**By Ludvig Hofsten**

### Fractal patterns

Nature is complex yet simple, full of variation yet repetitive. Let’s have a look at the shapes below:

All of the shapes above are based on a mathematical/natural phenomenon called fractals. A fractal is a pattern that repeats itself at different scales. In theory a fractal repeats itself forever. If the replication is exactly the same at every scale, it’s called a self-similar pattern. The interesting thing about fractals is that they are very complex but made by a simple process repeated over and over again. Every new branch on a tree, for example, is created in the same way as the old branches, but in a smaller scale.

### Lindenmayer systems – adding randomness to the fractal patterns

When studying how trees and plants grow, a fractal pattern with its repetition is a part of the explanation. However we can clearly see the difference between a drawn tree that strictly follows the fractal laws (below to the left) and a natural tree (below to the right).

Both of the trees follow the principles of a repeating pattern in different scales but the tree to the right seems to be slightly more random in its structure.

The principles behind the growth process of plants, morphogenesis, have been studied for a long time: “Morphogenesis is the development of structural features and patterns in organisms. Formal interest in this area probably dates to the 19th century, when it was observed that the phyllotaxis (leaf arrangement) of many plants could be predicted using the ratios of successive Fibonacci numbers. The field received a true jump-start in 1952, however, when Alan Turing published “The Chemical Basis of Morphogenesis” [2], his last publication before his death in 1954. Turing proposed a reaction-diffusion model as a possible basis for the development of many features, including the arrangement of tentacles on the hydra and the patterns of colour seen on animal skin.” (Jennings 2011)

One of the most most important contributors to the research was Aristid Lindenmayer, an Hungarian biologist who studied the growth process of the plant development in a mathematical way. In 1968 he developed a model to recreate this growth process. The system, called Lindenmayer system, or simply L-systems for short, may be described as a mathematical theory of plant development. Lindenmayer studied simple organisms such as yeast, fungi and various types of algae.

**Computing the growth of algae**

The L-system could best be described as a language. To set up an L-system in order to replicate a certain organism you set up an axiom, which is the starting point, like a seed, and a list of rules that has to be followed. For simple organisms like algae, it could look like this:

**Axiom: A (This is the starting point)**

**Rule 1: A –> DB**

**Rule 2: B –> C**

**Rule 3: C –> D**

**Rule 4: D –> E**

**Rule 5: E –> A**

If we start with the axiom A and follow the rules stated above we get the following structure:

The rules in the example above are quite simple. For example, the angles between the branches are not specified nor the length of each branch.

### Dragon curve

Let’s look into another example called “The Dragon Curve”. This time we include a rotation rule.

**Axiom: FX**

**Rule 1: X –> X+YF+**

**Rule 2: Y –> -FX-Y**

**Rule 3: F –> ε**

F=Move forward one string

– =turn right 90 degrees

+ = turn left 90 degrees

ε = empty string

This gives us the following pattern:

Stage 0: FX

Stage 1: X+YF+

Stage 2: X+YF++-FX-Y+

Stage 3: x+YF++-FX-Y++-X+YF+–FX-Y

Stage 4: X+YF++-FX-Y++-X+YF+–FX-Y++-X+YF++-FX-Y+–X+YF+–FX-Y+

Stage 5: X+YF++-FX-Y++-X+YF+–FX-Y++-X+YF++-FX-Y+–X+YF+–FX-Y++-X+YF++-F

X-Y++-X+YF+–FX-Y+–X+YF++-FX-Y+–X+YF+–FX-Y+

Which translates into a shape that eventually looks like something like this:

Below are some other tree shapes that have been computed using L-systems:

### Architecture without blueprints – structures that develop like nature

By studying the L-systems above we have seen that it is possible to construct algorithms that contain the “code” behind the growth patterns of various organisms.

Different codes give different trees and branching layouts. However, each code is characterized by a certain amount of randomness – before converting a code into a visual pattern, it’s impossible to know what it will look like.

Would it be possible to use some of these ideas in the architecture design process? Most buildings and structures are based on detailed drawings and exact blueprints. How would it be if a piece of architecture would grow and develop like the organisms above?

The architect could, instead of designing the building in detail, set up a number of “rules” that must be followed during the construction, much like the L-systems above. The structures could be built by a community together and the rules could make the collaboration easier. The resulting architecture will be an organic combination of the original axiom and the decisions taken by the community while constructing.

The images below show what this kind of organic, L-system inspired construction process could look like:

**References**

Chang, A; Zhang. T. 2016. *On the Fractal Structure of the Boundary of Dragon Curve*. [ONLINE] Available at: __http://poignance.coiraweb.com/math/Fractals/Dragon/Bound.html__. [Accessed 19 September 2016]

Hugh Pryor. 2016. *Fractal generating app*. [ONLINE] Available at: __http://www.hughpryor.co.uk/fractals/__. [Accessed 19 September 2016].

Jennings, C. 2011. *Lindenmayer Systems*. [ONLINE] Available at:__http://www.cgjennings.ca/toybox/lsystems/__. [Accessed 19 September 2016].

Martin J. M. de Boer, F. David Fracchia, and Przemyslaw Prusinkiewicz. 1992.A model for cellular development in morphogenetic fields. In *Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology*, pp. 351-370. Springer-Verlag.

Virtual Complexity Lab at Monash University. 2007. *L-Systems*. [ONLINE] Available at:__http://vlab.infotech.monash.edu.au/tutorials/l-systems/__. [Accessed 19 September 2016]

Wikipedia. 2016. *Fractals*. [ONLINE] Available at: __https://en.wikipedia.org/wiki/Fractal__. [Accessed 19 September 2016].