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.