L-systems for George
by Cynthia Beal
Posted to the Novelty List, 1997-07-12

Looking for
Timewave Zero?
I've been reading a wonderfully fascinating book called "The Algorithmic Beauty of Plants" by Przemyslaw Prusinkiewicz and Aristid Lindenmayer. If you [George] haven't seen or read it I highly recommend it to you. The book explores L-systems, and I'll give you a couple of random quotes:

Lindenmayer systems - or L-systems for short - were conceived as a mathematical theory of plant development ... The emphasis was on plant topology, that is, the neighborhood relations between cells or larger plant modules ... The central concept of L-systems is that of rewriting. In general, rewriting is a technique for defining complex objects by successively replacing parts of a simple initial object using a set of rewriting rules or productions. The classic example of a graphical object defined in terms of rewriting rules is the snowflake curve, proposed in 1905 by von Koch. Mandelbrot restates this construction [his fractal stuff, etc.] ...

The most extensively studied and best understood rewriting systems operate on character strings ... a wide interest in string rewriting was spawned in the late 1950s by Chomsky's work [context-free class of formal grammars]. He applied the concept of rewriting to describe the syntactic features of natural languages. A few years later Backus and Naur introduced a rewriting-based notation in order to provide a formal definition of the programming language ALGOL-60 ...

In 1968 a biologist, Aristid Lindenmayer, introduced a new type of string-rewriting mechanism [L-system] ... the difference between Chomsky grammars and L-systems is in the method of applying productions. In Chomsky grammars productions are applied sequentially, whereas in L-systems they are applied in parallel and simultaneously ... This difference reflects the biological motivation of L-systems. Productions are intended to capture cell divisions in multicellular organisms, where many divisions may occur at the same time ...

... The curves [in diagram] belong to the class of FASS curves (an acronym for space-filling, self-avoiding, simple and self-similar) which can be thought of as finite, self-avoiding approximations of curves that pass through all points of a square. McKenna [not The Terence, but it is a "coincidence" - cab] presented an algorithm for constructing FASS curves using edge replacement. It exploits the relationship between such a curve and a recursive subdivision of a square into tiles ...

If you're looking at the diagrams as you're following the discussion of the application of these algorithms throughout the book, you see that these various L-systems and their counterparts are tools of pattern generation. They allow for edge-creation (defining form), replication, growth, branching, destruction, cessation, and even chaos and randomity. The complexity of the modeling languages is fascinating and their output is inspiring - the book is filled with plants and flowers and landscapes generated by the formulas that are beautiful and realistic, given the relative sterility of their context - a couple of rules and no other influences - bugs, wind, sun, universe, etc.

Algorithms can be constructed and used to search a given space for all possible arrangements of potential FASS constructs - it's a short leap to imagine our material space as a huge exploratory process of matter uncovering and unfolding itself as the generated variety of Unpredictable Inevitabilities.

Once I was exposed to this book and then had an opportunity to see the 3-d Timewave on the Net, it was another easy think to see TW as a huge algorithm, a space-defining thing called a "frame" in this book:

Construction of the L-system generating the Hilbert curve can be extended to other FASS curves. Consider an array of m x m square tiles, each including a smaller square, called a frame. The edges of the frame run at some distance from the tile's edges. Each frame bounds an open self-avoiding polygon. The endpoints of this polygon coincide with the two contact vertices of the frame. Suppose that a single-stroke line running through all tiles can be constructed by connecting the contact vertices of neighboring frames using short horizontal or vertical line segments. A FASS curve can be constructed by the recursive repetition of this connecting pattern. To this end, the array of m x m connected tiles is considered a macrotile which contains an open polygon inscribed into a macroframe. An array of m x m macrotiles is formed, and the polygons inscribed into the macroframes are connected together. An array of m x m macrotiles is formed, and the polygons incribed into the macroframes are connected together. This construction is carried out recursively, with m x m macrotiles at level n yielding one macrotile at level n + 1.
Sounds like god to me, or at least one of its fingers ... :-)

To further your interest, the Table of Contents reveals the following explorations: Branching Structures, Modeling in three dimensions, Stochastic L-systems, Context-sensitive L-systems, Growth Functions, Parametric L-systems, Phyllotaxis, Animation, Symmetry and self-similarity, Iterated function systems ...

And, if you take this L-system stuff and re-examine Terence's Timewave Construct based on King Wen as described in "The Invisible Landscape", it suggests that his construct is only one of many that could be generated off the KW I Ching.

TM used the first order of difference (OOD) when he mapped the Timewave, in part (see p. 145, IL) because it had moments of singularity - necessary to close the "frame" of the construct. This seems a good assumption to make, since singularities are moments of definition and certainly describe the pulse of a thing.

It's unclear why the 1st OOD was so important to the Timemakers of King Wen. And the 1st OOD is only one of the fascinating or recursive features of the I Ching. Our fellow sometime poster, Silas Flood, has made mention of his I Ching matrix graphed on the Net. I looked at it and saw another incredibly valid pattern. I think that, if we're going to consider TW and I Ching and the power of Calendric definition over/throughout the unfolding of reality, we need to look much more closely at some of these original assumptions Terence made and probably add to them. L-systems may prove to be part of the answer, especially with respect to the complexities that seem to occur at nodes, places where the structure branches and changes direction. I think he's onto something but I won't have time to really think about it again till after the harvest. Perhaps I need my own La Chorrera ...

Anyhow, I think my point, George, is that there's a treasure chest of lenses in these L-system algorithms, and what I've seen is a peripheral demonstration of the Concilience of Induction that Tom Stringer has asked for in that the patterns described through the graphs of the computer can, indeed, reflect the world -

This from a letter to another friend: "What's interesting about this is that after studying the algorithms' basic structure and thinking through what the math pattern changes mean when they're translated into 3-space in the form of a leaf or a growing stem, for example, I was able to go into the greenhouse and, out of the blue, I looked at an unidentified plant (one of a couple hundred in the whole Collection) and, bang!, I knew what genus it was based on a computer generated picture of a cousin I'd seen in the algorithm book. The picture didn't look like the plant, per se, but the FORM did. There is an inherent form to the real plant, that the algorithm captured, tied to the genus and observable, out of context, even when the species I had was very different from the one in the book. I'm excited about really seeing the plants this way, formally, mathematically ... most people would say this is dry and dull and too stiff but I just see magnificent ripples on the lake of matter - ripples that grow and make flowers and attract bees and get powdery mildew ..."

Now, if we can look at an occurence of something, say the plant "Lychnis Cordata", describe regular features of it, model it into a form, and then take that picture out into a space (the World) and discover its previously unseen kindred through commonalities revealed by selecting the form based only on these screened features, we have a tool of conceptual unification, a filter for congruence, that may be needed to bring us more clearly into synch again. And perhaps that's, in part, what this thing of exploration through a biologically-oriented consciousness is about right now.

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