Diseñar fractales con sistemas iterativos de funciones hiperbólicas (IFS) y simetría Zn/DN
Configuraciones de Fractales Predefinidas
Libre
Sorpresa
Divinidad
Estrellas
Vórtices
Entropía
Diseñar el Fractal
Propiedades de Simetría de los Iconos Fractales
Loco
Trinidad
Cruz
Mañana
Amantes
Diablo
Justicia
Hermite
Roda de la Fortuna
Configuración manual (Matemáticos)
Projections on
Z-pole off
Affine Parameters
Iterated Function Set
Configuración del Trazador
Colors
Axis off
Export off
Otras Acciones
IFS data
Atràs
Diseñar
New
Redraw
Repaint
Añadir iteraciones
+10K
+100K
+1M
+10M
Close
Occurrences and Fractal Dimensions
G-Matrix: coefficients of the hyperbolic IFS
Atràs
Generación de Iconos Fractales con un Sistema Iterativo de Funciones Hiperbólicas
Este generador del fractal
Este software en línea gratuito permite de generar iconos fractales con formas, colores y simetrías diversas
Para generar un fractal, sólo tiene que pulsar el botón rojo Diseñar el Fractal
Configuraciones Fractales Predefinidas
Usted puede seleccionar algunas configuraciones predefinidas de fractales - Libre, Sorpresa, Divinidad, Estrellas, Vórtices, Entropía -
La configuración Sorpresa está configurada por defecto
The configuración Libre restablece el fractal a un simple triángulo de Sierpinski
Puede configurar manualmente la forma del fractal en cualquier momento (véase más adelante)
Propiedades de Simetría (Grupos Zn) de los Iconos Fractales
Puede seleccionar el grupo de simetría Zn del 3 al 9
El valor por defecto Loco significa que el Zn se elige aleatoriamente entre 3 y 9
La Roda de la Fortuna permite establecer una Zn aleatoria de alto rango, elegido entre 30 y 60
Puede configurar manualmente la simetría Zn del fractal en cualquier momento (ver más abajo)
Configuración manual: → Proyección, Z-Pole, Affine Parameters, Iterated Function Set
El botón Projection permite elegir si las proyecciones lineales - transformaciones lineales con determinante igual a cero - estan permitidas o no al azar - por defecto es sí
La opción Z-pole permite añadir una transformación lineal con un punto fijo de cero a cualquier serie de transformaciones
Los botones Parameters y Iterated Function Set abren las ventanas donde se puede configurar manualmente los parámetros de los conjuntos de funciones
Configuración del Trazador: → Colors, Axis, Export
The colors in the fractal icon image are proportional to the probability that a given point is visited during the iterations
Clicking the Colors button opens a window where you can select the color palette of the fractal icon
You can either select one of the three pre-defined color palettes at the bottom of the window, or define your own color palette
You define a color palette by choosing the hue, the saturation and the lightness of the two extreme colors of the palette
The program will then adjust the intermediate colors proportionally to the gradient between the two extreme colors you defined
By activating the axis option, you ask the program to plot also the coordinate axis of the space and the fixed points of each function in the iterated function set
The fractal icon images are plotted on a html canvas element: such element doesn't allow to copy and export the image
By clicking on the Export button, you convert the image into a png image: you can then right click on the image and save the image anywhere on your computer
Note that the image is exported without any background: you will need to add the background on any other application where you wish to display the exported image
Diseñar: → New, Redraw, Repaint
El butón New genera un nuevo icono fractal con la misma configuración que el actual: la anterior se borra
El botón Redraw rediseña el icono desde el principio
El botón Repaint permite aplicar nuevos ajustes de color para el icono fractal existente
Añadir iteraciones: → +10k, +100k, +1M, +10M
The initial fractal icon is plot after 10,000 iterations of the function set: this should prevent users with low calculation capacities (eg. smartphones) to be locked in too long waiting times
The initial fractal image is usually of poor quality, but allows you to identify nice icons
You can then add iterations to get a high quality icon and to finetune the fractal image
Note that a fractal is defined as the result of an infinity of iterations, what is obviously impossible on a computer
Usually you get most of the image after 1M iterations, but sometimes the image yet improves significantly up to 100M iterations
Be ready to wait if you launch fractal iterations over 100k without a powerful enough computer
IFS data (Matemáticos)
For the mathematicians, we plot in the IFS data all the most significant parameters of the iterated function set
A first matrix show (i) the number of times a point has been hit by an iteration divided in percentiles, and including the minimum and maximum of such hits numbers, (ii) the total number of hits (pixels points) and of iterations realized, (iii) an estimate of the mathematical ball-dimension of the fractal and (iv) the maximum radius of the fractal
In the left G-matrix you find the scale, stretch, rotation, shear, radius and phase of the fixed points, if the mirror was on - if mr=0 you get a projection, if mr=-1 you get a reflection -, the determinant and the trace of the transformation
In the right G-Matrix, for the same transformation, you find the standard parameters of the linear matrix (G), the translation parameter t and the applied probability of appearance of a transformation at each iteration
Guía de Matemática para la Generación de los fractales
An IFS is a set of affine transformations
Each of those affine transformation is a contraction - the absolute value of the scales in the x and y direction are less than unity - A set of affine contractions is called an Hyperbolic IFS
It is proven that any hyperbolic IFS has a unique set of fixed points, called the attractor A of the HIFS
Starting with a random complex number z (or a point in the plane) and applying iteratively a transformation randomly selected within the HIFS, after a transitory period (eg. 500 iterations) the application inevitably hits the attractor
Continuing to apply iteratively a transformation randomly selected within the HIFS, the program shows the attractor of the HIFS, coloring the pixel proportionally to the hits
The attractor of the HIFS is the fractal image
Seleccionadas Referencias en línea: → Transformaciónes afínes
