On Fractals: On Stretching

"Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals." (Györgi Ligeti, [interview] 1999)

"A chaotic system has sensitive dependence on initial conditions and has orbits that are bounded. Both of these properties arise from geometrical features of the dynamical system. In order for the system to have SDIC, the dynamical system must perform some sort of a stretch. This stretch has the effect of pulling apart nearby initial conditions, leading to SDIC. In order for the orbits to stay bounded, however, this stretching cannot occur indefinitely. Thus the dynamical system also needs to perform a fold that brings orbits back together so they do not grow without bound." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"Imagine two points in the dough that are initially close to each other: perhaps two adjacent specks of cinnamon. During each stretch, the distance between the points gets larger. During the fold, however, the two points might get closer together. This will occur if the two points are on opposite sides of the midpoint of the dough. So the stretching continually pushes the points apart, and the folding brings them closer if they are far apart and on opposite sides of the midpoint. In this way stretching and folding produce chaotic trajectories; the orbit is bounded and has sensitive dependence on initial conditions." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"This generalized chaos game is constructed as follows. Thus far each move in the game takes a single point and moves it to another location. Which location it gets moved to is determined by a random rule. We could also imagine a chaos game that uses slightly more complicated rules. It is easiest to think of these rules in terms of their effects on shapes rather than single points. To that end, consider operations that not only move a shape but also transform it - stretch, shrink, shear, and/or rotate it. The technical term for these sorts of transformation is affine. An affine transformation is any geometric transformation that keeps parallel lines parallel." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"The Newtonian universe is material in the sense that the world was viewed as being made up of stuff - tangible, real objects. It was argued that even forces like gravity that appear to act across empty stretches of space are conveyed by tiny particles, or corpuscles. Moreover, since the universe is material, its behavior can be predicted or understood. Things are they way they are for a reason or a cause. The Newtonian world is mathematical, in that it was viewed that the regularities or laws that describe or govern the world are mathematical in nature." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"[...] the Rössler attractor stretches and folds trajectories in phase space. [...] Stretching is responsible for the butterfly effect; when a stretch occurs, nearby trajectories are pushed farther apart. Folding of some sort is necessary to keep orbits bounded. If there was not any folding, orbits would tend toward infinity. It is thus not surprising that we observe stretching and folding in a three-dimensional chaotic system such as the Rössler equations in addition to the one-dimensional logistic equation." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)