their procedure is translatable by suitable rules into a game of chess. Should we still be inclined to say that they were playing a game? What right would one have to say so?

201. This is our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule. The answer was: if everything can be made out to accord with the rule, then it can also be made out to conflict with it. And so there would be neither accord nor conflict here.

It can be seen that there is a misunderstanding here from the mere fact that in the course of our argument we gave one interpretation after another; as if each one contented us at least for a moment, until we thought of yet another standing behind it. What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call “obeying the rule” and “going against it” in actual cases.

Of course Wittgenstein is writing about social phenomena where custom and training are factors, but the undecidability of rules is the point here. Socially dogmatic, we are dismissive of blatant divergence from consensus. Less dogmatic — but not free of dogma — science resorts to the metaphysical-aesthetic notion of Ockham’s razor with which to cut through the myriad of rules that might potentially be made to accord with observed behavior. Is there really a fractal in the tree’s DNA? The fractal pattern of tree growth is but an interpretation of the tree’s DNA — an interpretation that would be different given a differing machinery of RNA transcription, amino acid assembly, protein expression, etc.

Ouroboros to Mise en Abyme

A few unsystematic thoughts on Ouroboros and mise en abyme:

  • I almost mentioned Douglas Hofstadter’s book, I Am a Strange Loop, in last week’s post (“The Mythical Economy,” 23 July 2008). He could have gone with Ouroboros on the cover too, but instead he went with mise en abyme.

    Or maybe he couldn’t have gone with Ouroboros. While Ouroboros is, on a superficial level, obviously a strange loop and a symbol that could be seen as self-referential, a peek at the index of Hofstadter’s book at the entry for “video feedback” — a technological mise en abyme — shows that he has a thicker analogy in mind:

    video feedback, 65-71; as candidate for strange loop, 103, 187, 203, 361; epiphenomena in, 68, 70-71; fear of meltdown, 56, 57; fractalic gestalts of, 204; lack of “I” in, 203; lack of perception in, 75-77, 187, 203; lack of symbols in, 203; lack of thinking in, 203; locking-in of patterns in, 70; parameters of, 65-67, 69, 75; reverberation in, 67-68; two systems entwined in, 210-211, 253-254; vanilla loop in, 208

  • While I’m amused at the notion of an Ouroboros economy, I can’t really think of any real correlate to the slightly humorous image. Unless maybe something like a naturalistic notion of the human economy, wherein the human economy is nature parasitic upon itself. The destruction of the biological world as giving birth to the artifactual or the cybernetic world. Ouroboros reborn for the Twenty-first Century!

  • The thing that’s really causing me to bring up mise en abyme is some thoughts on how people think about the future. People are faced with the need to decide and nearly all decisions that people make are, when not completely about the future, at least future-oriented. People’s thoughts about the future are divided into two closely related activities, carried out in tandem: planning and prediction. Prediction occasionally becomes an activity of its own, but for the most part prediction is an activity carried out in service of the more pragmatic planning.

    Planning is a branching strategic game. It works like this. I have a goal whose attainment is not simple: it involves a number of steps and it could be thwarted at any one of them. I start with my known situation and have a vague idea what the path to my goal would be and I make a series of hypothetical decisions. I test the soundness of a hypothetical decision by predicting the outcome of such an action. That is, I imagine a potential future.

    In the first round, the one inaugurated in the present, I know what my options are because they are present. In the second round and in all subsequent rounds, I must employ prediction to imagine what options I will have to choose from because from then on I am dealing in an imagined future. I repeat the hypothetical decision, predict, test, simulate new options algorithm down until I reach the last round whose decision results in the attainment of the goal.

    When I make predictions about the future, I rarely make a single prediction, since prediction, especially the sort of intuitionistic variant that people employ for the purpose of most of their planning, is not a very reliable. So I predict a range of possible futures. And in each possible future I face a range of possible decisions that I can take. Predicting and planning branch. Most of these I abandon in favor of the most fruitful seeming paths. But if a path dead-ends, I back up until I find the probable fateful decision that sent me down the path to the dead end. I recollect the other options at that possible future and imagine my way down another branch. I also generally flag a number of contingency plans. I went with this predicted future, but as things actually unfold, if it turns out that I predicted wrong, I have a plan ready for that other branch too.

    When I have what I imagine to be a satisfactory path from present to goal, I lock in each decision hypothetically made into “what I’ve decided upon.”

    This is a pretty systematic model and not necessarily exactly how most people make plans. People rarely sit town and carry it out algorithmically from beginning to end. More frequently people engage in this activity in fits and starts, not taking the problem from start to finish, but working on pieces that strike them at various occasions throughout their day. They absentmindedly do it while at their work computer, or do it extremely quickly while laying a joint plan with a partner over the telephone. Or maybe they try to be thorough about it and make a list on a notepad so they can see what’s done and what still in need of attention. Whatever the case, I think that ultimately this is what people are doing.

    The important point for mise en abyme is that near future decisions can only be locked in once more distant future decisions have been validated. Each step is dependent on the one after it having been made first. One starts the planning and predicting from the present and works one’s way forward, but one decides, as it were, backward, from the future to the present. Predictions and plans regarding the immediate future include as a part within them predictions and plans regarding the immediate future, which in turn contain predictions and plans about the distant future and so on. My thoughts about the future are mise en abyme insofar as they contain within them further thoughts about more distant futures.

    What one is doing in this process of planning for the future is conducting is a depth first search of potential futures. And depth first search is canonically thought of as recursive.

  • Mise en abyme seems to have a lot more analogistic or systemizing potential. Scale symmetry (e.g. fractals) along with all the related phenomena that can be grouped under that pattern seem coterminous with mise en abyme. Hegel’s logical schema seems like a highly abstract instance of mise en abyme, where each intellectual system is subsumed into a higher order intellectual system.

  • Perhaps there is a historical development of the conceptual depth and sophistication of the idea of self-referentiality. Ouroboros is simple cyclicality, though extrapolated into a notion of infinity or eternity. Homunculus is a limited instance of scale symmetry. Modern formal recursion are the culmination.