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• Introduction to fractals: | ||
• Definitions for IFS fractals: | ||
• Definitions for complex fractals: | ||
• Definitions for Koch snowflakes: | ||
• Definitions for Peano maps: |
Next: Definitions for IFS fractals, Previous: fractals-pkg, Up: fractals-pkg [Contents][Index]
This package defines some well known fractals:
- with random IFS (Iterated Function System): the Sierpinsky triangle, a Tree and a Fern
- Complex Fractals: the Mandelbrot and Julia Sets
- the Koch snowflake sets
- Peano maps: the Sierpinski and Hilbert maps
Author: José Ramírez Labrador.
For questions, suggestions and bugs, please feel free to contact me at
pepe DOT ramirez AAATTT uca DOT es
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Some fractals can be generated by iterative applications of contractive affine transformations in a random way; see
Hoggar S. G., "Mathematics for computer graphics", Cambridge University Press 1994.
We define a list with several contractive affine transformations, and we randomly select the transformation in a recursive way. The probability of the choice of a transformation must be related with the contraction ratio.
You can change the transformations and find another fractal
Sierpinski Triangle: 3 contractive maps; .5 contraction constant and translations; all maps have the same contraction ratio. Argument n must be great enougth, 10000 or greater.
Example:
(%i1) load("fractals")$ (%i2) n: 10000$ (%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])$
Categories: Package fractals
3 contractive maps all with the same contraction ratio. Argument n must be great enougth, 10000 or greater.
Example:
(%i1) load("fractals")$ (%i2) n: 10000$ (%i3) plot2d([discrete,treefale(n)], [style,dots])$
Categories: Package fractals
4 contractive maps, the probability to choice a transformation must be related with the contraction ratio. Argument n must be great enougth, 10000 or greater.
Example:
(%i1) load("fractals")$ (%i2) n: 10000$ (%i3) plot2d([discrete,fernfale(n)], [style,dots])$
Categories: Package fractals
Next: Definitions for Koch snowflakes, Previous: Definitions for IFS fractals, Up: Top [Contents][Index]
Mandelbrot set.
Example:
This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.
(%i1) load("fractals")$ (%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5], [gnuplot_preamble, "set view map"], [gnuplot_pm3d, true], [grid, 150, 150])$
Categories: Package fractals
Julia sets.
This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.
Example:
(%i1) load("fractals")$ (%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5], [gnuplot_preamble, "set view map"], [gnuplot_pm3d, true], [grid, 150, 150])$
See also julia_parameter
.
Categories: Package fractals
Default value: %i
Complex parameter for Julia fractals.
Its default value is %i
; we suggest the values -.745+%i*.113002
,
-.39054-%i*.58679
, -.15652+%i*1.03225
, -.194+%i*.6557
and
.011031-%i*.67037
.
Categories: Package fractals
While function julia_set
implements the transformation julia_parameter+z^2
,
function julia_sin
implements julia_parameter*sin(z)
. See source code
for more details.
This program runs slowly because it calculates a lot of sines.
Example:
This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.
(%i1) load("fractals")$ (%i2) julia_parameter:1+.1*%i$ (%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3], [gnuplot_preamble, "set view map"], [gnuplot_pm3d, true], [grid, 150, 150])$
See also julia_parameter
.
Categories: Package fractals
Next: Definitions for Peano maps, Previous: Definitions for complex fractals, Up: Top [Contents][Index]
Koch snowflake sets. Function snowmap
plots the snow Koch map
over the vertex of an initial closed polygonal, in the complex plane. Here
the orientation of the polygon is important. Argument nn is the number of
recursive applications of Koch transformation; nn must be small (5 or 6).
Examples:
(%i1) load("fractals")$ (%i2) plot2d([discrete, snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])$ (%i3) plot2d([discrete, snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])$ (%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])$ (%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])$
Categories: Package fractals
Previous: Definitions for Koch snowflakes, Up: fractals-pkg [Contents][Index]
Continuous curves that cover an area. Warning: the number of points exponentially grows with n.
Hilbert map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.
Example:
(%i1) load("fractals")$ (%i2) plot2d([discrete,hilbertmap(6)])$
Categories: Package fractals
Sierpinski map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.
Example:
(%i1) load("fractals")$ (%i2) plot2d([discrete,sierpinskimap(6)])$
Categories: Package fractals
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