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An Aesthetic Exploration of Multivariate Polynomial Maps

this research report is still in progress.

Context

in 2001 i found a book by mathematician julien c. sprott called “strange attractors: creating patterns in chaos”. the book described the mathematics behind a class of fractals called strange attractors.

in the book sprott presents a computer program which can search for strange attractors by trying many randomly generated equations until one was discovered which met certain criteria. the results were clouds of points which formed interesting organic shapes. each set of equations created a unique set of points.

when i read this i was curious to see what would be the outcome of making small changes to the equations. i made a simple program which did this, and the result was clouds of points which would morph between many different shapes.

the simple program i made would move through the parameter space randomly and automatically, and so it only showed that there was some kind of continuity to the parameter space which the strange attractors occupied, but it did not provide any useful way of visualising this space.

in this research i would like to explore this space, and possibly map it to some degree.

i've always intended this exploration to be more aesthetic than mathematical. it is certainly informed by mathematics, but i feel that any great mathematical insights are unlikely to come from me. however, i maintain the vaguely plausible fantasy that my unscientific endeavours might inspire some more useful research by a suitably qualified mathematician at some point in the future.

Problem/Aim

the specific aims of this research project are to get some insight into the structure of the parameter space that these strange attractors occupy. the name of the project comes from what i hope is a correct name for the full class of equations which make up the parameter space i am interested in searching.

the level of complexity of the equations can be varied, but even in one of the more simple (yet still interesting cases) the equation is defined by 33 values. when considered as a space, this is a 33 dimensional space. this raises some problems of how to sensibly navigate or visualise such high dimensional spaces.

having more knowledge about the parameter space would assist in searching for visually, or perhaps even mathematically interesting attractors.

i expect there to be a tight feedback loop between initial observations and future explorations. it is very much like sending off a ship into unchartered waters. if you find an island, you will probably explore the island.

i'm expecting the parameter space to be a kind of meta-attractor, and to be interesting in similar ways to the attractors themselves.

Methods

this exploration is all driven by tools and technology. the discovery of fractals was catalysed by the availability of computers, as they require millions of calculations that just weren't feasible to be done by hand. even at the time that the sprott book was written, generating one of these attractors would take a few seconds, making the animation from my 'lung' applet infeasible. and to extend further, to map out the parameter space involves many times more calculations.

so, the main method of my research will be to make a tool for viewing and exploring the parameter space.

Solution/Results

so far, the main outcomes of my research have been a small number of maps of slices of parameter space. these maps suggest that the space where the attractors exist is aesthetically interesting, and invites further exploration.

Discussion

the features of the parameter space maps are as interesting as i had hoped.

i didn't expect to be making maps this early. i had not anticipated that the manual-searching part of the tool would be so difficult, and i also i had not realised that it would actually be easier to do the mapping. i had expected to move on to the mapping after having an 'explorer' of sorts.

i've yet to speak to anyone with a maths background to ask if this is of any relevance, or more realistically to find out that it is very boring mathematically. i can imagine that it might be considered a little boring because i am working with big complicated polynomials. most of the famous fractals are based on very simple equations.

i still want to continue, making a useable application for exploring the space. at the moment i am still driving it quite manually.

References


Libarynth > Libarynth Web > PixResearchReport r1 - 16 Jan 2007 - 08:44


pix_research_report.txt · Last modified: 2007/06/14 08:45 (external edit)