I’m interested in the pattern of chaos, quantifying and predicting organic growth patterns and using theory and potential equation of chaos. Creating something that could potentially be built from math patterns.

What's interesting about fractals is many of their basic beings are comprised of the addition of an imaginary number (as I’ve seen online) and a real number, a fractal is essentially a graph of an iterative process applied to complex numbers 4 - 6i, 2.1 + 4.7i, 1-I etc. A loose parallel to architecture (the mix of the real and the imaginary) it’s a mathematical way to describe things. Real factors such as site forces (wind, sun path etc), programmatic forces (number of people inhabiting building, internal loads, heat gain loss) can be translated to form by factoring them into an equation as variables and modeled with 3d programs. It is essentially giving physical form to factors that are traditionally perceived as intangible.

So for example say you have a certain word: a totally intangible non structural being that is purely a placeholder and unit of language. By assigning numerical values for letters in the alphabet, a=1, b=2, you can create the equation of the word E.G. Cat:  (C = 3) + (A = 1) + (T = 18) + i^2 = fractal generation. So the resulting fractal would be a physical interpretation of the word “cat”, a form from an imaginary thing. It’s almost bridging the two worlds of imaginary and buildable.  



I started with modeling basic functions to see if I could get them to weave and create three dimensional objects rather than just a planar set of curves. If possible, it would open up the door to be able to model any equation and see how crazy it would look in 3d. The equation I started with was the Lissajous curve (describes advances harmonic motion). The curve the equation generates bounces back and forth between two imaginary planes, and depending on how many steps it draws as the curve moves back and forth and along a linear path, it builds interesting wavy forms.  So in the pic below left you can see the x and y values as they bump against the top and bottom and left and right. The curve was expanded along +X axis as it was drawing so that’s why you get the cool tower-like form with the differing transparencies. The equation drew 2000 steps along the way. In the green pics on the right you can see the variation in steps and what it looks like when there are only a few steps recorded as the curve draws, and what it looks like when there are near 2000 steps drawn. The more steps, the more form-like it becomes. The diagrams on the right are not the lissajous curve equation, but same basic idea.

The pic above and two below are the basic Lissajous in 3d, it weaves through itself without ever touching. I can expand it in all axis and control how many lines, curves, bumps in x + y etc. The bottom two pics I swept a rail along the curve and extruded so it does touch itself but It looks pretty interesting, It could really go in some cool directions if I had some time to mess around with it.



I wanted to then take this idea further and see if the basic pattern or curve function could react to other variables I give it (so I kind of got side tracked from modeling fractals in 3d to modeling linear equations). The diagram above, along with the red ones I sent you a while back, imply time and movement by being linear, so I thought it would be interesting to interject major changes to the functions at certain points in time along its linear lifespan, much like an organic growth system. For example a tree: it has a basic growth function, the same as all other trees of its type, but what makes the tree unique is how the ‘tree growth equation’ reacts to major events and variables specific to where the tree lives (e.g. specific wind forces, a wall that is in the way of its linear growth vector, someone cutting branches off.) The ‘tree growth equation’ continues, just in an augmented and distorted fashion due to the interjections…. Although it may look like other trees, the unique characteristics and its reactions to the variables of its specific site and life make it a one-off individual.



To do this I need to extract a specific point in time from the equation as it moves through a linear 3 dimensional vector and somehow impact it drastically, mabey by moving it, scaling it… whatever really. How the growth system would then react and continue would be really interesting. I would have to spend some more time on the definition/script to do this and since my partner and I switched ideas and decided to go in a different direction with the project, I doubt I will be able to revisit this until next quarter. It would be great to incorporate this stuff into the project, but it has been difficult to work in anything along these lines with my partner, she’s not very open to this stuff. Mabey I can sneak it in later.

Here are some pics of the equations and grasshopper definition, you can get a sense of the equation and how its set up to be manipulated manually. It’s actually simpler than I thought, which was a nice surprise.