L-systems

Lindenmayer systems, or L-systems for short, are formal grammar, which were originally introduced by Aristid Lindenmayer in 1968 as a thesis of the research which investigated the development of simple multicellular organisms. Afterward, the systems were applied to higher plants and plant organs. At the circumstances, L-systems are well known as the plant modeling systems explaining the plant structures and its development processes, and computations of the processes are frequently manipulated in the field of digital biology. Additionally, architectural field is also one of such fields using L-systems, which might be comparatively emergent subject in the absolutely long history of architecture.

The followings are some examples of digital designs using L-systems. The first one is Phyllotactics designed Alisa Andrasek, biothing, and the second work is scripted by Marc Fornes, who is one of the youngest and the most energetic architects in the field of architectural digital design.

Phyllotactics: AlisaAndrasek

source: www.biothing.org

Marc Fornes

source: www.theverymany.net

There are many other interesting architectural works in relation to L-systems, and such applications of L-systems to architectural design would be researched and developed much more in future as digital design goes father.

As far as concerned about the L-systems, how does it work in practical? Before explaining its details, it is necessary to understand its grammatical structure, which is generated by context-free L-systems, well known as OL-systems.

G = { V, ω, P }

where, V is the alphabet of the system, which is denoted as V* and V+. V* is the set of all words over V and V+ is the set of all nonempty words over V. ω is called the axiom or initiator defined by V+. P is a finite set of productions derived from V and V*, and additionally the set, (a,x) is determined by P is explained as a → x, where a is called the predecessor and x is called the successor.

If an OL-system is deterministic, which means that there is only one rule of a → x, OL-system is called as DOL-system. On the other hand, rules of a → x are in dependency on a probability, the system is a stochastic L-system. Furthermore, the systems are categorized in terms of more detailed elements. The book called the Algorithmic Beauty of Plants written by Przemslaw Prusinkiewicz and Aristid Lindenmayer contains more details.

The followings are parts of examples of L-systems, which are very simple samples.

Example 1: Algae

V: a, b

ω: a

P1: a → ab

P2: b → a

The sequences generated by the above system are as follows.

a

ab

aba

abaab

abaababa

…

source from the book: The Algorithmic Beauty of Plants

Example 2: Koch Curve (Turtle Interpretation of Strings)

V: F, +, -

ω: F

P: F → F+F-F-F+F

where, F means to draw forward, + means to turn left at 90 degrees, and – means turn right at 90 degrees. Turtle interpretation of strings is explained in particular in the Algorithmic Beauty of Plants noted above.

source from the book: The Algorithmic Beauty of Plants

The results of the system are shown as follows.

F

F+F-F-F+F

F+F-F-F+F+ F+F-F-F+F- F+F-F-F+F- F+F-F-F+F+ F+F-F-F+F

…

source from the book: The Algorithmic Beauty of Plants

Examples above are only simple elements of L-systems, and it is necessary to understand detailed parts of the systems in order to generate digital organisms or plants.