.
.
VIDEO ART
Fractals
Art Conceptual Praxis-
Fractals by C.N.Couvelis
[Χ.Ν.Κουβελης]
music Karlheinz Stockhausen-Steve Reich
.
.
.
.
.
.
.
FractalFrom Wikipedia, the free encyclopedia
A fractal is "a rough or fragmented geometric shape that can be
split into parts, each of which is (at least approximately) a redu-
ced-size copy of the whole,"[1] a property called self-similarity.
Roots of the idea of fractals go back to the 17th century, while
mathematically rigorous treatment of fractals can be traced back
to functions studied by Karl Weierstrass, Georg Cantor and Felix
Hausdorff a century later in studying functions that were continu-
ous but not differentiable; however, the term fractal was coined by
Benoît Mandelbrot in 1975 and was derived from the Latin fractus
meaning "broken" or "fractured." A mathematical fractal is based
on an equation that undergoes iteration, a form of feedback based
on recursion.[2] There are several examples of fractals, which are
defined as portraying exact self-similarity, quasi self-similarity, or
statistical self-similarity. While fractals are a mathematical cons-
truct, they are found in nature, which has led to their inclusion in
artwork. They are useful in medicine, soil mechanics, seismology,
and technical analysis.
A fractal often has the following features:[3]
It has a fine structure at arbitrarily small scales.
It is too irregular to be easily described in traditional Euclidean
geometric language.
It is self-similar (at least approximately or stochastically).
It has a Hausdorff dimension which is greater than its topological
dimension (although this requirement is not met by space-filling
curves such as the Hilbert curve).[4]
It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals
are often considered to be infinitely complex (in informal terms).
Natural objects that are approximated by fractals to a degree in-
clude clouds, mountain ranges, lightning bolts, coastlines, snow
flakes, various vegetables (cauliflower and broccoli), and animal
coloration patterns. However, not all self-similar objects are fractals
—for example, the real line (a straight Euclidean line) is formally
self-similar but fails to have other fractal characteristics; for instance,
it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal-generating software.
Images produced by such software are normally referred to as being
fractals even if they do not have the above characteristics, such as when
it is possible to zoom into a region of the fractal that does not exhibit
any fractal properties. Also, these may include calculation or display
artifacts which are not characteristics of true fractals
The mathematics behind fractals began to take shape in the 17th centu-
ry when a mathematician and philosopher Gottfried Leibniz considered
recursive self-similarity (although he made the mistake of thinking that
only the straight line was self-similar in this sense).
It was not until 1872 that a function appeared whose graph would today
be considered fractal, when Karl Weierstrass gave an example of a fun-
ction with the non-intuitive property of being everywhere continuous
but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with
Weierstrass's abstract and analytic definition, gave a more geometric
definition of a similar function, which is now called the Koch curve.[5]
Wacław Sierpiński constructed his triangle in 1915 and, one year later,
his carpet. The idea of self-similar curves was taken further by Paul
Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Sur-
faces Consisting of Parts Similar to the Whole described a new fractal
curve, the Lévy C curve. Georg Cantor also gave examples of subsets
of the real line with unusual properties—these Cantor sets are also now
recognized as fractals.
Iterated functions in the complex plane were investigated in the late 19th
and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and
Gaston Julia. Without the aid of modern computer graphics, however,
they lacked the means to visualize the beauty of many of the objects
that they had discovered.
In the 1960s, Benoît Mandelbrot started investigating self-similarity in
papers such as How Long Is the Coast of Britain? Statistical Self-Simila-
rity and Fractional Dimension,[6] which built on earlier work by Lewis
Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal"
to denote an object whose Hausdorff–Besicovitch dimension is greater
than its topological dimension. He illustrated this mathematical defini-
tion with striking computer-constructed visualizations. These images
captured the popular imagination; many of them were based on recur-
sion, leading to the popular meaning of the term "fractal".[7]
A class of examples is given by the Cantor sets, Sierpinski triangle and
carpet, Menger sponge, dragon curve, space-filling curve, and Koch
curve. Additional examples of fractals include the Lyapunov fractal
and the limit sets of Kleinian groups. Fractals can be deterministic (all
the above) or stochastic (that is, non-deterministic). For example, the
trajectories of the Brownian motion in the plane have a Hausdorff dimen-
sion of 2.
Chaotic dynamical systems are sometimes associated with fractals. Ob-
jects in the phase space of a dynamical system can be fractals (see attra-
ctor). Objects in the parameter space for a family of systems may be fra-
ctal as well. An interesting example is the Mandelbrot set. This set con-
tains whole discs, so it has a Hausdorff dimension equal to its topologi-
cal dimension of 2—but what is truly surprising is that the boundary of
the Mandelbrot set also has a Hausdorff dimension of 2 (while the topo-
logical dimension of 1), a result proved by Mitsuhiro Shishikura in 1991.
A closely related fractal is the Julia set.
Five common techniques for generating fractals are:
Escape-time fractals – (also known as "orbits" fractals) These are defi-
ned by a formula or recurrence relation at each point in a space (such
as the complex plane). Examples of this type are the Mandelbrot set,
Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov
fractal. 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.
Iterated function systems – These have a fixed geometric replacement
rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch
snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge,
are some examples of such fractals.
Random fractals – Generated by stochastic rather than deterministic
processes, for example, trajectories of the Brownian motion, Lévy flight,
percolation clusters, self avoiding walks, fractal landscapes and the Brow-
nian tree. The latter yields so-called mass- or dendritic fractals, for exa-
mple, diffusion-limited aggregation or reaction-limited aggregation
clusters.
Strange attractors – Generated by iteration of a map or the solution of a
system of initial-value differential equations that exhibit chaos.
L-systems - These are generated by string rewriting and are designed
to model the branching patterns of plants.
Classification
Fractals can also be classified according to their self-similarity. There
are three types of self-similarity found in fractals:
Exact self-similarity – This is the strongest type of self-similarity; the
fractal appears identical at different scales. Fractals defined by iterated
function systems often display exact self-similarity. For example, the
Sierpinski triangle and Koch snowflake exhibit exact self-similarity.
Quasi-self-similarity – This is a looser form of self-similarity; the fra-
ctal appears approximately (but not exactly) identical at different scales.
Quasi-self-similar fractals contain small copies of the entire fractal in
distorted and degenerate forms. Fractals defined by recurrence relations
are usually quasi-self-similar. The Mandelbrot set is quasi-self-similar,
as the satellites are approximations of the entire set, but not exact copies.
Statistical self-similarity – This is the weakest type of self-similarity; the
fractal has numerical or statistical measures which are preserved across
scales. Most reasonable definitions of "fractal" trivially imply some form
of statistical self-similarity. (Fractal dimension itself is a numerical mea-
sure which is preserved across scales.) Random fractals are examples of
fractals which are statistically self-similar. The coastline of Britain is
another example; one cannot expect to find microscopic Britains (even
distorted ones) by looking at a small section of the coast with a magni-
fying glass.
Possessing self-similarity is not the sole criterion for an object to be ter-
med a fractal. Examples of self-similar objects that are not fractals include
the logarithmic spiral and straight lines, which do contain copies of them-
selves at increasingly small scales. These do not qualify, since they have
the same Hausdorff dimension as topological dimension.
Approximate fractals are easily found in nature. These objects display
self-similar structure over an extended, but finite, scale range. Examples
include clouds, river networks, fault lines, mountain ranges, craters,[8]
snow flakes,[9] crystals,[10] lightning, cauliflower or broccoli, and sy-
stems of blood vessels and pulmonary vessels, and ocean waves.[11]
DNA and heartbeat[12] can be analyzed as fractals. Even coastlines
may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer
by using a recursive algorithm. This recursive nature is obvious in these
examples—a branch from a tree or a frond from a fern is a miniature re-
plica of the whole: not identical, but similar in nature. The connection
between fractals and leaves is currently being used to determine how
much carbon is contained in trees.[13]
In 1999, certain self similar fractal shapes were shown to have a pro-
perty of "frequency invariance"—the same electromagnetic properties
no matter what the frequency—from Maxwell's equations (see fractal
antenna).[14]
Further information: Fractal art
Fractal patterns have been found in the paintings of American artist
Jackson Pollock. While Pollock's paintings appear to be composed of
chaotic dripping and splattering, computer analysis has found fractal
patterns in his work.[15]
Decalcomania, a technique used by artists such as Max Ernst, can pro-
duce fractal-like patterns.[16] It involves pressing paint between two
surfaces and pulling them apart.
Cyberneticist Ron Eglash has suggested that fractal-like structures are
prevalent in African art and architecture. Circular houses appear in cir-
cles of circles, rectangular houses in rectangles of rectangles, and so on.
Such scaling patterns can also be found in African textiles, sculpture,
and even cornrow hairstyles.[17][18]
In a 1996 interview with Michael Silverblatt, David Foster Wallace ad-
mitted that the structure of the first draft of Infinite Jest he gave to his
editor Michael Pietsch was inspired by fractals, specifically the Sier-
pinski triangle (aka Sierpinski gasket) but that the edited novel is
"more like a lopsided Sierpinsky Gasket".[19
Applications
Main article: Fractal analysis
As described above, random fractals can be used to describe many
highly irregular real-world objects. Other applications of fractals
include:[20]
Classification of histopathology slides in medicine
Fractal landscape or Coastline complexity
Enzyme/enzymology (Michaelis-Menten kinetics)
Generation of new music
Signal and image compression
Creation of digital photographic enlargements
Seismology
Fractal in soil mechanics
Computer and video game design, especially computer graphics for
organic environments and as part of procedural generation
Fractography and fracture mechanics
Fractal antennas – Small size antennas using fractal shapes
Small angle scattering theory of fractally rough systems
T-shirts and other fashion
Generation of patterns for camouflage, such as MARPAT
Digital sundial
Technical analysis of price series (see Elliott wave principle)
Fractals in networks
.
.
.
.
VIDEO ART
Fractals
Art Conceptual Praxis-
Fractals by C.N.Couvelis
[Χ.Ν.Κουβελης]
music Karlheinz Stockhausen-Steve Reich
.
.
.
.
.
.
Fractal
A fractal is "a rough or fragmented geometric shape that can be
split into parts, each of which is (at least approximately) a redu-
ced-size copy of the whole,"[1] a property called self-similarity.
Roots of the idea of fractals go back to the 17th century, while
mathematically rigorous treatment of fractals can be traced back
to functions studied by Karl Weierstrass, Georg Cantor and Felix
Hausdorff a century later in studying functions that were continu-
ous but not differentiable; however, the term fractal was coined by
Benoît Mandelbrot in 1975 and was derived from the Latin fractus
meaning "broken" or "fractured." A mathematical fractal is based
on an equation that undergoes iteration, a form of feedback based
on recursion.[2] There are several examples of fractals, which are
defined as portraying exact self-similarity, quasi self-similarity, or
statistical self-similarity. While fractals are a mathematical cons-
truct, they are found in nature, which has led to their inclusion in
artwork. They are useful in medicine, soil mechanics, seismology,
and technical analysis.
A fractal often has the following features:[3]
It has a fine structure at arbitrarily small scales.
It is too irregular to be easily described in traditional Euclidean
geometric language.
It is self-similar (at least approximately or stochastically).
It has a Hausdorff dimension which is greater than its topological
dimension (although this requirement is not met by space-filling
curves such as the Hilbert curve).[4]
It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals
are often considered to be infinitely complex (in informal terms).
Natural objects that are approximated by fractals to a degree in-
clude clouds, mountain ranges, lightning bolts, coastlines, snow
flakes, various vegetables (cauliflower and broccoli), and animal
coloration patterns. However, not all self-similar objects are fractals
—for example, the real line (a straight Euclidean line) is formally
self-similar but fails to have other fractal characteristics; for instance,
it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal-generating software.
Images produced by such software are normally referred to as being
fractals even if they do not have the above characteristics, such as when
it is possible to zoom into a region of the fractal that does not exhibit
any fractal properties. Also, these may include calculation or display
artifacts which are not characteristics of true fractals
The mathematics behind fractals began to take shape in the 17th centu-
ry when a mathematician and philosopher Gottfried Leibniz considered
recursive self-similarity (although he made the mistake of thinking that
only the straight line was self-similar in this sense).
It was not until 1872 that a function appeared whose graph would today
be considered fractal, when Karl Weierstrass gave an example of a fun-
ction with the non-intuitive property of being everywhere continuous
but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with
Weierstrass's abstract and analytic definition, gave a more geometric
definition of a similar function, which is now called the Koch curve.[5]
Wacław Sierpiński constructed his triangle in 1915 and, one year later,
his carpet. The idea of self-similar curves was taken further by Paul
Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Sur-
faces Consisting of Parts Similar to the Whole described a new fractal
curve, the Lévy C curve. Georg Cantor also gave examples of subsets
of the real line with unusual properties—these Cantor sets are also now
recognized as fractals.
Iterated functions in the complex plane were investigated in the late 19th
and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and
Gaston Julia. Without the aid of modern computer graphics, however,
they lacked the means to visualize the beauty of many of the objects
that they had discovered.
In the 1960s, Benoît Mandelbrot started investigating self-similarity in
papers such as How Long Is the Coast of Britain? Statistical Self-Simila-
rity and Fractional Dimension,[6] which built on earlier work by Lewis
Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal"
to denote an object whose Hausdorff–Besicovitch dimension is greater
than its topological dimension. He illustrated this mathematical defini-
tion with striking computer-constructed visualizations. These images
captured the popular imagination; many of them were based on recur-
sion, leading to the popular meaning of the term "fractal".[7]
A class of examples is given by the Cantor sets, Sierpinski triangle and
carpet, Menger sponge, dragon curve, space-filling curve, and Koch
curve. Additional examples of fractals include the Lyapunov fractal
and the limit sets of Kleinian groups. Fractals can be deterministic (all
the above) or stochastic (that is, non-deterministic). For example, the
trajectories of the Brownian motion in the plane have a Hausdorff dimen-
sion of 2.
Chaotic dynamical systems are sometimes associated with fractals. Ob-
jects in the phase space of a dynamical system can be fractals (see attra-
ctor). Objects in the parameter space for a family of systems may be fra-
ctal as well. An interesting example is the Mandelbrot set. This set con-
tains whole discs, so it has a Hausdorff dimension equal to its topologi-
cal dimension of 2—but what is truly surprising is that the boundary of
the Mandelbrot set also has a Hausdorff dimension of 2 (while the topo-
logical dimension of 1), a result proved by Mitsuhiro Shishikura in 1991.
A closely related fractal is the Julia set.
Five common techniques for generating fractals are:
Escape-time fractals – (also known as "orbits" fractals) These are defi-
ned by a formula or recurrence relation at each point in a space (such
as the complex plane). Examples of this type are the Mandelbrot set,
Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov
fractal. 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.
Iterated function systems – These have a fixed geometric replacement
rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch
snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge,
are some examples of such fractals.
Random fractals – Generated by stochastic rather than deterministic
processes, for example, trajectories of the Brownian motion, Lévy flight,
percolation clusters, self avoiding walks, fractal landscapes and the Brow-
nian tree. The latter yields so-called mass- or dendritic fractals, for exa-
mple, diffusion-limited aggregation or reaction-limited aggregation
clusters.
Strange attractors – Generated by iteration of a map or the solution of a
system of initial-value differential equations that exhibit chaos.
L-systems - These are generated by string rewriting and are designed
to model the branching patterns of plants.
Classification
Fractals can also be classified according to their self-similarity. There
are three types of self-similarity found in fractals:
Exact self-similarity – This is the strongest type of self-similarity; the
fractal appears identical at different scales. Fractals defined by iterated
function systems often display exact self-similarity. For example, the
Sierpinski triangle and Koch snowflake exhibit exact self-similarity.
Quasi-self-similarity – This is a looser form of self-similarity; the fra-
ctal appears approximately (but not exactly) identical at different scales.
Quasi-self-similar fractals contain small copies of the entire fractal in
distorted and degenerate forms. Fractals defined by recurrence relations
are usually quasi-self-similar. The Mandelbrot set is quasi-self-similar,
as the satellites are approximations of the entire set, but not exact copies.
Statistical self-similarity – This is the weakest type of self-similarity; the
fractal has numerical or statistical measures which are preserved across
scales. Most reasonable definitions of "fractal" trivially imply some form
of statistical self-similarity. (Fractal dimension itself is a numerical mea-
sure which is preserved across scales.) Random fractals are examples of
fractals which are statistically self-similar. The coastline of Britain is
another example; one cannot expect to find microscopic Britains (even
distorted ones) by looking at a small section of the coast with a magni-
fying glass.
Possessing self-similarity is not the sole criterion for an object to be ter-
med a fractal. Examples of self-similar objects that are not fractals include
the logarithmic spiral and straight lines, which do contain copies of them-
selves at increasingly small scales. These do not qualify, since they have
the same Hausdorff dimension as topological dimension.
Approximate fractals are easily found in nature. These objects display
self-similar structure over an extended, but finite, scale range. Examples
include clouds, river networks, fault lines, mountain ranges, craters,[8]
snow flakes,[9] crystals,[10] lightning, cauliflower or broccoli, and sy-
stems of blood vessels and pulmonary vessels, and ocean waves.[11]
DNA and heartbeat[12] can be analyzed as fractals. Even coastlines
may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer
by using a recursive algorithm. This recursive nature is obvious in these
examples—a branch from a tree or a frond from a fern is a miniature re-
plica of the whole: not identical, but similar in nature. The connection
between fractals and leaves is currently being used to determine how
much carbon is contained in trees.[13]
In 1999, certain self similar fractal shapes were shown to have a pro-
perty of "frequency invariance"—the same electromagnetic properties
no matter what the frequency—from Maxwell's equations (see fractal
antenna).[14]
Further information: Fractal art
Fractal patterns have been found in the paintings of American artist
Jackson Pollock. While Pollock's paintings appear to be composed of
chaotic dripping and splattering, computer analysis has found fractal
patterns in his work.[15]
Decalcomania, a technique used by artists such as Max Ernst, can pro-
duce fractal-like patterns.[16] It involves pressing paint between two
surfaces and pulling them apart.
Cyberneticist Ron Eglash has suggested that fractal-like structures are
prevalent in African art and architecture. Circular houses appear in cir-
cles of circles, rectangular houses in rectangles of rectangles, and so on.
Such scaling patterns can also be found in African textiles, sculpture,
and even cornrow hairstyles.[17][18]
In a 1996 interview with Michael Silverblatt, David Foster Wallace ad-
mitted that the structure of the first draft of Infinite Jest he gave to his
editor Michael Pietsch was inspired by fractals, specifically the Sier-
pinski triangle (aka Sierpinski gasket) but that the edited novel is
"more like a lopsided Sierpinsky Gasket".[19
Applications
Main article: Fractal analysis
As described above, random fractals can be used to describe many
highly irregular real-world objects. Other applications of fractals
include:[20]
Classification of histopathology slides in medicine
Fractal landscape or Coastline complexity
Enzyme/enzymology (Michaelis-Menten kinetics)
Generation of new music
Signal and image compression
Creation of digital photographic enlargements
Seismology
Fractal in soil mechanics
Computer and video game design, especially computer graphics for
organic environments and as part of procedural generation
Fractography and fracture mechanics
Fractal antennas – Small size antennas using fractal shapes
Small angle scattering theory of fractally rough systems
T-shirts and other fashion
Generation of patterns for camouflage, such as MARPAT
Digital sundial
Technical analysis of price series (see Elliott wave principle)
Fractals in networks
.
.
.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου