FRACTALS
When you want to measure a fractal line with a unit or tool you'll always find thinner and thinner objects that will escape your unit or tool's sensibility. Besides, as the sensibility of your tool increases so does the length of the line you're trying to measure! The fractal dimension allows us to measure the degree of complexity in an object by evaluating the frequency with which our measurements grow bigger or smaller according to the scale applied.
We know that in classical geometry a segment has a dimension of 1, a circle has a dimension of 2 and a sphere has a dimension of 3. To be consequent with this idea, a fractal line must have a dimension between 1 and 2 [it doesn't occupy the whole plane]. For instance, Cantor's triadic set doesn't occupy the whole segment and Koch and Peano's lines are larger than a line. The same thing happens with fractal planes and their extension between dimensions 2 and 3.
:: properties of fractals
In his work Fractal Geometry: Mathematical Foundations and Applications [1990] Kenneth Falconer describes the properties that a certain fractal "F" should possess:
· F possesses details in each scale of observation; that is, it doesn't have a particular scale but all scales are valid and optimal to its representations.
· It is impossible to describe F in euclidean geometry, neither locally nor globally.
· F possesses some kind of self·similarity.
· F's fractal dimension is bigger than its topological dimension. [see Haussdorff dimension]
· The algorithm that describes F is very simple and possibly, of a recursive nature.
: self·similarity
An object's measurement depends on the chosen scale in which the observation is carried out and for fractal dimensions, scale implies self·similarity. They present such a perfect self·similarity that it would be impossible to distinguish a fractal's image in scale 1 than other in scale 200, since they are built up from copies more or less exact of parts of themselves.
In general, F is a self·similar structure when built as the sum of other structures that are copies of F in a reduced size.
To illustrate this property scientists usually make use of Barnsley's Fern:
Each leaf of this fern is identical to the original fern. If we zoomed in on one of its leafs we would find an identical fern only much smaller, and we could repeat this process to infinity. This means that each sub·group of leafs is identical to the original fern.
. . . . . . . . . . . . . . . . . . . . . . . .
:: Koch's curve
Each step in its generation process the curve increases a third of its length [each curve's length is 4/3 of the previous one]:
Its dimension is 1.2618
. . . . . . . . . . . . . . . . . . . . . . .
:: Peano's curve
It receives its name from the Italian mathematician Giuseppe Peano. It's a curve which, in its limit, occupies the whole plane. Its properties are quite curious:
· It never passes twice over the same point.
· It's continuous and converges uniformly.
· The function that defines the curve is injective and homemorfa in a certain interval, however, its limit has a higher dimension.
. . . . . . . . . . . . . . . . . . . . . . .
:: Cantor's triadic set
This group is considered the precursor of all fractals. It was defined by Georg Cantor in 1983 and possesses a series of notorious metrical properties. It is a group difficult to accept conceptually speaking because it vanishes progressively until it becomes invisible, even though on the other hand it is accepted as an infinite succession of segments of a length different to zero.
A segment of a fixed length divides itself into three parts and the central segment is suppressed. This process repeats itself in the segments obtained from each division. As you can see this procedure is recursive and the aspect of a Cantor's triadic set of high level is always the same despite the level of construction in which you find it.
. . . . . . . . . . . . . . . . . . . . . . .
:: Sierpinski's triangle
Sierpinski's triangle is a popular geometrical group introduced by the famous Polish mathematician Waclack Sierpinski [1882-1969]. It's a deterministic fractal that can be generated in various ways. The usual one consists of, starting with an equilateral triangle, marking the medium points of its sides and erasing the interior triangle [considered as an open group]. The same process is applied to the three obtained triangles and so on [technically speaking, Sierpinski's triangle is defined as the intersection of the closed groups that are generated during each generation process].
a=3, s=(1/2), its dimension is 1.5850
· How to build a Sierpinski's triangle
. . . . . . . . . . . . . . . . . . . . . . . .
:: Highway·Lévy's dragon
This curve [the images' silhouette] was built around 1967 by NASA's physicist John E. Heighway. Heighway illustrated its construction through a proper bending of a sheet of paper [so it's topological dimension is 1].
. . . . . . . . . . . . . . . . . . . . . . .
:: Benoit Mandelbrot and IBM
IBM were getting really mad at the noise in the telephone lines of their computer network. That noise was unavoidable, they could attenuate it by amplifying the signal, but they always found interferences producing continous errors in the data stream.
This tragedy reached Mandelbrot's ears, who thought of a method that describes the erroneus distribution of the data stream: in periods of apparition followed by errors [as minnimal as they were] there were always periods of clean transmission. Mandelbroot's geometrical intuition led him to the discovery of a direct relationship between the periods of errors and the periods of clean transmission that very much resembled Cantor's triadic set!
Mandelbrot's group is the most known and studied of all fractal groups today. It's defined on a complex plane as:
If the succession is limited, then it is determined that c belongs to Mandelbrot's group, and if not, it's excluded from it.
In this image all black dots belong to the group and the coloured ones don't. The colours indicate the speed with which the succession diverges [tends to infinity]: in dark green, after a few calculations it becomes evident that the point doesn't belong to the group whether as in white, it has taken much longer to prove it. Since you cannot calculate infinite values it is necessary to set a limit and determine that if the first points in the succession are limited then it is considered that the dot belongs to the group. When increasing the value of p the image precision increases as well.
Mandelbrot's group at different scales
When you want to measure a fractal line with a unit or tool you'll always find thinner and thinner objects that will escape your unit or tool's sensibility. Besides, as the sensibility of your tool increases so does the length of the line you're trying to measure! The fractal dimension allows us to measure the degree of complexity in an object by evaluating the frequency with which our measurements grow bigger or smaller according to the scale applied.
We know that in classical geometry a segment has a dimension of 1, a circle has a dimension of 2 and a sphere has a dimension of 3. To be consequent with this idea, a fractal line must have a dimension between 1 and 2 [it doesn't occupy the whole plane]. For instance, Cantor's triadic set doesn't occupy the whole segment and Koch and Peano's lines are larger than a line. The same thing happens with fractal planes and their extension between dimensions 2 and 3.
:: properties of fractals
In his work Fractal Geometry: Mathematical Foundations and Applications [1990] Kenneth Falconer describes the properties that a certain fractal "F" should possess:
· F possesses details in each scale of observation; that is, it doesn't have a particular scale but all scales are valid and optimal to its representations.
· It is impossible to describe F in euclidean geometry, neither locally nor globally.
· F possesses some kind of self·similarity.
· F's fractal dimension is bigger than its topological dimension. [see Haussdorff dimension]
· The algorithm that describes F is very simple and possibly, of a recursive nature.
: self·similarity
An object's measurement depends on the chosen scale in which the observation is carried out and for fractal dimensions, scale implies self·similarity. They present such a perfect self·similarity that it would be impossible to distinguish a fractal's image in scale 1 than other in scale 200, since they are built up from copies more or less exact of parts of themselves.
In general, F is a self·similar structure when built as the sum of other structures that are copies of F in a reduced size.
To illustrate this property scientists usually make use of Barnsley's Fern:
Each leaf of this fern is identical to the original fern. If we zoomed in on one of its leafs we would find an identical fern only much smaller, and we could repeat this process to infinity. This means that each sub·group of leafs is identical to the original fern.
. . . . . . . . . . . . . . . . . . . . . . . .
:: Koch's curve
Each step in its generation process the curve increases a third of its length [each curve's length is 4/3 of the previous one]:
Its dimension is 1.2618
. . . . . . . . . . . . . . . . . . . . . . .
:: Peano's curve
It receives its name from the Italian mathematician Giuseppe Peano. It's a curve which, in its limit, occupies the whole plane. Its properties are quite curious:
· It never passes twice over the same point.
· It's continuous and converges uniformly.
· The function that defines the curve is injective and homemorfa in a certain interval, however, its limit has a higher dimension.
. . . . . . . . . . . . . . . . . . . . . . .
:: Cantor's triadic set
This group is considered the precursor of all fractals. It was defined by Georg Cantor in 1983 and possesses a series of notorious metrical properties. It is a group difficult to accept conceptually speaking because it vanishes progressively until it becomes invisible, even though on the other hand it is accepted as an infinite succession of segments of a length different to zero.
A segment of a fixed length divides itself into three parts and the central segment is suppressed. This process repeats itself in the segments obtained from each division. As you can see this procedure is recursive and the aspect of a Cantor's triadic set of high level is always the same despite the level of construction in which you find it.
. . . . . . . . . . . . . . . . . . . . . . .
:: Sierpinski's triangle
Sierpinski's triangle is a popular geometrical group introduced by the famous Polish mathematician Waclack Sierpinski [1882-1969]. It's a deterministic fractal that can be generated in various ways. The usual one consists of, starting with an equilateral triangle, marking the medium points of its sides and erasing the interior triangle [considered as an open group]. The same process is applied to the three obtained triangles and so on [technically speaking, Sierpinski's triangle is defined as the intersection of the closed groups that are generated during each generation process].
a=3, s=(1/2), its dimension is 1.5850
· How to build a Sierpinski's triangle
. . . . . . . . . . . . . . . . . . . . . . . .
:: Highway·Lévy's dragon
This curve [the images' silhouette] was built around 1967 by NASA's physicist John E. Heighway. Heighway illustrated its construction through a proper bending of a sheet of paper [so it's topological dimension is 1].
. . . . . . . . . . . . . . . . . . . . . . .
:: Benoit Mandelbrot and IBM
IBM were getting really mad at the noise in the telephone lines of their computer network. That noise was unavoidable, they could attenuate it by amplifying the signal, but they always found interferences producing continous errors in the data stream.
This tragedy reached Mandelbrot's ears, who thought of a method that describes the erroneus distribution of the data stream: in periods of apparition followed by errors [as minnimal as they were] there were always periods of clean transmission. Mandelbroot's geometrical intuition led him to the discovery of a direct relationship between the periods of errors and the periods of clean transmission that very much resembled Cantor's triadic set!
Mandelbrot's group is the most known and studied of all fractal groups today. It's defined on a complex plane as:
If the succession is limited, then it is determined that c belongs to Mandelbrot's group, and if not, it's excluded from it.
In this image all black dots belong to the group and the coloured ones don't. The colours indicate the speed with which the succession diverges [tends to infinity]: in dark green, after a few calculations it becomes evident that the point doesn't belong to the group whether as in white, it has taken much longer to prove it. Since you cannot calculate infinite values it is necessary to set a limit and determine that if the first points in the succession are limited then it is considered that the dot belongs to the group. When increasing the value of p the image precision increases as well.
Mandelbrot's group at different scales