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An overview of Algorithmic Art (AA) and its dependence on computers is presented; it is clearly specified that AA uses mathematics although it is not always explicitly mathematical. Some technical details involved in the creation of images of AA are detailed. The fact that some impressive AA images created with early personal computers and, which are now impossible to reproduce due to the current super speed of PCs, is highlighted.
The VirtualDynamics Intelligencer Math-Algorithm-Art Oct 2006
The Breathtaking Algorithmic Art
Javier Montenegro Joo
VirtualDynamicsSoft ----- jmj@VirtualDynamics.Org
An overview of Algorithmic Art (AA) and its dependence on computers is presented; it is clearly specified that AA uses
mathematics although it is not always explicitly mathematical. Some technical details involved in the creation of images
of AA are detailed. The fact that some impressive AA images created with early personal computers and, which are now
impossible to reproduce due to the current super speed of PCs, is highlighted.
Keywords: Art, mathematics, algorithms, computing, Digital.
Algorithmic Art as an expression of Digital Art
Algorithmic Art (AA) is one of the several
manifestations of Digital Art or Art by Computers,
which owes its existence to that of computers.
Among the expressions of the AA are the beautiful
and stunning images created from computation
protocols (algorithms) on mathematical
expressions as equations, formulas, logical rules,
Algorithmic art makes use of mathematics
AA however, is not limited to the mathematical
evaluation of equations; the AA use math just as
what they are, a tool. For example, in those
pictures that result from the replication of a
transformation, this is repeated very many times.
And while it is true this replication is run by a
series of mathematical transformations, the end
result is not essentially mathematical.
As an analogy, consider the design of an aircraft.
Mathematics is required to calculate the power of
the engine and its relationship with the weight of
the same aircraft and the load that it will transport
and the resistance of the materials to be used in
its construction. Even more, in the case of a high-
performance jet, its aerodynamics is calculated
with mathematics but based on physical
principles. However the plane is not a product of
mathematics, although it may be of physics. The
same applies when calculating an anti-seismic
structure of a modern building.
Computers are essential for AA
Computers are indispensable to create works of
AA, because with them it is possible to operate
quickly and with no errors, on all the pixels (points
of the plane) that make up the image.
The VirtualDynamics Intelligencer Math-Algorithm-Art Oct 2006
It is very important to keep in mind that not all
images of algorithmic art must be colored; many
could be in strictly black and white, especially
those that are based on geometric
transformations. However, if the image is colored,
the color of each pixel should be calculated. For
example, a 60 cm x 60 cm image with a rather low
density (90 pixels/inch), has about 4'665,600
pixels, and when creating the image, the color of
each one of these pixels must be determined. The
higher the resolution (pixels/inch) the better the
image quality, because the image includes more
information, more detail.
In some algorithms a formula or mathematical
equation is used to calculate the color of each
pixel and in these cases the image generation is
rather quick and simple. In other algorithms the
color is obtained by operating on mathematical or
logical expressions several times, making a slight
variation each time and in this way generating the
corresponding graphical representation, this
requires iterations, that is, processing several
times each pixel across the image and, obviously
this modality demands more computer time and
greater number of calculations. Obviously, to run
all this work manually, i.e., without computers,
would result extremely tedious, slow and subject
to errors.
In general, the beauty and visual impact that
produces an image is completely independent on
the number of calculations to obtain it or on the
complexity of the algorithm that generates it.
The beauty of mathematics
In the past, before the Algorithmic Art, it was
common to hear about the "beauty of
mathematics", especially in those universities
where sophisticated mathematics was handled in
one way or another. Then the beauty of
mathematics referred to the elegance and
perfection of some of its topics, features which not
all areas of mathematics possess. Nobody
foreshadowed then that "The beauty of
Mathematics" would apply in the future to the
creation of beautiful and stunning images,
abstract and not so abstract, symmetrical and not
so symmetrical.
The experience of the author (JMJ) of this
document is that some mathematical expressions
(equations) really hide beauty. The challenge is to
identify or build these equations and create a
method (algorithm) to extract and visualize that
beauty. It is the case of the images shown in this
document, JMJ had to design the corresponding
equations and create the algorithm to externalize
the beauty that these equations kept hidden.
If someone devises how to extract the information
that some equations hide, he will discover
amazing symmetries in them, he will even identify
some mathematical expressions that convey no
symmetry but are impressive.
Images from the viewpoint of computers
The VirtualDynamics Intelligencer Math-Algorithm-Art Oct 2006
Computationally speaking, an image is a three-
dimensional matrix of components (x, y, z), whose
first two components x and y indicate the position
of a point in the plane of the image, and the value
of z indicates the color that appears at that point.
The value acquired by z depends on the algorithm
that is applied to create the image.
In strict black and white images, the value of z can
acquire either of two values, 0 or 1, representing
white or black, respectively. In grey leveled
images, z can take any value between 0 and 255,
which makes 256 possible shades of grey
between strict black and strict white. In color
images, one of the most common color
representations is the RGB (Red, Green, Blue),
indicating the combination of shades of red, green
and blue that may be stored in each z. In this case
z has 3 components: R, G and B. Each of these 3
colors can take any value between 0 and 255.
This makes a total of
      possible
colors in each z, this is, in each pixel (x,y) of the
image. Note that although the human eye is
unable to identify this millionaire amount of colors,
the algorithms are able to generate them, and in
practice when working with colors, it is very likely
that all of them are be generated; unhappily that
high color resolution is impossible to detect with
the naked eye. Color images included in this
document might include the millionaire amount of
mentioned colors, but human eyes are not
prepared either to distinguish them or to identify
As an example of the RGB representation,
consider color RGB = (128, 200, 86). In this case
the color is one that results from combining 128,
200 and 86 shares of red, green and blue,
Shocking images as a result of logical
As already mentioned the images of AA are not
necessarily the result of mathematical
calculations. Some stunning and interesting
images result from logical protocols, in which
logical transformations, such as translations,
rotations, changes of scale, etc, take place.
Obviously these transformations are made
through mathematics, but the essence of the
transformation is not mathematical, it is logical. A
simple example of a logical transformation is the
one that consists in placing three dots around the
place where before there was only one dot, next
rotate and resize the set. This protocol may be
replicated several times, depending on the
algorithm. The procedure is executed with
mathematics, but the operation is logical. This
simple operation is the basis of tessellations
(generation of amazing mosaics).
Traditional graphic artists paint images with the
help of utensils like paintbrushes, sharpened
canes, rags and nails; they use these implements
as painting tools. Some artists even use their own
blood to paint. Similarly, Algorithmic Art uses
mathematics as a tool.
Algorithmic art techniques
Not all the techniques used to create works of AA,
were created expressly for that purpose, many
techniques have been adapted from other areas
such as physics, biology, mathematics, etc.
Some well-known techniques that are commonly
used to create works of algorithmic art are
Generative Art, fractal modeling, Iterated Function
Systems, Cellular Automata Tesellations, Polar
inversion, Recursion, etc.
The VirtualDynamics Intelligencer Math-Algorithm-Art Oct 2006
Obviously there must be some other techniques of
AA, which are not known and are therefore of very
limited use. In these cases, only the creator of the
algorithm can use it to create images, which can
be stunning and with great visual impact.
Time Evolution as AA
Some works of algorithmic art are evolutionary.
They show the time-evolution of certain
silhouettes, shapes and colors. The final resulting
image could be very interesting and visually
shocking, but the series of transformations taking
part in their generation, this is, the evolution that
generates the final product, may also be
fascinating. This may be appreciated when the
author of the artwork makes the computer to
slowly execute and display graphically on the
computer monitor the results as time elapses. For
this reason an algorithmic artwork can also be
rendered as a film or video.
Some artworks of AA are nowadays
impossible to re-create
There exist certain old cinema films -which are
true jewels of cinematography- that have been
lost because they cannot be reproduced any
more. Due to some technical reason it is
impossible to render them in current video
formats. The same happens with some artworks
of algorithmic art.
Since the advent of computers, especially the
personal ones, the creators of AA in the whole
world have created some really impressive
images, which are literally unique because they
cannot be reproduced anymore, unfortunately
they cannot be any longer copied or re-created.
The main reason is that the current computers are
too fast and do not allow old algorithms to be
executed. In cases like these, attempts were
made so as to adapt the old algorithm to modern
technology, but this has not always worked.
Years ago, when the operating system of personal
computers used to be the DOS (Disk Operating
System), authors of computer art created some
amazing images. As time went by, the upgrading
of computers switched from DOS to Windows and
Linux; but these modern operating systems are
super-fast compared to DOS and, as a
consequence the early algorithms created for
DOS could no longer be executed. As a result
those beautiful images created under DOS were
lost, some still exist nowadays but cannot be re-
created, literally. In some way more than a few of
these images are like those classic paintings
which art collectors and museums would like to
Sometimes the situation is even worse, because
from the impressive images created under DOS,
only a remembrance remain today. In these cases
not only the printing has been lost, but even the
algorithm has been lost.
The VirtualDynamics Intelligencer Math-Algorithm-Art Oct 2006
In the past due to the slowness of computers, if
desired, it was possible to display on the monitor
the step-by-step evolution of an algorithm; then it
was possible to interrupt the development of an
image and get some extraordinary image that
eventually showed up on the way towards the
objective image. Currently modern computers
execute algorithms so swiftly that its author only
sees the final image it creates. Obviously, this is a
point where the impressive speed of modern
computers has disadvantages.
As an interesting example, many years ago the
author of this document, created an algorithm
showing on the monitor of the computer, the
evolution of a group of ellipses that were rendered
as a knot of random colored ellipses. This was
created in DOS which now is no longer used as it
is very slow for today's computing standards.
Nowadays to execute this algorithm, it has had to
be adapted to modern fast computers and, even
when executed very slowly, it turns out very quick
to the human eyes, and only the final knot is
appreciated. In this case the captivating beauty of
the algorithm used to be the visualization of the
slow step-by-step evolution of the knot of ellipses
and, this has been lost.
The images included in this document
Images included in this document have all been
created with algorithms developed expressly for
AA by the author (JMJ). Here images are small,
low-resolution and in compressed format, so its
quality is not optimal. When any of these images
is to be printed, it is developed to measure not
less than 60 cm x 60 cm, and in non-compressed
format so that it has a very high quality.
Some of the images included in this document are
symmetrical, others show inverted symmetry (they
are anti-symmetric) and yet others have no
symmetry at all, some are even abstract.
Obviously in order to create these images each
one of them had to be planned before its
development. If the image was to be symmetrical,
details associated to symmetry had to be included
in the algorithm, when anti-symmetry was the
objective, an algorithm that introduced inverted
symmetry had to be included in the development
of the image. All the process is like the one
followed when constructing a house to dwell in
Sometimes, and just like when an architect
constructs his own house, some modifications
were introduced at construction time. These
modifications were added because the obtained
images were not as fascinating as the expected
ones. The colors and degrees of contrast among
them were also calculated and they were
sometimes modified as well, so as to achieve a
more impressive and captivating image.
This paper has signaled that the algorithmic
creation of images depends strongly on both,
The VirtualDynamics Intelligencer Math-Algorithm-Art Oct 2006
computers and mathematics. The fact that
mathematics is used to communicate computers
how to proceed so as to create an artwork does
not necessarily mean that algorithmic artworks are
in essence, mathematical; indeed, many times
these images are result of analogical
transformations. This paper has mentioned some
known techniques used to create algorithmic art
images; also some technical details involucrate in
the planning and the development of digital
images have been exposed. It has been explained
why due to the high speed of modern computers,
many algorithmic artworks, created originally in
ancient DOS, definitively disappeared, since they
cannot be reproduced anymore.
This document is an adaptation from the original
in Spanish.
[1] Herbert W. Franke, (1986), Refractions of
Science into Art, The Beauty of Fractals.
[2] Jaap A. Kaandorp, (1994), Fractal Modelling,
Growth and Form in Biology. Springer-Verlag.
[3] Przemyslaw Prusinkiewicz Aristid
Lindenmayer (1990), The Algorithmic Beauty of
Plants. Springer.
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[5] Michael Barnsley (1988), Fractals Everywhere,
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ResearchGate has not been able to resolve any citations for this publication.
Art critics in the centuries to come will, I expect, look back on our age and come to conclusions quite different than our own experts. Most likely the painters and sculptors esteemed today will nearly have been forgotten, and instead the appearance of electronic media will be hailed as the most significant turn in the history of art. The debut of those first halting and immature attempts to achieve that ancient goal, namely the pictorial expression and representation of our world, but with a new media, will finally be given due recognition.
A largely phenomenological study of two-dimensional cellular automata is reported. Qualitative classes of behavior similar to those in one-dimensional cellular automata are found. Growth from simple seeds in two-dimensiona! cellular automata can produce patterns with complicated boundaries, characterized by a variety of growth dimensions. Evolution from disordered states can give domains with boundaries that execute effectively continuous motions. Some global properties of cellular automata can be described by entropies and Lyapunov exponents. Others are undecidable.
1 Graphical modeling using L-systems.- 1.1 Rewriting systems.- 1.2 DOL-systems.- 1.3 Turtle interpretation of strings.- 1.4 Synthesis of DOL-systems.- 1.4.1 Edge rewriting.- 1.4.2 Node rewriting.- 1.4.3 Relationship between edge and node rewriting.- 1.5 Modeling in three dimensions.- 1.6 Branching structures.- 1.6.1 Axial trees.- 1.6.2 Tree OL-systems.- 1.6.3 Bracketed OL-systems.- 1.7 Stochastic L-systems.- 1.8 Context-sensitive L-systems.- 1.9 Growth functions.- 1.10 Parametric L-systems.- 1.10.1 Parametric OL-systems.- 1.10.2 Parametric 2L-systems.- 1.10.3 Turtle interpretation of parametric words.- 2 Modeling of trees.- 3 Developmental models of herbaceous plants.- 3.1 Levels of model specification.- 3.1.1 Partial L-systems.- 3.1.2 Control mechanisms in plants.- 3.1.3 Complete models.- 3.2 Branching patterns.- 3.3 Models of inflorescences.- 3.3.1 Monopodial inflorescences.- 3.3.2 Sympodial inflorescences.- 3.3.3 Polypodial inflorescences.- 3.3.4 Modified racemes.- 4 Phyllotaxis.- 4.1 The planar model.- 4.2 The cylindrical model.- 5 Models of plant organs.- 5.1 Predefined surfaces.- 5.2 Developmental surface models.- 5.3 Models of compound leaves.- 6 Animation of plant development.- 6.1 Timed DOL-systems.- 6.2 Selection of growth functions.- 6.2.1 Development of nonbranching filaments.- 6.2.2 Development of branching structures.- 7 Modeling of cellular layers.- 7.1 Map L-systems.- 7.2 Graphical interpretation of maps.- 7.3 Microsorium linguaeforme.- 7.4 Dryopteris thelypteris.- 7.5 Modeling spherical cell layers.- 7.6 Modeling 3D cellular structures.- 8 Fractal properties of plants.- 8.1 Symmetry and self-similarity.- 8.2 Plant models and iterated function systems.- Epilogue.- Appendix A Software environment for plant modeling.- A.1 A virtual laboratory in botany.- A.2 List of laboratory programs.- Appendix B About the figures.- Turtle interpretation of symbols.
  • Michael Barnsley
Michael Barnsley (1988), Fractals Everywhere, Academic Press
  • Stephen Wolfram
Stephen Wolfram ( ), Computer Software in Science and Mathematics, Scientific American [7] Norman H. Packard – Stephen Wolfram, (1985), Two-Dimensional Cellular Automata, Journal of Statistical Physics, Vol 38.
The Beauty of Fractals
  • W Herbert
  • Franke
Herbert W. Franke, (1986), Refractions of Science into Art, The Beauty of Fractals.