FRACTAL GALLERY 1 (1995)

Fractal Gallery is small exhibition of color fractal images of various kinds which has been built starting from different mathematical problems and techniques, as an example of the ways a computer machine can be forced to generate very elegant shapes which are also of special interest for mathematicians. In the following you will find images of:

Fractals generated by the iteration of a geometric procedure like the curve of Von Koch;

Fractals generated by iteration of complex functions like the Mandelbrot set and Julia sets;

Fractals reproducing realistic shapes like trees, ferns, mountains and so on.

Together with the images you can find some information about the mathematical problems and computer programs which can generate such plots.

Enjoy a nice tour to Fractal Gallery!

The curve of Von Koch

D = log N / log K

The shape becomes more complete if we start from a regular triangle instead of a single segment.

The previous plot is easily obtained, on your screen or printer output if you have a Postscript interpreter, by the very simple PostScript program KochX1.ps.

Nesting the same procedure you can generate higher order approximations of the curve of Von Kock which is the fractal limiting curve. Here you see the second, third and forth order approximations resembling snow christals. If you are interested to the corresponding PostScript programs click on the row at the right each figure.

You must click on each image if you want to see it in full size.

The Mandelbrot set

in which the starting term for n = 0 is fixed (usually = 0) and the parameter c runs across all the complex plane. Numeric approximation is obtained by computer replacing the series with a finite summation of terms. Usually the escape time method is used. A simple program written for MacIntosh in Quick Basic language which generates plots in 8 colors is MandelbrotX1.bas.

Click that image to see it in full size.

An alternative and generally better way to build up color images of the Mandelbrot set consists in writing down a prgram, in some suitable language (like MandelbrotX2.bas in Quick Basic for MacIntosh), which creates a sequential text file containing the escape speeds, i.e. the number of terms needed for the partial sum of the series to become greater than the fixed upperbound. The text file can be read by a program, like e.g. NCSA DataScope for MacIntosh (a Unix version is also available), which converts each number into a colour density plot according to a chosen color palette. One of the advantage of such a procedure is related to the capability of changing the choice of the color palette in order to reach the best chromatic effect. The files can be saved in the significantly shorter format HDF. A collection of images is presented here.

Click that image to see it in full size.

A zoom in Mandelbrot set shows another Mandelbrot

Mandelbrot set is selfsimilar along its symmetry axis

A zoom in the Seahorse Valley

A deeper zoom requires double precision calculations

Mandelbrot seahorse eye

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The Julia sets

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The program JuliaX2.bas generates text files which can be read by NCSA DataScope and saved in HDF format. Here are some plots.

The seahorse Julia set

Julia sets when c is moving along the boundery of Mandelbrot set

Julia sets when c is moving along an axis of Mandelbrot set crossing the Seahorse valley

A zoom into the seahorse Julia set

Julia seahorse eye

Click on each image to see it in full size.

Three dimensional plots

Three dimensional plots of Mandelbrot and Julia sets are presented here in which the quote is proportional to the escape speed of convergence of the truncated series.

A night view of the Mandelbrot set

Julia dentrite like sets

A 3D view of Julia seahorse

A 3D view of Mandelbrot seahorses

The Mandelbrot set like a gothic cathedral

Mandelbrot set like an iced lake

Another view of the iced lake

Realistic shapes

Fractals reproducing a realistic shape can be generated in several ways. The simplest way is that of iterating an assigned function many many times and to plot the result at each iteration; the choice of the colour may be random or other. A more refined procedure is obtained by the iteration of one or more affine transformations of the type

The plot tends to stabilize around as many attractors as the number of affine transformations. The collage of the attractors may generate the desired shape. The form of the attractors depend on the choice of the coefficients a, b, c, d, e, f which generate each transformation. This method is known as IFS (Iterated Function System) method. Here are some examples of both fractal types and you can find also the Quick Basic programs for MacIntosh by which they have been generated.

Fractals generated by iterated functions

Program InsectX1.bas A small insect

Program InsectX2.bas Another example of small insect

Program WebX.bas A shape resembling a web

Program FlowersX.bas Fractal flowers

Program GalaxyX.bas A galaxy

You must click on each image if you want to see it in full size.

Fractals generated by iterated affine transformations

Program TreeX.bas The fractal tree

Program FernX.bas An example of fern

Another example of fern

An autumnal leaf

Program MountainsX.bas A mountain chain with snow at high quote

Another mountain chain example

You must click on each image if you want to see it in full size.

You can continue by yourself to explore the wonderful universe of fractals on your personal computer. If you have a fast machine and a good screen resolution you will get better and better images! Some of them will be realistic, other will be only of mathematical interest, some other will be elegant and decorative...

While waiting that the Fractal Gallery become larger and recher of pictures you may build up your own.