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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 entitled "Strange
-Attractors: Creating Patterns in Chaos" (CPiC). The book describes the
-mathematics behind a class of fractals called strange attractors.
-The bulk of CPiC deals with a sub-class of strange attractors called iterated
-maps. An iterated map is an equation, or often a system of equations, which are
-evaluated iteratively. Iteration in this case means that we take the result of
-evaulating the equations with, and feed this back into the same equations as
-inputs for the next evaluation.  With most randomly chosen equations, the set
-of results produced by succesive iterations is not particularly interesting,
-perhaps rapidly climbing to infinity, collapsing to 0 or some other fixed
-value. For a small number of equations however, the set of results carves out a
-region of number space which exhibits interesting fractal properties. This
-region of number space is called an attractor. The complete term "strange
-attractor" refers also to the chaotic propeties of these attractors (CPiC p10,
-CaTSA p127).
-In CPiC Sprott presents a computer program which can search for the equations
-which produce strange attractors by trying many randomly generated equations
-until one is discovered which meets certain criteria. The results are clouds of
-points which formed interesting shapes, many with an organic quality. Each set
-of equations created a unique set of points.
-The organic quality of the images is particularly intriguing as they are the
-result of a relatively simple mathematical process. In fact it is this
-complexity, emerging from apparently simple equations, which has attracted
-mathematicians.
-The book describes a range of different classes of equations which exhibit the
-property of producing strange attractors. One class of equations which
-particularly caught my attention was the polynomial map.  Polynomials are
-interesting as they are easy to generalise, meaning one can easily add or
-remove terms from the equation to experiment with systems of greater or lesser
-complexity.  Additionally, this flexibility is easily parameterised, making it
-possible to automate. Finally, there are a number of properties of polynomials
-that makes their evaluation on computers convenient.
-With each class of strange attractor producing equations presented in the book,
-there were a number of parameters (coefficients, constants) that could be a
-adjusted to produce different attractors. Often, only a small percentage of the
-possible constant values would give rise to a strange attractor. The program
-that Sprott develops in the course of the book randomly tests many different
-randomly generated parameters, and displays the strange attractor produced by
-those which exhibited favourable properties.  Since the values were chosen
-randomly from the infinite range of possible values, it gives the impression of
-the different attractors lying spread out discretely across number space,
-isolated from each other like stars in the night sky.
-I was curious to see what would be the outcome of making small changes to the
-chosen values, to see if the attractors did not exist as pinpricks in number
-space, but possibly as extended areas, possibly even connected.  I made a
-simple program which did this, starting from one known strange attractor and
-making small changes to the parameters, and plotting the resulting attractor.
-The result was the familiar clouds of points, slowly morphing between many
-different shapes.
-The simple program I made would move through the parameter space randomly and
-automatically. It showed that there was a continuity to the parameter space
-which the strange attractors occupied. It did not however, provide any useful
-way of visualising the path it was taking through this space.
-In this research I would like to explore this space in a more controlled
-manner, and possibly map it to some degree. The mapping of fractal spaces is
-not unheard of in mathematics. As a simple example, a region of the Lyapunov
-fractal has been dubbed "Zircon Zity" by mathematician, Alexander Dewdney.
-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. I maintain however, 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 specific system of equations which I used in my initial simple program is
-shown below:
-X<sub>n+1</sub> = a<sub>1</sub> + a<sub>2</sub>X<sub>n</sub> +
-a<sub>3</sub>X<sub>n</sub><sup>2</sup> +
-a<sub>4</sub>X<sub>n</sub>Y<sub>n</sub> +
-a<sub>5</sub>X<sub>n</sub>Z<sub>n</sub> + a<sub>6</sub>Y<sub>n</sub> +
-a<sub>7</sub>Y<sub>n</sub><sup>2</sup> +
-a<sub>8</sub>Y<sub>n</sub>Z<sub>n</sub> + a<sub>9</sub>Z<sub>n</sub> +
-a<sub>10</sub>Z<sub>n</sub><sup>2</sup>
-Y<sub>n+1</sub> = a<sub>11</sub> + a<sub>12</sub>X<sub>n</sub> +
-a<sub>13</sub>X<sub>n</sub><sup>2</sup> +
-a<sub>14</sub>X<sub>n</sub>Y<sub>n</sub> +
-a<sub>15</sub>X<sub>n</sub>Z<sub>n</sub> + a<sub>16</sub>Y<sub>n</sub> +
-a<sub>17</sub>Y<sub>n</sub><sup>2</sup> +
-a<sub>18</sub>Y<sub>n</sub>Z<sub>n</sub> + a<sub>19</sub>Z<sub>n</sub> +
-a<sub>20</sub>Z<sub>n</sub><sup>2</sup>
-Z<sub>n+1</sub> = a<sub>21</sub> + a<sub>22</sub>X<sub>n</sub> +
-a<sub>23</sub>X<sub>n</sub><sup>2</sup> +
-a<sub>24</sub>X<sub>n</sub>Y<sub>n</sub> +
-a<sub>25</sub>X<sub>n</sub>Z<sub>n</sub> + a<sub>26</sub>Y<sub>n</sub> +
-a<sub>27</sub>Y<sub>n</sub><sup>2</sup> +
-a<sub>28</sub>Y<sub>n</sub>Z<sub>n</sub> + a<sub>29</sub>Z<sub>n</sub> +
-a<sub>30</sub>Z<sub>n</sub><sup>2</sup>
-The terms a<sub>1</sub> though to a<sub>30</sub> are the parameters which
-define a particular attractor. When I refer to "parameter space", I am refering
-collectively to these values. In the same way that you could take two numbers
-to be coordinates on a two-dimensional map, it can be useful to imagine a long
-list of parameters as coordinates into a space of considerably higher
-dimension, in this case, a 30 dimensional space. This is not to suggest that
--dimensional space is easily comprehensible, but by extrapolation from a two
-dimensional plane, through to a three dimensional cube and beyond, it is
-possible imagine at least some of the properties of this space.
-Perhaps the most important aspect to comprehend is that sets of parameters
-where the corresponding values in each set differ only slightly from each other
-(eg, a<sub>7</sub> in each set) are in some sense adjacent.
-The right hand side of equation is simply a polynomial of degree 2 in
-X<sub>n</sub>, Y<sub>n</sub> and Z<sub>n</sub>. As mentioned in the previous
-section, it is easy to generalise these equations and replace them with
-polynomials of higher or lower degree (greater or fewer numbers of term).  With
-higher degree polynomials, the number of coefficients (parameters) rapidly
-increases.
-The level of complexity of the equations can be varied, but even in this simple
-(yet still interesting) case the equation is defined by 30 values. If you
-consider the 3 starting values of X, Y and Z, then it 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.
-[ automated help looking at the space, fractal dimension, lyapunov ]
-It is hoped that the ability to explore and derive a structural understanding
-of the number space, may assist in searching for visually or perhaps even
-mathematically interesting regions in space.
-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. Similar relationships
-are already known to exist between better understood fractals. Each point in
-the well known Mandelbrot Set fractal corresponds to a particular configuration
-of the similarly popular Julia Set fractal in which the solution is bounded
-(Sprott p.353). The exact definition of boundedness in this case is beyond the
-scope of this report, but it is sufficient to understand that the Mandelbrot
-Set can be understood as a map describing the behaviour of the Julia Set over a
-range of different parameters. Similarly, I would like to generate a similar
-map describing the presence of Strange Attractors across a range of parameters.
-==== Methods ====
-This exploration is driven by tools and technology. The discovery of fractals
-was catalysed by the availability of computers, as they require large numbers
-of calculations and their exploration was not feasible without computational
-assistance. Even at the time that the Sprott book was written, generating one
-of these attractors would take a several seconds, making the animations of my
-early exploratory program infeasibly time consuming. The mapping of the
-parameter space involves many times more calculations again, making efficient
-implementation a reasonably high priority.
-In addition to the raw computational challenges, the other important problem to
-face was the construction of a suitable interface for conveniently navigating
-the high dimensional number spaces.
-[ display issues ... when did i switch to soya? .. not documented, just before
-getting the dotplot working i guess ]
-My early plans were to make a neat, self contained application, displaying the
-navigation interface alongside a 3D rendering of the current strange attractor.
-An early problem I needed to solve was the problem of integrating a 3D
-rendering into that same window as the navigation interface.
-The common methods for producing real-time 3D graphics on computers are biased
-towards applications in which everything takes place in the 3D render window,
-for example, a 3D game, which takes over the complete display of your computer.
-I was to discover that relegating that 3D window to a region within another
-application window is not always a simple task. The problem was further
-complicated the particular 3D library I had hoped to use (OGRE) and the
-language in which I had hoped to do the interface programming (Python).
-[ summary of the problem? ]
-After a number of different trial and error approaches to the problem, I gave
-up on the dream of a neat, single-windowed application and fell to the less
-problematic approach of having the 3D render and the navigation interface
-appear as two distinct windows.
-With this initial structural decision made, I moved to implementing the code to
-evaluate the equations which would generate the attractor. I decided to use
-SciPy, a python library for performing scientific calculations. It emphasises
-the use of matrix operations, and so I arranged the evaluation so that the bulk
-of the calculation would be done as simple operations on large matrices. This
-method was chosen in the interests of efficiency. While it was not possible to
-easily test how efficient the matrix operations in SciPy were, it is reasonable
-to assume that they are optimised to some degree.
-Soon after completing an initial implementation of the evaluation code, I was
-somewhat suprised to discover that using functions provided by SciPy,
-generating maps of the parameter space would be much easier than I had
-imagined. In fact, I was able to render my first parameter maps long before I
-had a working renderer for the 3D dot plots of the attractors themselves.
-[ randomly chosen attractor 168coeffs.. is that degree 3? ]
-The first plots were quite time consuming. For each point on the map I was
-calculating many iterations of the equations, only stopping if the values
-became incalculably large (infinte). During the ample opportunity for
-reflection afforded by the long rendering times, I was reminded of the way in
-which the popular images of the Mandelbrot set are typically generated.
-Most images of the Mandelbrot set are called "Escape Time" plots. Each pixel in
-the image represents a unique starting value which is fed into an iterative
-equation. The black region in the centre of the image are those starting
-conditions for which the succesive values produced by the iteration stay close
-to their initial value, and define the Mandelbrot set proper. The coloured
-bands surrounding this region represent starting values for which sucessive
-iteration causes the values to spin off towards infinity. The different colours
-represent the number of iterations required for the values to cross some
-arbitrary threshold.
-For my purposes, escape time plots had several benefits. Firstly, there is
-generally less calculation to do for each point, as the values will cross the
-chosen escape threshold long before my old limit of infinity.
-Secondly, while not technically representing attractors themselves, the shape
-of the colour bands tend to give clues about the presence of nearby attractors,
-acting as a kind of mathematical aura (the allusion to pseudo-science is
-particularly appropriate).
-[ animations ]
-[ optimisations (psyco, liboil, handcoded c) ]
-[ dotplot renderer, interface ]
-[ basin of attraction 0,0,0 assumption ]
-[ sprott mention of robustness ]
-==== Solution/Results ====
-[ intial plots of basin of attraction (accidentally) .. ]
-For historical value, below is the first map I rendered. Mistakenly, it was the
-result of somewhat misplaced ambition. It was intended to be a plot of the
-varying fractal dimension across a two-dimensional slice of the parameter
-space. Due to a conceptual error, it is actually a rendering of the varying
-fractal dimension across a two-dimensional slice of the basin of attraction.
-Upon realising this error, the slight variance in the values towards the center
-became quite surprising.
-[ desc of basin of attraction, why the factal dimension shouldn't change much ]
-highf-100.png (accidental basin plot)
-[ low order ]
-[ fractal dimension plot + composite ]
-[ basin of attraction animations ]
-[ render errors? ]
-==== 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.
-[ benefits of flexible lib vs app ]
-i still want to continue, making a useable application for exploring the space. at the moment i am still driving it quite manually.
-[ basin plots are truly 3d (not slices), lend themselves to 3d printing ]
-[ ability to click on an map and see the attractor ]
-[ sonification ]
-[ reactions from DE ]
-==== References ====
-  * http://www.rawilson.net/chaos/lyapunov/index.html
-  * http://sprott.physics.wisc.edu/sa.htm
-  * http://ibiblio.org/e-notes/MSet/Logistic.htm
-  * images and animations at  [[Pix Strange Attractor]]