Member 2743
14 entries
8233 views

 (M)Pecs, HU Immortal since Sep 3, 2010 Uplinks: 0, Generation 4
erdos’ favorites
 From BenRayfieldPhysics and Computing...
Now playing SpaceCollective
Where forward thinking terrestrials share ideas and information about the state of the species, their planet and the universe, living the lives of science fiction. Introduction
Featuring Powers of Ten by Charles and Ray Eames, based on an idea by Kees Boeke.

Lindemayer systems were invented by the Hungarian biologist and botanist Aristid Lindenmayer in 1968. He used this grammar system to model growth of plants and other natural processes.

An L-system is a special formal grammar with parallel rewriting rules. Its formal specification consists of rewriting rules in X -> (Y|t)+ form ( where X, Y are nonterminals, t is a terminal symbol) and an axiom (initial state). In this context, nonterminals will be interpreted as pseudoterminals to let the grammar produce finite words over the alphabet. Parallel rewriting in this context means that all possible rules shall be applied on each step.

The Spidron system was first modelled by the Hungarian graphics artist Daniel Erdely in 1979. He presented a paper model as a homework exam in the design class of Erno Rubik. The mathematical model was described by Lajos Szilassi.

The Spidron system is a spiral-like fractal system of folding that can be deformed in the 3d space without torsion. The base of the system are symmetric concentric similiar polygons with constant scale ratio and rotation. Folding an outer polygon (so called ring) forces the inner polygon to fold.

Lindenmayer spidron

We can easily create a graphical representation of l-systems by giving special meanings to terminals in the word, like matrix and stack operations, etc..

in the followings we will create a single spidron-arm with an L-system:

constants (terminals):

• + rotate by 30 degrees

• - rotate by -60 degrees

• * scale by 1/sqrt(3)

• E, F move forward

variables (pseudoterminals): S (starting symbol)
axiom: S
rules: S -> s + F - F + + + + F - - F - - - S

full spidron

We can create a full spidron by extending two matrix operators:

• [ push the current transformation matrix to the stack

• ] use transformation matrix removed from the top of the stack

axiom for the spidron: [ S ] F - - - S
It is also possible to create a hornhedron by introducing mirroring or an additional rewrite rule.

spidronized spidron

You may have observed that any two points of the plane can be connected by a spidron arm. This gives the idea of replaceing the building up line segments of a spidron arm by other -smaller- spidron arms. We can repeat this process infinite times and get a much more complex fractal shape: an increasing yet bounded point cloud.

Let's extend our l-system with the following rule: F -> [S]E
The cover image shows such an l-system with a specific axiom. The following images show the evolution of the basic spidron-lindenmayer system.

Sun, Jul 22, 2012  Permanent link
Categories: Fractal, l-system, spidron, generative, mathematics, linguistics