Barnsley fern
Encyclopedia
The Barnsley Fern is a fractal
named after the British mathematician
Michael Barnsley
who first described it in his book Fractals Everywhere. He made it to resemble the Black Spleenwort, Asplenium adiantum-nigrum
.
sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the Sierpinski triangle
, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's book about fractals is based on the course which he taught for undergraduate and graduate students in the School of Mathematics, Georgia Institute of Technology
, called Fractal Geometry. After publishing the book, a second course was developed, called "Fractal Measure Theory". Barnsley's work has been a source of inspiration to graphic artists
attempting to imitate nature with mathematical models.
The fern code developed by Barnsley is an example of an iterated function system
(IFS) to create a fractal. He has used fractals to model a diverse range of phenomena
in science and technology, but most specifically plant structures.
--Michael Barnsley et al.
Barnsley's fern uses four affine transformation
s. The formula for one transformation is the following:
Barnsley shows the IFS code for his Black Spleenwort fern fractal as a matrix of values shown in a table. In the table, the columns "a" through "f" are the coefficients of the equation, and "p" represents the probability factor.
These correspond to the following transformations:
The first point drawn is at the origin (x0 = 0, y0 = 0) and then the new points are iteratively computed by randomly applying one of the following four coordinate transformations:
ƒ1
This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem. This part of the figure is the first to be completed in during the course of iterations.
ƒ2
This coordinate transformation is chosen 85% of the time and maps any point inside the leaflet represented by the red triangle to a point inside the opposite, smaller leaflet represented by the blue triangle in the figure.
ƒ3
This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (it flips it).
ƒ4
This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (without flipping it).
The first coordinate transformation draws the stem. The second generates successive copies of the stem and bottom fronds to make the complete fern. The third draws the bottom frond on the left. The fourth draws the bottom frond on the right. The recursive nature of the IFS guarantees that the whole is a larger replica of each frond. Note that the complete fern is within the range −2.1818 = x = 2.6556 and 0 = y = 9.95851.
One experimenter has come up with a table of coefficients to produce another remarkably naturally looking fern however, resembling the Cyclosorus or Thelypteridaceae
fern. These are:
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
named after the British mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Michael Barnsley
Michael Barnsley
Michael Fielding Barnsley is a British mathematician, researcher and an entrepreneur who has worked on fractal compression; he holds several patents on the technology. He received his Ph.D in Theoretical Chemistry from University of Wisconsin–Madison in 1972...
who first described it in his book Fractals Everywhere. He made it to resemble the Black Spleenwort, Asplenium adiantum-nigrum
Asplenium
Asplenium is a genus of about 700 species of ferns, often treated as the only genus in the family Aspleniaceae, though other authors consider Hymenasplenium separate, based on molecular phylogenetic analysis of DNA sequences, a different chromosome count, and structural differences in the rhizomes...
.
History
The fern is one of the basic examples of self-similarSelf-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the Sierpinski triangle
Sierpinski triangle
The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...
, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's book about fractals is based on the course which he taught for undergraduate and graduate students in the School of Mathematics, Georgia Institute of Technology
Georgia Institute of Technology
The Georgia Institute of Technology is a public research university in Atlanta, Georgia, in the United States...
, called Fractal Geometry. After publishing the book, a second course was developed, called "Fractal Measure Theory". Barnsley's work has been a source of inspiration to graphic artists
Graphics
Graphics are visual presentations on some surface, such as a wall, canvas, computer screen, paper, or stone to brand, inform, illustrate, or entertain. Examples are photographs, drawings, Line Art, graphs, diagrams, typography, numbers, symbols, geometric designs, maps, engineering drawings,or...
attempting to imitate nature with mathematical models.
The fern code developed by Barnsley is an example of an iterated function system
Iterated function system
In mathematics, iterated function systems or IFSs are a method of constructing fractals; the resulting constructions are always self-similar....
(IFS) to create a fractal. He has used fractals to model a diverse range of phenomena
in science and technology, but most specifically plant structures.
--Michael Barnsley et al.
Construction
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s. The formula for one transformation is the following:
Barnsley shows the IFS code for his Black Spleenwort fern fractal as a matrix of values shown in a table. In the table, the columns "a" through "f" are the coefficients of the equation, and "p" represents the probability factor.
| w | a | b | c | d | e | f | p |
|---|---|---|---|---|---|---|---|
| ƒ1 | 0 | 0 | 0 | 0.16 | 0 | 0 | 0.01 |
| ƒ2 | 0.85 | 0.04 | −0.04 | 0.85 | 0 | 1.6 | 0.85 |
| ƒ3 | 0.2 | −0.26 | 0.23 | 0.22 | 0 | 1.6 | 0.07 |
| ƒ4 | −0.15 | 0.28 | 0.26 | 0.24 | 0 | 0.44 | 0.07 |
These correspond to the following transformations:
Computer generation
Though theoretically, Barnsley's fern could be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer mandatory. Popular with mathematicians, there are probably as many computer models of Barnsley's fern today as there are fern varieties. As long as the math is programmed correctly using Barnsley's matrix of constants, the same fern shape will be produced.The first point drawn is at the origin (x0 = 0, y0 = 0) and then the new points are iteratively computed by randomly applying one of the following four coordinate transformations:
ƒ1
- xn + 1 = 0
- yn + 1 = 0.16 yn.
This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem. This part of the figure is the first to be completed in during the course of iterations.
ƒ2
- xn + 1 = 0.85 xn + 0.04 yn
- yn + 1 = −0.04 xn + 0.85 yn + 1.6.
This coordinate transformation is chosen 85% of the time and maps any point inside the leaflet represented by the red triangle to a point inside the opposite, smaller leaflet represented by the blue triangle in the figure.
ƒ3
- xn + 1 = 0.2 xn − 0.26 yn
- yn + 1 = 0.23 xn + 0.22 yn + 1.6.
This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (it flips it).
ƒ4
- xn + 1 = −0.15 xn + 0.28 yn
- yn + 1 = 0.26 xn + 0.24 yn + 0.44.
This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (without flipping it).
The first coordinate transformation draws the stem. The second generates successive copies of the stem and bottom fronds to make the complete fern. The third draws the bottom frond on the left. The fourth draws the bottom frond on the right. The recursive nature of the IFS guarantees that the whole is a larger replica of each frond. Note that the complete fern is within the range −2.1818 = x = 2.6556 and 0 = y = 9.95851.
Mutant varieties
By playing with the coefficients, it is possible to create mutant fern varieties. In his paper on V-variable fractals, Barnsley calls this trait a superfractal.One experimenter has come up with a table of coefficients to produce another remarkably naturally looking fern however, resembling the Cyclosorus or Thelypteridaceae
Thelypteridaceae
Thelypteridaceae is a family of about 900 species of ferns.The ferns are terrestrial, with the exception of a few which are lithophytes . The bulk of the species are tropical, although there are a number of temperate species....
fern. These are:
| w | a | b | c | d | e | f | p |
|---|---|---|---|---|---|---|---|
| ƒ1 | 0 | 0 | 0 | 0.25 | 0 | −0.4 | 0.02 |
| ƒ2 | 0.95 | 0.005 | −0.005 | 0.93 | −0.002 | 0.5 | 0.84 |
| ƒ3 | 0.035 | −0.2 | 0.16 | 0.04 | −0.09 | 0.02 | 0.07 |
| ƒ4 | −0.04 | 0.2 | 0.16 | 0.04 | 0.083 | 0.12 | 0.07 |