Chiara Bova
Claudio Di Filippo
Paola Gigliotti
Tutor: Prof.ssa Anna Alfieri
Liceo Scientifico L. SicilianiCatanzaro
Fractals and art
European Student Conference in MathematicsEUROMATH-2012
21 - 25 March, 2012 Sofia, Bulgaria
Summary • Introduction: What is
a fractal?• Fractals in art:
Fractals in architecture
Fractals in painting• Creating fractal
images by Apophysis
Introduction
A fractal is a geometric figure in which an identical motif is repeated in a
continuously reduced scale.
The word fractal (from the Latin fractus, fragmented, interrupted) was first
introduced by B. Mandelbrot, stating that an object has the characteristic of being extremely
irregular as shape.
The main properties of fractals are:
• Self-similarity: If details are observed at different scales, there is always a rough resemblance to the original fractal.
The main properties of fractals are:
• Indefinite Resolution : it is not possible to define clear and absolute boundaries of the whole (the edges of the image).
The main properties of fractals are:
• Fractional dimension: although they may be represented in a two or three conventional space dimensions , their size is not full, or better, what measures the degree of its irregularity, is a fractional number.
Fractals and Architecture
The balance of proportions between the parties is very important in the works of art.
In the field of architecture, a lot of forms follow fractal geometry.
The fractal connect shapes through their two main features, the self-similarity and fractional dimension, in a continuous succession
of steps to reach the minimum of material.
Medieval Art
Study of the Cathedral of Barnsley:Michael Barnsley, in 1987, introduced a fractal known as Cathedral. It recalls to our minds the twelfth-thirteenth century Gothic cathedrals.
Using the mathematical software Maple 10, I studied the construction of this fractal, whose realization is the only repeatition of the main structure at smaller and smaller scales.
The starting figure is the
isosceles triangle ABC,
which four geometric
transformations are applied
to, respectively:
T1, T2, T3, T4.
B
C
A
The first geometric transformation that changes the triangle ABC in the triangle AED is a contraction (a homothety).
yy
xxT
5
4'
3
1'
1
A E
D
The second geometric transformation changes the original triangle in the triangle GBF. One way to achieve this transformation is to reduce (by a homothety) and apply a translation.
yy
xxT
5
4'
3
2
3
1'
2
F
G B
The third geometric transformation changes the original triangle in the triangle EGH. One way to achieve this transformation is to reduce (by a homothety) and to apply a translation.
yy
xxT
5
1'
3
1
3
1'
3 H
E G
The fourth geometric transformation that changes the initial triangle in the triangle ILM is the composition of a homothety and a translation on the x and y axes.
5
4
50
23'
3
1
3
1'
4
yy
xxT
M
LI
5/450/230
3/103/1
05/1005/4005/40
3/103/13/203/1003/1T
Genetic code of the cathedral of Barnsley
Here there are some examples of Gothic cathedrals:
Milan Cathedral
Notre Dame of Reims Notre Dame of Paris
Renaissance Art• During Renaissance age we see that there is the recovery of balance between the parts of a building and its geometric shapes.
This is the Ovate Stair designed by Andrea Palladio. It is located in the castle of Duino in Trieste.It is a shaped elliptical spiral staircase.
0.250000 0.000000 0.000000 0.250000 0.000000 0.500000 0.822978 -0.475000 0.474955 0.822724 0.301140 -0.173839
The fractal structure corresponding to that is the SPIRAL:
Santa Maria Novella , Firenze.
L. Battista Alberti Facade
I made three pictures with the Geogebra program, in order to study its geometric proportions.
The entire facade of S. Maria Novella fully fits into a square, and three squares, whose sides are equal to the half side of the greater one, circumscribe the central parts, that is two squares circumscribe the lower part and the third square circumscribe the upper middle part.
In particular, we can see how the invention of the volutes connecting the top to the bottom become a decorative element, which repeats the geometric rhythm of the two parts of the facade.
The form of the square is also repeated in the sub-modules; it can be seen that there is diagonally a correspondence between geometric shapes and symmetry.
Modern ArchitectureAmong the modern architects, many artists have been
inspired by fractal geometry.
Among these, I have taken into account:
Frank Lloyd Wright
(8 june 1867 –9 april 1959)
Frank O. Gehry (Toronto, 28
february 1929).
Frank Lloyd WrightPalmer House (Ann Arbor, Michigan)
Palmer House is a residence designed in 1952 for William Palmer. The plant is based on the model of an equilateral triangle. The Palmer House exemplifies Wright's organic architecture American, in which all parts are connected to the whole and are related to the environment, with an adaptation to the forms of nature.
[0.1,0.3,0,0.4,-0.4,0.2],[0.8,-0.4,0.4,0.8,-0.06,0.05]
As in fractal geometry the spiral is multiplied through cyclical pattern, here the form of an equilateral triangle is self-similar. It
opens an evolving curve.
Marin County Civic Center San Rafael, California
Here Wright uses the form of the cycloid on different levels, proposing an increasing lowering on different scales.
O sR
r
C
2,50 cm
c1
c2
T
ZP
Cicloide Accorciata
1
1
1-(2/3)*cos(t)y(t)=
Structure of cycloid and shortened cycloid :
Frank O. Gehry“Guggenheim Museum” , Bilbao
Museum of Contemporary Art opened in 1997.It consists of a series of complex volumes, interconnected in a spectacular way.The imposing structure blends with the environment thanks to its simple elegance also due to the materials of which it is coated.
In this work there is the recovery of 'organic architecture. The structure is almost a sculpture surrounded by the landscape, so that it is the nature itself that unconsciously produces fractal forms.
0.340621 -0.071275 0.071284 0.340623 0.000000 0.000000 0.166667 0.037463 -0.345977 0.345978 0.037471 0.341000 0.071000 0.166667 0.340621 -0.071275 0.071284 0.340623 0.379000 0.418000 0.166667 -0.233669 0.257876 -0.257882 -0.233675 0.720000 0.489000 0.166667 0.173052 0.301926 -0.301922 0.173045 0.486000 0.231000 0.166667 0.340621 -0.071275 0.071284 0.340623 0.659000 -0.071000 0.166665
Gala spheres
Salvador Dalì
• Salvador Dalì (Figueres -May 11, 1904 – Figueres -January 23, 1989)
• and the self-similarity
• Jackson Pollock (Cody -January 28, 1912 –Long Island- August 11, 1956)
• and the fractal dimension
Salvador Dalì
His painterly skills are often attributed to the influence of Renaissance masters.Dalí's expansive artistic repertoire includes film, sculpture, and photography, in collaboration with a range of artists in a variety of media.
Dali’s signature
The Face of War(1940)
This painting inspired Mandelbrot, with its self-similarity of faces within faces, to infinity
Jackson Pollock The Shaman Artist
Jackson Pollock was an influential American painter and a major figure in the abstract expressionist movement. He was well known for his uniquely defined style of drip painting.
Pollock’s signature
Drip Painting
• Drip painting is a form of abstract art in which paint is dripped or poured onto the canvas.
Action painting sometimes called "gestural abstraction", is a style of painting in which paint is spontaneously dribbled, splashed or smeared onto the canvas, rather than being carefully applied
« My painting does not come from the easel. I prefer to tack the unstretched canvas to the hard wall or the floor. I need the resistance of a hard surface. On the floor I am more at ease. I feel nearer, more part of the painting, since this way I can walk around it, work from the four sides and literally be in the painting.This is akin to the methods of the Indian sand painters of the West.»
Fractal Dimension
• One of the peculiarities of the fractal figures is the fractal dimension: in fact, the figures of Euclidean geometry are full size, while the complex shapes of fractal geometry have non-integer dimension
First MethodSecond MethodBox Counting
As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases
First Method
There are two main approaches to generate a fractal structure. One is growing from a unit object
Second Method
..the other is to construct the subsequent divisions of an original structure, like the Sierpinski triangle.
Richard Taylor• Richard Taylor is currently Professor of Physics at the
University of Eugene. He had the intuition that Pollock adopted «rhythms of nature»
Blue Poles and Richard Taylor- Tate Gallery
Blue Poles, number 11
For Example Lavander Mist Nr. 1 / 1950
Richard Taylor, Box Counting
In fractal geometry, the Minkowski–Bouligand dimension or box-counting dimension, is a way of determining the fractal dimension
Box Counting Dimension
Suppose that N(ε) is the number of boxes of side length ε required to cover the set. Then the box-counting dimension is defined as:
Estimating the box-counting dimension of the coast of Great Britain
Another mathematic example
• The fractal dimension of the triangle of Sierpinsky is:
This process…
• Can be applied to :
This painting (Blue Poles) divided by a Cartesian coordinate system is made up of 56 squares
Single Square : 1m₂Complete painting 42 m₂
• D = log N / log (1/e)
In the end…
The fractal dimension of Jackson Pollock’s painting is…
Blue PolesLavander Mist
Artists examined:
Fernando Garbellotto
Kerry Mitchell
Apophysis is one of the many freeware software used to create fractal images.
The interface displayed at the beginning of the
program
After you have chosen the start figure, through the window
Mutation, it could be possible to chose one of the many
mutations of the same fractal where afterwards work, from
the voice Trend you can assign to the picture the main
characteristic that also lodge when you change the
coefficients of the transformation.
The voice Trend: By there is open a pull-down menu
Possible mutations of
the fractal
In the Editor window the picture is divided in many triangles, every of which shows a precise transformation. By changing the coefficients put in a table or just by moving the selected triangle in the level you obtain a new picture with certain parameters. Afterwards using other voices of the menu it is possible to change futhermore the picture by giving it originality.
The diffent triangles that
detect the transformation
A table where you can insert
the coefficients
A preview of the picture that we are
creating
After you have realized the fractal image it is possible to change the
combination of the colours through the window Gradient and
clicking on the voice Preset it opens a pull-down menu where it
could be possible choose the combination of the more suitable
colours.
The voice Preset:By clicking it opens the menu to choose the many combinations.
After you have realized the fractal image you also need to know also how save it. Through the voice Render it could be possible
to do it because the pictures are saved in the required format, for example .jpg or .png
This is where it could be possible
adjust the size of the picture
This is where it is displayed the name with the relative path of the picture.
This is where it could be
possible to modify the
quality of the picture.
Transform: -0.65106/-0.94386/ -0.41886/0.719811/4.3605/1.09089
Transform:0.33275/0.645358/0.406388 /0.480645/0.519377/0.271655
We learnt… Properties of fractals,made by software: IFS Kit,
Apophisys, Maple10; Mathematical structure of fractals; The employment of fractals for the study of certain
architectural works; References to painters, architects and professional
men and women that made use of fractal models; The study of pictorial techniques with historical and
mathematical outlines; The creation of our particular fractal pictures.
Thank you for the
attention
