Geometry – science and art

Luminiţa – Dominica Moise1, Ruxandra Cristea

1

(1) "Nicolae Kretzulescu" Commercial High School,

Str. Hristo Botev 17, sect. 3, Bucharest, ROMANIA

E-mail: dominic_moise @yahoo.com, cristea.ruxandra@gmail.com

Abstract The paper presents some ways of using the new technologies to help the students understand

classic geometry, as well as the bases of fractal geometry. As it is dedicated to the famous

Romanian geometer Gheorghe Ţiţeica upon the anniversary of 140 years since his birth, the

paper also presents a few of his results in classical geometry. The results are of high interest

for the students and are presented by using new educational technologies.

Keywords: Fractal geometry, New educational technologies, Ţiţeica surface, Architectural

process

„The scientific study of natural phenomena aims to

take a mathematical form and this study is complete

when the mathematical form has been found. Born in

parallel with the Greek art, mathematics has kept in its

internal fabric a certain affinity with art.”

Gheorghe Ţiţeica

Contents

1. Introduction

2. The art of geometry vs. the geometry of art

3. Fractal geometry, a new dimension of human knowledge

4. Gheorghe Ţiţeica - a world famous Romanian geometrician

5. New technologies to support geometry

1. Introduction

Lines or curves, planes or surfaces, symmetries and translations, scales or proportions, they are

concepts that appear in both mathematics and the visual arts. Geometry can thus be considered a

bridge between art and science.

2. The art of geometry vs. the geometry of art

Over time geometry has been considered as a paradigm of reason, having as tools abstraction,

perfection, ordering, analysis. Moreover, through its clarity, elegance and brevity, geometry can be

considered an art of reason.

Pure art, "cleaned" of previous constraints, especially religious ones, is exemplified in

movements like constructivism, purism, cubism, expressionism, surrealism, deconstructionism.

Geometric order governs here and everything relates to it.

Purists regard art from a psychological, scientific perspective. What impresses the individual at

the most profound level must be the base of new art. Based on a generalization of certain results in

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experimental psychology, it was meant to found an art that will scientifically correspond to man’s

spiritual needs, the key to adaptation being the elementary geometric shapes: the circle or the

sphere, a square or a cube, triangle or cone. The order which these embody is considered to be a

human constant, a premise of emotion: "All plastic satisfactions result from the system of

geometry" or "Man is a geometric animal" or "The shapes of simple geometry produce the purest

effect on us. Thus we have physiological keyboard whose sensitive properties we know." (Le

Corbusier). After the geometric order is enthroned, everything relates to it. Some art critics do not

consider as sufficient a justification like "As with other architectural idols, the conclusion also

fulfils the role of hypothesis, preceding the demonstration."

The strongest connection between geometry and art is observed in architecture. Geometry is a

fundamental discipline for architecture because it allows a spatial representation of the ideas of the

architect.

The fundamental principles of architecture were set out by Vitruvius in "The Ten Books of

Architecture":

1. Architecture depends on Order, Arrangement, Eurythmy, Symmetry, Quality and

Economics;

2. Order is selecting a standard module and building the whole starting from it;

3. Arranging is putting the parts in the appropriate place;

4. Eurythmy is the beauty and suitability of the parties together;

5. Symmetry is the relationship between the parties and between them and the whole;

6. Quality is the unified style;

7. Economy is the management of materials and construction.

More recently, in the late twentieth century, we see new attempts to revive the notion of

"architecture". For example M. Drăgănescu introduced a new concise and general definition of the

architecture of an object (1971): "The architecture of an object of any kind, can be defined as a

triplet <Af,Ao,Ag> where: Af is the functional architecture of the object (generally, of the external

functions, but also of some internal functions if they have a special role); Ao represents the parts

(or the main relevant parts) of the object; Ag is how the object is perceived by an external observer

or by the object itself."

Based on the works of M. Drăgănescu, Gorun Manolescu [7] proposed a new hierarchy of the

levels of an architecture by introducing two additional levels: the formative invariants and the

physical, concrete structure:

a) A first structure, directly visible - the physical structure;

b) A second structure, more profound – the organizational and functional structure;

c) A third even more profound structure – the structure of the formative invariants.

What could mean the structure of the formative invariants? Obviously we think of the

Erlangen program of Felix Klein (1872).

"Formative invariants contain 'in nuce' both the backbone of the future artifact and the

associated dynamics (evolution), just as it happens with another formative invariant, this time used

by Nature, namely the DNA. Furthermore, we believe that we are not wrong saying that similarly

to how from primitive forms like 'the sphere with handles' and 'the sphere with Moebius bands'

from the Topology applied to figures, through successive topological transformations derive other

highly sophisticated shapes, in the same way, through successive transformations and detailing,

from a (synthetic) structure of formative invariants, there can be generated other structures:

functional-organizational and physical, in the case of any artefact or natural object (as the

formative invariants preserve themselves). Finally, one can say that the sustainability and

efficiency of the ultimate structure (the physical one) of a natural or artificial object depends on

correctly intuiting the structure of the formative invariants."[7]

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Symmetry, rhythm, harmony are specific to architecture in general, but their opposites -

asymmetry, arrhythmia, dissonance – are met since the beginnings.

If modern purism chose the orderly side - symmetry (nineteenth century), modern architecture

has broken with the past, as modernism opted for the chaotic side. An apparent chaos, modernism

is however governed by formulas, mathematical equations, even new mathematical theories such

as fractal geometry, catastrophe theory, chaos theory or topology.

3. Fractal geometry, a new dimension of human knowledge Fractal geometry is a new language used to describe, analyze and model the complex shapes in

nature. If the traditional elements of the language of Euclidean geometry are forms visible as lines,

circles, spheres, the new language elements are algorithms that can be converted into shapes and

designs only with the help of computers. The algorithm is a powerful descriptive tool. For

example, if such a language should be formalized - say experts - we could describe the formation

of a cloud as simple and accurate as an architect who describes a house using traditional geometry

elements.

We will further illustrate the generation of natural elements with the help of fractal geometry,

as well as the definition of fractals as attractors of systems of iteration functions. We need some

theoretical preliminaries

.

Let (X, d) be a complete metric space.

Definition: The set H (X) consists of non-empty compact subsets of X.

Definition: Let xX and BH(X).

d(x, B) = min {d(x, y)/ yB} represents the distance from x to B.

Definition: Let A and B from H(X).

d(A, B) = max{d(x, B)/xA} represents the distance from A la B.

h(A, B) = max{d(A, B), d(B, A)}

It can be shown that h defines a metric on H(X), called the Hausdorff -Pompeiu metric.

Theorem: (H(X)) is a complete metric

space.

The metric space H(X) is the space

where the fractals are found.

Definition: Let (X, d) be a metric space.

The function f: X→X is named a

contraction function if there exists k [0,1)

so that d(f(x),f(y))≤ kd(x, y), for every x,

yX.

Contraction principle (Banach)

Theorem: Let (X, d) a complete metric space and f: X X a contraction function. Then:

1) f has a unique fixed point u and

2) for any x0 X , the sequence f(n)(

x0) converges to u.

Hutchinson Operator

We consider 2R (the Euclidean plane) as a complete metric space with the usual distance

(Euclidean). Let n be a fixed natural number (not zero) and let for any {1,2,,..., }j n , a

Figure 1. The successive transformations of a square of

sides 1 after some contractions

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contraction 2 2:jW R R having the contraction factor jk . If A is a subset of

2R , we note

with ( )jW A the image o the set A by the function jW .

Definition: We define the application (Hutchinson operator):

H : H(R2) → H(R

2) , H(A)=W1 (A)U W2 (A)U ... U Wn (A).

We will note: 1 2( , ,..., )nH W W W .

Also (2R , W1, W2,...,Wn ) is called iterative function system (IFS).

Theorem: Hutchinson's operator is a contraction on the complete metric space of the compact

plan parts H(R2) with the Hausdorff distance. In addition, the contraction factor k is the largest

element of the set 1 2{ , ,..., }nk k k .

Definition: The fixed point FH(R2) of the Hutchinson operator (it exists and it is a unique

according to the contraction mapping principle) is called the attractor of the iterative function

system (deterministic fractal) and it is the limit of the string Hn(A), for every A H(R2).

These notions, which may seem too technical for students in secondary education, can be

understood because they are richly illustrated with programs made in the LabVIEW environment,

a software package that we have called Fractall.

Figure 2.The Sierpinski triangle as attractor of an iterative system

The four geometrical transformations which applied successively to a square lead to a fern are:

0.00 0.00( , ) ;

0.00 0.16

xf x y

y

0.85 0.04 0.00

( , ) ;0.04 0.85 1.60

xf x y

y

0.20 0.26 0.00( , ) ;

0.23 0.22 1.60

xf x y

y

0.15 0.28 0.00( , ) .

0.26 0.24 0.44

xf x y

y

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Figure 3. The fern as attractor

Mathematics is the domain of freedom and fantasy for architecture - says Michele Emmer from

the University La Sapienza, Rome:

"There are probably two words that express the capacity of contemporary architecture to

enrich the capacity of representation and modelling of space: fantasy and freedom, and they are

conferred by the new geometries, topology and computer graphics programs."

The best known example of applied fractal

geometry is the Guggenheim Museum in Bilbao,

Spain, conducted by Frank Gehry (1990), which

removes the limits imposed by classicism and brings

fractal shapes in the building’s design. Here 26 petals

are developed and twisted, resulting from the

recursive application of an iteration function system.

4. Gheorghe Ţiţeica – a world known Romanian

geometrician

At the end of the nineteenth century, the famous

Erlangen program of Felix Klein gave the idea of studying

geometry by certain groups of transformations. Following

Klein's ideas, Gh. Ţiţeica studied certain curves and

surfaces, discovering some affine, centro-affine and

projective properties of objects subject to transformations.

In 1907 he discovered a class of surfaces in three-

dimensional space, today examples of affine fields, thus

becoming the first geometrician who studied the affine

spheres using Euclidean invariants. A hipersurface Ţiţeica

can be characterized as the locus of points that are at a

affine distance fixed from a centre point (considered as the

origin). The software can decide whether a surface can be

a Ţiţeica surface and there can also be made graphic

representations of the surfaces.

5. New technologies to support geometry

"The task of education and training based on the new information and communication

technologies isn’t to demonstrate that it has immediate results in a race with other types of

Figure 4. Guggenheim Museum, Spain

Figure 5. Ţiţeica surface [6]

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educational systems, but to replace some of the existing structures with a new, probably

superior spectrum of performances, to meet the inherent changes that occur in culture and

civilization. "(Istrate, Olimpius. Remote education. Designing materials, Agata Publishing,

2000, p 39.)

Since geometry also involves creating an intuitive support for the mathematical notions, the

educational software - very diverse - allows a better understanding of the concepts in the

educational process. The software used in teaching has distinctive design features, but what is

giving it the quality of an educational software is the teaching strategy underlying its design and

use.

Figure 6. Illustration of two results from classical geometry using the GeoGebra program: The ”5 lei

coin” problem of Ţiţeica and Euler Cercle.

Figure 7. The Sierpinski triangle as a Figure 8. The Lorenz fractal generated

Geometric fractal (GeoGebra) with the Fractall program package

Figure 9. Fractal art realized in school fractal projects

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"Mathematics is a way of expressing natural laws, it is the easiest and best way to describe a

general law or the flow of a phenomenon, it is the most perfect language in which one can

narrate a natural phenomenon." Gheorghe Ţiţeica

6. References

Books:

Barnsley, M. (1993): Fractals everywhere, Second Edition, Academic Press Professional.

Moise, D. L., Bogdan, B and Druţă, D. (2007): Algoritmi, numere şi fractali, editura Printech, Bucureşti.

Andersen, K. (2007): The Geometry of an Art: The History of the Mathematical Theory of Perspective

from Alberti to Monge, Springer Verlag.

Peitgen, H.O., Jurgens, H. and Saupe, D. (1992): Chaos and New Frontiers of science, Springer Verlag.

Istrate, O. (2000): Educaţia la distanţă. Proiectarea materialelor, Editura Agata.

Theses:

Bobe, A. (2005): Studiul unor algoritmi de algebra si geometrie computaţională.

Journal Articles:

Manolescu, G. (2002): Consideraţii asupra noţiunilor de arhitecturǎ, proces architectural, Qualia

arhitecturalǎ, Noema vol. I, nr. 1, 101.

Conference Proceedings:

Moise, D. L. and Druţă, D. (2012): Fractall - pachet de programe pentru studiul geometriei fractale. In

Proceedings of The 10th National Conference of Virtual Learning, Brasov, 215-222.