Euromath 2012 Sofia Bulgaria

1. Chiara Bova
Claudio Di Filippo
Paola Gigliotti
Tutor: Prof.ssa Anna Alfieri
Liceo Scientifico L. Siciliani
Catanzaro
Fractals and art
European Student Conference in Mathematics
EUROMATH-2012
21 - 25 March, 2012 Sofia, Bulgaria

3. Summary
• Introduction: What is
a fractal?
• Fractals in art:
Fractals in architecture
Fractals in painting
• Creating fractal
images by Apophysis

4. 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.

5. 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.

6. 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).

7. 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.

8. 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.

9. 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.

10. The starting figure is the
isosceles triangle ABC,
which four geometric
transformations are applied
to, respectively:
T1, T2, T3, T4.
B
C
A

11. The first geometric
transformation that
changes the triangle ABC
in the triangle AED is a
contraction (a
homothety).






=
=
=
yy
xx
T
5
4
'
3
1
'
1
A E
D

12. 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
xx
T
5
4
'
3
2
3
1
'
2
F
G B

13. 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
xx
T
5
1
'
3
1
3
1
'
3 H
E G

14. 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
xx
T
M
LI

15. 







5/450/230
3/103/1
05/1005/4005/40
3/103/13/203/1003/1
T
Genetic code of the cathedral of Barnsley

16. Here there are some examples
of Gothic cathedrals:
Milan Cathedral

17. Notre Dame of Reims Notre Dame of Paris

18. 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.

19. 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:

20. Santa Maria Novella ,
Firenze.
L. Battista Alberti Facade
I made three pictures with​​
the Geogebra program, in
order to study its
geometric proportions.

21. 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.

22. 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.

23. 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.

24. Modern Architecture
Among 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).

25. Frank Lloyd Wright
Palmer 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.

26. [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.

27. 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.

28. O sR
r
C
2,50 cm
c1
c2
T
ZP
Cicloide Accorciata
1
1
A T4,23 cm
Structure of cycloid and shortened cycloid :

29. 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.

30. 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

31. Gala spheres
Salvador Dalì

32. • 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

33. 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

34. The Face of War
(1940)
This painting
inspired
Mandelbrot,
with its self-
similarity of
faces within
faces, to
infinity

35. 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

36. 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

37. « 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.»

38. 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 Method
Second Method
Box Counting
As the length of the measuring stick is scaled smaller and
smaller, the total length of the coastline measured
increases

39. First Method
There are two
main
approaches to
generate a
fractal
structure. One
is growing
from a unit
object

40. Second Method
..the other is to
construct the
subsequent
divisions of an
original
structure, like
the Sierpinski
triangle.

41. 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

42. Blue Poles, number 11

43. For Example Lavander Mist Nr. 1 / 1950

44. Richard Taylor, Box Counting
In fractal geometry, the
Minkowski–Bouligand
dimension or box-
counting dimension, is
a way of determining
the fractal dimension

45. 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:

46. Estimating the box-counting dimension of the
coast of Great Britain

47. Another mathematic example
• The fractal dimension of the triangle of Sierpinsky is:

48. This process…
• Can be applied to :
This painting (Blue Poles)
divided by a Cartesian
coordinate system is made
up of 56 squares
Single Square : 1m₂
Complete painting 42 m₂
• D = log N / log (1/e)

49. In the end…
The fractal dimension of Jackson Pollock’s painting is…
Blue Poles
Lavander Mist

50. Artists examined:
Fernando Garbellotto
Kerry Mitchell

53. Apophysis is one of the many freeware software
used to create fractal images.
The interface displayed
at the beginning of the
program

54. 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

55. 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

56. 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.

57. 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.

58. Transform: -0.65106/-0.94386/
-0.41886/0.719811/4.3605/1.09089

59. Transform:0.33275/0.645358/0.406388
/0.480645/0.519377/0.271655

60. 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.

62. Thank you
for the
attention