Although this work is ambitious in scope, I feel this is an important element of the writing, to present an overview such that the reader is able to see the inter-relations of many ideas relevant to design.
Introduction[ edit ] The word "fractal" often has different connotations for laypeople than for mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception.
The mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background.
The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure.
If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. The difference for fractals is that the pattern reproduced must be detailed. Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived.
In contrast, consider the Koch snowflake. It is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable ".
In a concrete sense, this means fractals cannot be measured in traditional ways.
But in measuring a wavy fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.
Bytwo French mathematicians, Pierre Fatou and Gaston Juliathough working independently, arrived essentially simultaneously at results describing what are now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors i.
In  Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations.
These images, such as of his canonical Mandelbrot setcaptured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". Authors disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and an unusual relationship with the space a fractal is embedded in.
Koch snowflake Quasi self-similarity: A consequence of this structure is fractals may have emergent properties  related to the next criterion in this list. Irregularity locally and globally that is not easily described in traditional Euclidean geometric language.
For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls". A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimensionand is fully defined without a need for recursion.
Because of the butterfly effect a small change in a single variable can have a unpredictable outcome. Iterated function systems — use fixed geometric replacement rules; may be stochastic or deterministic;  e.
The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points or pixel data are passed through this field repeatedly. A fractal generated by a finite subdivision rule for an alternating link Finite subdivision rules use a recursive topological algorithm for refining tilings  and they are similar to the process of cell division.
A fractal flame Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space.- Elementary Arithmetic - High School Math - College Algebra - Trigonometry - Geometry - Calculus But let's start at the beginning and work our way up through the various areas of math.
We need a good foundation of each area to build upon for the next level.
Fractal Science Kit fractal generator Fractal Equations. Fractal Equations. Initial Z is a numeric textbox used to enter a complex value for the Initial Z required for Mandelbrot fractals.
See Solver Functions for a detailed description of attheheels.comoPolynomial and the SolverMethodOptions. Various Number Theorists' Home Pages/Departmental listings Complete listing [ A | B | C | D | E | F | G | H | I | J | K | L | M] [ N | O | P | Q | R | S | T | U | V.
HOME. Lancaster University. Department of Independent Studies. A Pattern Language of Sustainability Ecological design and Permaculture. By Joanne Tippett. April, Fractal Geometry, Complex Dimensions and Zeta Functions Geometry and Spectra of Fractal Strings. Complex Dimensions and Zeta Functions, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics.
The book is self containing, the material organized in chapters preceding by an introduction . This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics.
The significant studies and problems illuminated in this work may be used in a .