mathematics, arts and photography by dr marcella lorenzi, university of calabria

8/8/2019 Mathematics, arts and photography by Dr Marcella Lorenzi, University of Calabria

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MATHEMATICS, ART

AND PHOTOGRAPHY

(from Antiquity to Futurism)Talk at the Final Event: SCIENAR Project

The Dutch Academy of Science

Amsterdam, 15 October 2010

Marcella Giulia LORENZI

(Universit della Calabria)(in collaboration with Mauro FRANCAVIGLIA)

Part I: Geometry in Art untilXX Century


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GEOMETRY AND ART from Antiquity to Futurism

Agreement Number 2008-2254/001-001 CTU-MECOAN

SCIENtific Scenarios and theARts

Culture Project

PARTNERS

University of Calabria - Italy (Coordinator)

Institute for Computers - Romania

Virtual Image - England

Electronic Media Reporting - NL

Polytechnic School - Slovakia

www.scienar.eu/main

Project SCIENAR


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Project SCIENAR

Art and Science: they shared deep relationships over the centuries

from ancient Greek to nowadays

Interdisciplinary Approach: scientific and humanistic domains are part of

A SINGLE AND UNIQUE CULTURE

Today: Importance of communicating and diffusing Science through emotion

Possibility of involvement of public and policy makers, more funding

GEOMETRY AND ART from Antiquity to Futurism


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Katherine Hayles says that: artistic expression and Science are cultural products that,

at the same time, express and contribute to form the matrix of the culture itself out of

which they emerge.

Observing reality through the

algorithmic formulation means to

contemplate the true image ofreality, to grasp its mechanisms,

to understand its secret codes.

[] Accordingly, all constraints,

all obstacles, all bridgeless

abyssae between Art and Science

disappear (from an interview

with Haebel, i.e. AntonioDAnna, one of the major

exponents of Fractalism in Italy).

Geometry & Art - A Parallel Development I

Art & Science as a Whole

William Latham- Internal Shape


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The new relation between Art and Science has to be

understood in terms of dynamical complexity: well

established although unpredictable. One looks for a way to

escape cultural addiction, rigid and pre-established models;Fractal Geometry is at the same time an instrument and a result

for this. Art Making can be therefore realized within a pensive

and careful feeling of contemporaneity, animated by a cognitive

pull made actual by appropriate modes and means. (Giudi

Scotto Rosati - 2006).

Geometry & Art - A Parallel Development II

The Common Language of Art & Science

Haebel- Ripetitivit Incostante

A. Einstein (XX Century) Where the world ceases to be the

scene of our personal hopes and wishes, where we face it asfree beings, admiring, asking and observing, there we enter

the realm of Art and Science.

Galileo Galilei(Il Saggiatore, XVI Century)One cannot understandthe Universe if one does not learn before to know the characters by which

it is written. The Universe is written in mathematical language and its

characters are triangles, circles and other geometrical figures.


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Geometry & Art - A Parallel Development III

The Four Eras in the Geometry of Art & Space

All Geometries - the Euclidean one, the Geometry of Perspective, non-EuclideanGeometries, as well as more modern and mathematically formal ones, like Topology,

Riemannian Geometry or Fractal Geometry - share striking relations with Art:

Euclidean Geometry Interprets and describes ordinary space

for all Classical forms of Art.

Projective GeometryPainters use it as Perspective, as they want

to reproduce exactly what the eye sees.

Riemannian GeometryHas to do with XIX Century Art (Cubism,

Impressionsim) in coincidence and in full

consonance with the historical, perceptive andinterpretative developments of observed reality.

Fractal Geometry,

T o p o l o g y ,

Dynamism

Have to do with Modern and Contemporary Art,

where new research is performed on Form, Shape

and Deconstruction (Cubism, Futurism).


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Mathematics & Art in the Antiquity I

Pythagorean Arithmetic and the Beauty of Cosmos. 1

The Mathematics of Antiquity is conventionally divided into Arithmetic and Geometry.Arithmetic is the Science of Numbers it has to do with the human need of

counting. Geometry is the Science of Shapes it has to do with the human need of

measuring. The two needs are of course deeply intertwined; they are developed

together with the need of understanding and representing Space.

The Pythagorean School developsNumber Theory and

attributes to them the task ofregulating beauty.

Pythagoras of Samo


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Mathematics & Art in the Antiquity II

Pythagorean Arithmetic and the Beauty of Cosmos. 2

Astronomical observations of the three major naturalcycles(Earth, Moon, Sun) lead mankind to develop

the idea of Day, Moon Period and Year the ratio

between the approximate durations of these cycles

lead to attribute special meanings to a few Numbers,

considered as magic because of their relations with

the periodicity of Nature: twelvelunar cycles (in oneYear) and twenty-eight Days in each Lunar Cycle

(Lunar Calendars). Sinceperiodicity imposes four

periods also the Numbers three and sevenacquire

thence a special role (the Months in each Season and

the Week). The twelve Months in a Year lead

eventually to subdivide theCelestial Sphereinto

twelve parts and introduce what we call the Zodiac.

12


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Mathematics & Art in the Antiquity III

Space and Euclidean Geometry

The knowledge about the properties of Space and itsmeasures are eventually encoded into the famous

treatise The Elements (of Geometry) ofEuclid.

Euclidean Geometry describes a Space formed by

points, lines, circles and all geometrical shapes

obtained by using a ruler and a compass.

A central role is attributed to the existence ofparallel lines. They are postulated to exist, even if

vision tells us thatparallel lines are seen to converge

somewhere

Euclid of Gela


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Mathematics & Art in the Antiquity IV

Euclidean Geometry and the Dichotomy: Continuum or Discrete?

While for Euclid the point is immaterial, thePythagorean school attributes instead materiality to

the point: Pythagoreans points have a dimension.

The Greek thinkers speculate aroundInfinity andInfinitesimals; their speculations will eventually form

the basis of modern Science and still form a

challenging table of debate. Even if post-EuclideanScience will favor the idea that Geometrical Space

is Continuous the dichotomy between continuity and

discreteness will continue forever.

Democrituswill introduce theidea ofAtomism.

The Physics of XX Century will eventually revitalize the dichotomy

because of the advent of

Quantum Mechanics.

Democritus of AbderaEuclid of Gela

Pythagoras of Samo


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Mathematics & Art in the Antiquity V

Pythagoras Theorem


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Mathematics & Art in the Antiquity VI

Platonic Solids

In Ancient Greece The Theory of Proportionswas a canon for Science and Art; Mathematical Objects such as Platonic Solids were symbols of

classical beauty and harmony.

The most beautiful objects in

the Universe

(Plato)

Exahedron

Tetrahedron

Octahedron

IcosahedronDodecahedron


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The Persistence of Forms in

Geometry & Architecture II

Fractalized structures in Roman time

Bridges and in modern constructions. 1)

The Marine County Civic Center; 2)

Aqueduct on Gard River (France)

F.L. Wright- Marine County Civic Center

Aqueduct of Gard


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Even if the correct equation of catenaries was in fact derived in 1691 by Leibniz,Christiaan Huygensand Johann Bernoulli, catenaries begun early to be used in theconstruction of Mesopotamic arches (already at the time of pre-Greek and pre-Roman

Architecture). In antiquity the curvature of the (inverted) catenary was in fact

intuitively discovered and understood to be useful in the construction of stable arches

and vaults. Greeks and Romans preferred instead to use the much less efficient

curvature of the circle, both in circular arches and semi-spherical vaults. Catenaries in Taq-i Kisra in

Ctesiphon (Mesopotamia)

Parabolas and catenaries in Gaud

Architecture (XX Century)

The Persistence of Forms in Geometry & Architecture III


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Castel del Monte and its Geometry

The Persistence of Forms in Geometry & Architecture IV

Geometrical Shapes in Romanesque Architecture. 1

Photo by M.G. Lorenzi


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Castel del Monte and its Geometry

The Persistence of Forms in Geometry & Architecture V

Geometrical Shapes in Romanesque Architecture. 2

Photo by M.G. Lorenzi


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The Persistence of Forms in

Geometry & Architecture VILisbon

from Mosteiro dos Jeronimos

to Calatravas Station of Oriente

Photos by M.G. Lorenzi


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Throughout his life Gaud studied natural angles andcurves and tried to incorporated them into his

sculptured particulars, into mosaics and into the

designs of his Architecture. Hyperbolic Paraboloids from

the Exterior of the Vaults of

La Sagrada Famila

Instead of relying directly on thesimple shapes of Geometry herather mimicked Nature by subtly

combining them; rotational surfaces

having a peculiar role, such as

hyperboloids and paraboloids, were in

fact borrowed from Nature, so to

allow his work to resemble

environmental elements. He saidonce: Those who look for the laws of

Nature as a support for their new

works collaborate with God.

The Persistence of Forms in Geometry & Architecture VII

Revolution Surfaces in ModernArchitecture (Gaud)


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The Harmonic RatioTake a segment AB and divide it into two parts AC and CB, so that the whole AB

contains the larger part AC as many times as the part AC contains the smaller part CB

AB : AC = AC : CB

AB = AC + CB

implies

(AC + CB) : AC = AC : CB

(AC + CB)/AC = AC/CB

setting AC/CB =

1 + 1/ = ===> = (1 + 5)/2The Golden Mean is an Irrational Number

The Canon of Beauty: The Golden Mean IWhat is the Golden Mean?

A________________C_________B


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The Harmonic Ratio was called Golden Mean by Luca PacioliTake a segment AB and divide it into two parts AC and CB, so that the whole AB

contains the larger part AC as many times as the part AC contains the smaller part CB

AB : AC = AC : CB

AB = AC + CB

implies



setting AC/CB =

1 + 1/ = ===> = (1 + 5)/2Luca Paciolisaid: the Golden Mean is Divina Proportione

The Canon of Beauty: The Golden Mean IIThe Golden Mean


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The Golden Mean pervades Art since the Antiquity it is supposed to exist in the proportions of many Egyptian monuments.

Gizah (Egypt) - The Pyramids of Saqqara

The Canon of Beauty: The Golden Mean IIIThe Golden Mean in Egyptian Pyramids


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The Golden Mean pervades Art since the Antiquity as a canon of beauty it is easy to recognize it in many pieces of Art.

Athinai (Greece) - The Parthenon

The Canon of Beauty: The Golden Mean IVThe Golden Mean in Greek Architecture


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Johannes Kepler

Harmonices Mundi

Johannes Keplersaid: the Golden Mean is one among two jewels of Euclidean Geometry

The Canon of Beauty: The Golden Mean VJohannes Kepler and The Golden Mean in Cosmology


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The Golden Mean in NatureThe Golden Mean is in fact intimately related with several growth phenomena

as it was recognized byLeonardo Fibonacci. It belongs to Nature..

AB : AC = AC : CB

AB = AC + CB

implies



setting AC/CB =

1 + 1/ = ===> = (1 + 5)/2

The Canon of Beauty: The Golden Mean VIThe Golden Mean in Nature


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Amazingques0on:WasLowerPaleolithicMan

masteringGeometry?

Hand axes from Lower Paleolithic

Reproduced after in the Acheullian-John Feliks-University of Michigan Courtesy ofGheorghe Samoila

The Canon of Beauty: The Golden Mean VIIThe Golden Mean in Proto-Geometry. 1

Acheullian-Kilombe

hand-axes seem to

respect the proportions of

the Golden Mean


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The Canon of Beauty: The Golden Mean XThe Golden Mean in Human Proportions. 2

CourtesyofGheorgheSamoila


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Leonardo da

Vinci

Homus

Vitruvianus

The Canon of Beauty: The Golden Mean XIThe Golden Mean in Human Proportions. 3


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Symmetry, after Vitruvius, resides inthe correlation by measurement between

various elements of the plan, and between

each of these elements and the whole As

in the human body

The Canon of Beauty: The Golden Mean XIIfrom the Golden Mean to Symmetry

Itproceeds from proportion and it achieves

consonance between every part and the whole

This symmetry is regulated by the modulus, the

standard of common measure which the Greeks

called the number


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Since the antiquities, Symmetry hasplayed a fundamental role in the

construction of classical beauty - in

good Sculpture, in good Painting

and especially in good Architecture.

The old meaning of symmetry

had to do mainly with the staticnotion of proportion; the present

notion of symmetry, on the

contrary, has a dynamical

interpretation, having to do with

motions or dilatations that

allow parts to superimpose exactly

to other parts of a given object.Exactly in this sense symmetry

enters the new and more dynamical

forms of Art.

Symmetry as a Mirror Game.

This mathematical exhibit is due to the

Department of Mathematics of the

University of Milano.

Art & Symmetry - IThe Paradigm of Symmetry


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Narciso (1579-1599) by Caravaggio (1573 1610)

In this famous painting one can immediately

recognize the existence of an axial symmetry.

Art & Symmetry - IIThe Paradigm of Symmetry in Painting


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Symmetry is present in an essential and practically universal way in Architecture, starting fromMesopotamic, Egyptian, Greek and Roman buildings; it helped in structural calculations and

contributed to construct elegantly and in a structurally stable way.

Town of Palmanova

Unusual 9-folded symmetry

Art & Symmetry - IIIThe Paradigm of Symmetry in Architecture

Bramante - San Pietro Planimetry

Example of 4-folded or 8-folded symmetry


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Ahlambra in Granada

Art & Symmetry IVSymmetry in Arab Architecture. 1


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Art & Symmetry VSymmetry in Arab Architecture. 2

Ahlambra in Granada


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Rosettes represent invariance groups of geometrical figures, that do not contain: i)

rotations of exact sub-multiples of 2 (cyclic groups); ii) rotations and reflections(diedral groups).

Rosettes existing in some Medieval Churches

Art & Symmetry VIRosettes


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Friezes are infinite sets of forms that repeat and show invariance with respect to

translations in a single direction.

The 7 types of Friezes

Art & Symmetry VIIFriezes


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Mosaics i.e. crystallographic groups of

the plane (forming a family of exactly 17

elements) - are infinite groups of forms that

repeat by showing a translational invariance

in two or more independent directions.

Mosaic in Via Tiberti (Cesena)

Art & Symmetry VIIIMosaics


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Knots in Mathematics: they are

images of a circle, i.e. closed

curves embedded into 3-

dimensional Euclidean Space.

Art & Symmetry IXKnots. 1


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Examples ofKnots in Art1) Isis Knot;

2) Savoia Knot;

3) Longobard Knot;

4) An example ofQuipu.1

2

3

4

Art & Symmetry XKnots. 2


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Other Tessellations of the Euclidean Plane.

Art & Symmetry XIITessellations. 2


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Examples ofColored Tessellations in the Plane.

Art & Symmetry XIIITessellations. 3


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Examples ofTessellations

of curved compact and

non-compact surfaces.

Art & Symmetry XIVTessellations. 4


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Symmetries in the Euclidean Plane

Symmetry in Advertising?Andy Warhol

Art & Symmetry XVSymmetry in Modern Art


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Aprimordial form of Perspective

already existed in Greek and Roman

Painting, as well as in Gothic:

inMasaccio (1401-1428) (frescoes ofCappella del Carmine, in

Firenze);

in Giotto (frescoes ofCappella degliScrovegni, in Padova, as well as in

Madonne in Trono).

In a sense an elementary perspective

was adopted, with more than one point

at infinity; it is like having more than

one point of observation in the painting,

each one corresponding to a different

viewpoint

From the V Postulate of Euclid

to Perspective and Projective Geometry - I

Several convergence lines,

more than one point at infinity

Perspective in Giotto is not exactly

codified and there is no single point of

convergence


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A primordial form of Perspective

already existed in Greek and Roman

Painting, as well as in Gothic:

inMasaccio (1401-1428) (frescoes ofCappella del Carmine, in

Firenze);

in Giotto (frescoes ofCappella degliScrovegni, in Padova, as well as in

Madonne in Trono).

In a sense an elementary perspective

was adopted, with more than one point

atinfinity; it is like having more than

one point of observation in the painting,

each one corresponding to a different

viewpoint


to Perspective and Projective Geometry - II

While in Renaissance, Painters and

Mathematicians begin to abandon empirism

and look for general methods, that later one

will be codified in systematical treatises on

Perspective (rules for Buona Pittura).

Thanks to Filippo Brunelleschi, LeonBattista Alberti (in Architecture) and Piero

della Francesca (in Painting), the Theory of

Good Proportions becomes a general

instrument, giving to painters rules that he

has to follow to obtain that what is seen is also

painted.


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to Perspective and Projective Geometry - III

While in Renaissance, Painters and

Mathematicians begin to abandon empirism

and look for general methods, that later one

will be codified in systematical treatises on

Perspective (rules for Buona Pittura).

Thanks to Filippo Brunelleschi, LeonBattista Alberti (in Architecture) and Piero

della Francesca (in Painting), the Theory of

Good Proportions becomes a general

instrument, giving to painters rules that he

has to follow to obtain that what is seen is also

painted.

Perspective in Piero della

Francesca: one point at infinity.


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Geometry is recognized as an essential tool for Artists.

() E sappi che cosa niuna dipinta mai parr pari alla vera, dove non sia certadistanza a vederle. ()

Pertanto affermo sia necessario al pittore imprendere geometria. ()

Perch ho dicto dato lochi, se intende essere posto al vedere in quelloluogho, dove tu vai stare a vedere il piano assegnato

Leon Battista Alberti

Piero Della Francesca

De prospectiva pingendi

(a mathematical treatise with

good rules for painting).


to Perspective and Projective Geometry - IV


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??.







convergence




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??.







convergence




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Botticelli Piero dellaFrancesca Leonardo da Vinci

Painting in Renaissance

Symmetry, Perspective and Proportions - I


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This famous masterpiece by Raffello

Sanzio is essentially based on a

clever use of both Perspective and

the Golden Mean.

The Painting occupies a Golden

Rectangle; in the square at the basis

of this rectangle there is a circle,

that inscribes two squares to form

an octagon. Prolonging its sides one

can determine the radius of two

further inscribed circles.

Raffaello Sanzio - La Trasfigurazione


Symmetry, Perspective and Proportions - II


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Leonardo da Vinci -Mona Lisa (Harmonic Analysis)


Symmetry, Perspective and Proportions - III



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Twohorizontalsuperposedgoldenrectangles.Goldensec0ondividestherectanglesatleFandright.

Parallelswithgoldenrectanglesdiagonalsaredrawn.

Athalfthebase,ameridianmarksLedasbodywhichisperfectlyframedbytheGoldensec0onmarkers.

Goldensec0onrectanglediagonalsareDrawn.Theircrossingmarksimportantfocal

pointsofthecomposi0on.

Ahorizontallinemarkstheheadschin.Theeyelineismarkedbythesideofthetrianglebuilthavingthepreviouslineasheightand

Mediator.

*ReproducedaFerMa0laGhykaTheGeometryofArtandLife(1)


Symmetry, Perspective and Proportions - III



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K. F. Gauss

(1827, Disquisitiones

supra Superficies Curvas)

B. Riemann (1854, On the Hypotheses that lie at the bases of

Geometry, published postmously in 1866)

If the independence of bodies from position does not exist, we

cannot draw conclusions from metric relations of the great to

those of the infinitely small; in that case the curvature at each

point may have an arbitrary value in three directions, provided

that the total curvature of every measurable portion of space does

not differ sensibly from zero [] The question of the validity of

the hypotheses of geometry in the infinitely small is bound up with

the question of the ground of the metric relations of space.

From Linearity to Curvature in XIX Century

Non-Euclidean Geometry


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The discussion about the validity of Euclids fifth postulate lead, however, also to

Hyperbolic Geometry and, in the artistic field, that paved the way for the development

of Impressionism. It turns out, in fact, that our visual space is hyperbolic rather thanEuclidean (Lunemberg, 1947), so that it is not surprising that artists in the XIX Century

begun to represent what the eye actually sees rather than what the eye ispretendedto see

in a fully Euclidean world. The XIX and XX Century will finally see the introduction of

time as a fourth sensibile dimension alongside and on equal footing with height,

lenght and depth (Einsteins Theory of General Relativity is at the top of this line of

thought); motion and curvature become part of the World and not something which is

embedded into the World. This will form the subject of the second half of this Lecture.

From Linearity to Curvature in XIX Century

Non-Euclidean Geometry, Impressionism and Fourth Dimension

MATHEMATICS ART


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MATHEMATICS, ART

AND PHOTOGRAPHY

(from Antiquity to Futurism)Talk at the Final Event: SCIENAR ProjectThe Dutch Academy of Science

Amsterdam, 15 October 2010

Marcella Giulia LORENZI

(Universit della Calabria)(in collaboration with Mauro FRANCAVIGLIA)

Part II: Space, Time and Motion in XX Century Art

A & S i i XX C R d I i 1


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At the end of XIX Century and during the XX Century Science becomes in a number of

occasions a source of inspiration for Art.

Examples begin with the Impressionists desire to decompose colors according to their light

decomposition in fundamental components (Divisionism, Pointillism)

Later on Picasso and Braque with Cubism will introduce different viewpoints into painting,

destroying classical perspective and letting extra dimensions enter the scene of Art

The need of representing dynamism and motion will lead Futurists to their ideasEventually motion will be captured by Cinema and later on by Computer Graphics

Art Nouveau will introduce new geometric shapes in all forms of Art, from Painting to

Architecture to all Arts Decoratifs

Geometric Forms and also Fractals will be used as subjects for Art

Photography and Digital Photography will produce new forms of Art

in which Science can play a predominant role

Art & Science in XX Century a Renovated Interaction. 1

A & S i i XX C R d I i 2


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Dynamics is impossibile to represent in just two

dimensions, yet artists (e.g., Balla, Boccioni,

Duchamp) tried to provide pictorialrepresentations of movement. Dynamism in

Modern Art is achieved through the moving

camera and Cinema, but the idea of time-flowing

sections may be also used in figurative arts (think

of Picasso or Dal, a 4-dimensional hypercube

that opens up in 3-dimensional space). The new

Mathematics of the XX Century is also the

Mathematics of Manifoldness, Curvature,

Discreteness, Fractals and Chaos, and some

modern ways of painting reflect these new ideas

(Picasso, Pollock). Also the famous artist Escher,

the unchallenged inventor of impossibile objects

and imaginary world, was influenced by theMathematics of Poincar and Penrose in creating

his striking pieces of Art. The connection existing

between Art and Mathematics is universally

recognized and needs no more examples.

Picasso - Still LifeThe substantial monochromy of theensemble, the unususally elliptic shape

(shared by other works of the Cubist

period) and the simple rope used as aframe represent an intentional recalling

of shape that transfigures in matter,

matter thakes new shapes, which do

n o t b e l o n g t o i t s o w n

nature(Itinerario nell'arte. vol. III).

Art & Science in XX Century a Renovated Interaction. 2

A t & S i i XX C t I i i d P i tilli


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Seurat

Le pont de

Courbevoie

Notice thepersistence

of the

Golden

Mean

Impressionism generated in XIX Century a great innovation in composition

in particular, studies on the decomposition of colors lead to Pointillism

Art & Science in XX Century - Impressionism and Pointillism

Th D t ti f P ti I


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Perspective - so long looked for inRenaissance, in order to reproduce exactly

reality, by respecting the mechanisms of

vision - was rejected in the XIX Century Art:

With Post-Impressionism Artists create paintingsin which different parts respect a rigorous

perspective, although several different viewing angles

are mixed-up together;

In a sense, Perspective gets deformed but notcompletely abandoned by Artists.

With Impressionism Artists begun to renounce theolder rules of good painting, by first deforming and

later cancelling totally the fundamental principles of

Renaissance perspective canons;

House of the Hung Man Paul Czanne

The Big Bathers Paul

Czanne

The Deconstruction of Perspective - I

Paul Czanne

Th D t ti f P ti II


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Pablo Picasso - demolishes Perspective and paintsfragments of reality as seen from different angles or in

different planes, later mixed-up

Many elements of Euclidean Geometry and Optics arerejected;

Euclidean Distance looses its meaning with stronginfluence on the final result: particulars that are far from

each other are seen to be near, whilst particulars that are

not near appear to be contiguous in the painting.

The represented reality cannot be always detected directly; one has to recognize different

pieces, glueing them together in its own perception - as in a Manifold or in a Kaleidoscope.

The artistic message is thus reconstructed in an independent way by each onlooker. The

Artistic piece becomes thus a Topological Manifold, that can be reconstructed only by

chart glueing.

Ambroise Voilard - Picasso

The Deconstruction of Perspective - II

Picasso and Cubism Manifolds and Fourth Dimension. 1

Th D t ti f P ti III


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Guernica -PicassoThe famous painting Guernica is formed by several independent portions, that intersectand overlap without an apparently clear and immediate logical order. Looking carefully at

this painting, however, one can realize that the painting is in fact formed by an Atlas,

Is it a Manifold or a Piece of Art..?

The Deconstruction of Perspective - III

Picasso and Cubism Manifolds and Fourth Dimension. 2

Breaking the Euclidean Symmetries

G t I i ti f M d A t I


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Golden Mean and Brancusis works

25 ofBrncuis ovoid works respect the rule of the Golden Mean. Out of

these, 5 display the ratios of 0,617 and 0,620, which compare with theideal ratio of 0,618. Other 13 display this ratio with a slight variation of

1%, and in only 4 there is a deviation of 2-3% from the nominal

value. (taken from a note oftefan Georgescu Gorjan)

Geometry as an Inspiration for Modern Art - I

The Golden Mean in Brncui



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Salvador Dali - Leda

Persistence of the Golden

Mean

Internetimage

Geometry as an

Inspiration for Modern

Art - II

Salvador Dali. 1



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An hypercube in four dimensions can be constructed out of 8 ordinary 3-dimensionl cubes in ordinary Euclidean Space, in much the same way in which anordinary cube in three dimensions can be constructed out of 6 ordinary squares (thatare ordinary cubes in 2-dimensional Euclidean Space) and an ordinary square can beconstructed out of 4 ordinary cubes in 1-dimensional Space, i.e. four segments.

This construction was adopted bySalvador Dal to represent a skillful

Crucifixion, in which the Holy Cross is infact the 3-dimensional representation of anhypercube. This introduces de facto anextra dimension in the painting, which -in a sense - acquires also an extension intime, to represent the flow of events thathas accompanied the Passion of Christ.

Geometry as an Inspiration for Modern Art - III

Salvador Dali. 2

Geometry as an Inspiration for Modern Art IV


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The beginning of XX Centurymarks the eve of new forms of

Art. Art Nouveau is one of the

most important ones (also called

LibertyorJugendstil)

One of the most importantcharacteristics of thisNew Style

resides in its continuous

reference to Nature, from which

the structural elements are

derived and given dynamical

undulate contours.

The clever use of geometrical structures inspired by such curves would produce

shapes in the form of trees and flowers, a reason for which the Style was also

called Floral. Some of the curved lines used became clichs, to be adopted by

artists worldwide.

G. Klimt The Tree of Life

Geometry as an Inspiration for Modern Art - IV

Art Nouveau and Geometrical Forms

Geometry as an Inspiration for Modern Art V


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Fig. 2 La Sagrada Famila, photo byMarcella Giulia Lorenzi

Geometry as an Inspiration for Modern Art - V

Art & Mathematics in Antoni Gauds Architecture. 1

La Sagrada Famlia

Geometry as an Inspiration for Modern Art VI


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Cubic Elements for the Pinnacles of LaSagrada Famila, in the Museum of the

Church,

photo by Marcella Giulia Lorenzi

Geometry as an Inspiration for Modern Art - VI


Geometrical Forms for La Sagrada Famlia (1)

Geometry as an Inspiration for Modern Art - VII


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Platonic Solids and otherGeometrical Shapes for the

Pinnacles of La Sagrada

Famila, in the Museum of

the Church,photo by Marcella

Giulia Lorenzi

Geometry as an Inspiration for Modern Art - VII



Geometry as an Inspiration for Modern Art - VIII


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Other Elements for the Pinnacles of La Sagrada Famila,

in the Museum of the Church,

photo by Marcella Giulia Lorenzi

Geometry as an Inspiration for Modern Art - VIII



Geometry as an Inspiration for Modern Art - IX


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photo by Marcella

Giulia Lorenzi

Geometry as an Inspiration for Modern Art - IX


The Geometrical Columns of La Sagrada Famlia. (4)

Geometry as an Inspiration for Modern Art - X


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Columns resemble Trees,from the Museum of the

Church,photo by Marcella

Giulia Lorenzi

Geometry as an Inspiration for Modern Art - X


The Geometrical Columns of La Sagrada Famlia (5)

Geometry as an Inspiration for Modern Art - XI


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F.A. Alsbachs Disced field at down - Harmonic Analysis

Reproducedwithar0stspermission

Geometry as an Inspiration for Modern Art XI

The Golden Mean in Modern Painting. 1


Geometry as an Inspiration for Modern Art - XII


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IdousetheGoldenMeanasastar1ngpointandtoarriveatbasicdimensions.The

designbeginswiththeGoldenMean,butIalsoplayedwiththeGeometrysomehow

asIpainted.FloydAlsbachEvening Canticle

Reproducedwithar0stspermission

Geometry as an Inspiration for Modern Art XII

The Golden Mean in Modern Painting. 2


Geometry as an Inspiration for Modern Art - XIII


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Vassily Kandinski- Black Square (1923)

In the Peinture ofXX Century some artistic movements begun to use Geometryas a mean to construct pieces of Art. Among the great names of these currents

we may just quote Vassili Kandinsky or Max Bill, but many others could be

recalled. In their paintings and sculptures the presence ofGeometry (Euclidean

or not) is not accidental but represents in fact an artistic theme.

Max Bill- Endless Ribbon

Geometry as an Inspiration for Modern Art XIII

Patterns and Topological Forms. 1

Geometry as an Inspiration for Modern Art - XIV


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Numeri innamoratiG. Balla

Disco Simultaneo -Robert Delaunay

Geometry as an Inspiration for Modern Art XIV


Geometry as an Inspiration for Modern Art - XV


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Emblematic examples of broken symmetry

may be found in the sculptures ofArnaldo

Pomodoro, that are placed along the streets of

the Swiss town ofLugano: they represent objects

that are symmetric only if seen from suitable

angles or positions.

Geometry as an Inspiration for Modern Art XV


From Order to Disorder - I


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One of the most fascinating conquers of XX Century is the notion ofFractal. Introduced at the end of XIX Century as curves that fill-upportion of the plane and have an intermediate dimension larger

than one but smaller than two (e.g., the Peano Curve or the Koch

Curve), they represent stimulating objects related with iteration and

self-similarity at scales smaller and smaller (coasts, mountains,

vegetation, nervous system are good examples).

Haebel- Isole

From Order to Disorder I

Fractals in Mathematics. 1

From Order to Disorder - II


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Worthwhile to mention is the revolutionary idea that continuity anddimension may be different from the notions that resisted since the time of

Greeks, through their advent.

B. Mandelbrot

Fractals

Peano Curve


Koch Curves

From Order to Disorder - III


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Mandelbrot Sets

The Mandelbrot Sets aredefined starting

from a Julia set, by the recursive formula

(x,y) (x 2 - y2+ x, 2xy + y)


From Order to Disorder - IV


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The advent of computer has certainlygiven a great impetus to Fractal

Painting - also called Fractalism - but

also the literary tradition of XX Century

had a propositive role. Mandelbrot

speaks of representations of infinitythrough finite and of non integer

dimensions. In parallel, several artists

of XX Century had liberated themselves

from predetermined models and have

focused their activity on the search of

unpractioned and unexperienceddimensions, so determining new

viewpoints and new models through

which one can experiment reality.

Paolo Barlusconi-Ineffabili Stati dellEssere

Fractalism in Art. 1

From Order to Disorder - V


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Haebel states moreover about Fractal Art:The scientific component is present only at

the level of study [] the creative act is by

no means mortified [] and it remains the

undiscussed protagonist [] it would be

impossible to repeat a painting, while

repetitivity is a founding value of scientific

work.

Haebel-Attrattore in Espansione

Again with Haebel we can say that: Fractal Geometry

configures a new theoretical and methaphorical field. The

rigid boundaries that have traditionally separated Art from Science are falling down one by one.

Interdisciplinarity is extremely open minded: it has no

limits, no boundaries, it is infinite as Fractal Geometry is.

Gianni DAnna-Fleur Fractal

Fractalism in Art. 2

F.L. Wright - From Order to Disorder - VI


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Fractalized structures in Roman time

Bridges and in modern constructions. 1)

The Marine County Civic Center; 2)

Aqueduct on Gard River (France)

F.L. Wright Marine County Civic Center

Acqueduct of Gard

Fractals in Architecture. 1

D. Libeskind- V & A Museum, London From Order to Disorder - VII


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Fractalized structures are often encountered inModern Architecture, as these examples show(below, a view of Guggenheim Museum at

Bilbao; on the left, the Victoria & Albert

Museum; below, a modern structure at an

exhibition centre in Denver. But also in

relatively older buildings or even in Roman

constructions, as in the next slide.

Daniel Libeskind-Art Museum, Denver F.O. Gehry - Guggenheim Museum, Bilbao

Fractals in Architecture. 2

From Order to Disorder - VIII


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Escher found inspiration in Alhambras Mosaics and realized tessellations of theplane through symmetrical schemes taken from Geometry (later replaced by Angels,

Demons, Ghosts, Reptiles, Fishes and Imaginary Beasts so creating unique artworks

based on Non-Euclidean Geometry or even Impossibile Geometries. The following

images live in the so-called Poincare Disk.

The dimension of reptiles decrease movingtowards the centre

The dimension of reptiles increase movingtowards the centre

Fractals and Hyperbolic Geometry in Eschers Art. 1

From Order to Disorder - IX


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Escher and Poincar Disk

From Order to Disorder IX

Fractals and Hyperbolic Geometry in Eschers Art. 2

Time and Motion in Modern Art - I


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Dynamics of a dog on a landGiacomo Balla

Nude descending

a staircase Marcel Duchamp

As we said, across XIX and XX Century: The idea ofSpace changes, together with its description; Non-Euclidean Geometries are introduced; Time becomes a fourth

sensible dimension; Technology grows rapidly; Velocity becomesessential

Artists of XX Century begin trying to represent motion pictorially

The Futurism. 1

Time and Motion in Modern Art - II


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The Futurism is an Art movement that started in Milano, 100 years ago(in 2009)

Futurists concept of movementand their attempts to capture a sense ofmovement in painting, sculpture and photography.

The Futurism. 2

Time and Motion in Modern Art - III


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The Aesthetic of Velocity generates a prevalence of truly dynamical

elements; motion involves both the object depicted and the space in

which motion takes place.

Trains, Cars, Airplanes are peculiar subjects, but also human figures

(dogs, dancers, children) animated by multicolored and polyphonic

brush touches, aimed at putting into evidence the propulsive push of

moving forms. The difference of velocity (higher or lower) is usually

represented by using either broken and rough-edged lines or more

harmonious and fluid linear brushstrokes.

Futurism is well described by the own words ofBoccioni(1913):

I want to render the fusion of a head with its environment.I want to render the prolongation of objects in space.I want to model light and the atmosphere.I want to transfix the human form in movement.I want to synthesize the unique forms of continuity in space.

The Futurism and the Aesthetic of Velocity

Time and Motion in Modern Art IV


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At the same time in which Marinetti, Tato, Boccioni, Balla,

Bragaglia, etc. where trying to insert movement in their

art, including photography

Muybridge used photography as ascientific instrumentto

study animal and human movement imperceptible to the

human eye

Futurism and Photography

Time and Motion in Modern Art V


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The Fotodinamismo of Futurists. 1

Time and Motion in Modern Art VI


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Manifesto della Fotografia Futurista

The Fotodinamismo of Futurists. 2

Time and Motion in


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Modern Art VII

The Fotodinamismo of

Futurists. 3

Time and Motion in Modern Art VIII


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Time and Motion in Modern Art - IX


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Generative Artis a new method of making Art. The term refers to how the Artis made, and does not take into account why it was made or what the content of theartwork is: Artworks, in Generative Art, can be identified in the creative processes

and not only in the results. Painting with Light: it is a form of Generative Art that through DigitalPhotography allows one the setting in motion of the fourth dimension

Generative Art Painting with Light

photo by Marcella

Giulia Lorenzi


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Future of Futurism Project


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On the 100th anniversary of the Futurist art movement whichstarted in Milano, we want to examine the aims and ideasof Futurist imagery, then proceed to a discussion of howtoday's digital photography can achieve many of the goalsof the Futurist artists.

Digital photography can record pictures that are very similarto the Futurists' vision of depicting a world that is alwaysin motion.

An Installation/Exhibition was held in Milano in 2009 and inBratislava in 2010; it will be expanded and presented in

other cities (Buenos Aires, Bucharest, etc.)

Book catalogue and scientific and historical hints

Future of Futurism Project

Doble, Francaviglia and Lorenzi


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Youll find more in Rick Dobles new book

EXPERIMENTAL DIGITAL ART PHOTOGRAPHY

Lark Books, April 2010

The Art Work of Rick Doble10 Years On The Internet

The Culmination Of Almost 40 Years Of A Life In Art

---- 2000+ Images ----

Including 1500+ Digital Photos

Plus Experiments, Art/Science Images, Essays, Lifestory,More

F i


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Futurism


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Optics and photography


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p p g p y

Camera Oscura is an optical instrument theprinciples of which are at the base of Photography andCinematography. Light is a physical phenomenon of electromagneticnature; emitted by some sources (for example, the Sun,fire, lamps, etc.) it propagates into transparent media

and is reflected, diffused or absorbed by matter.

Starting from the laws of Visual Perception, from theanathomy and physiology of the eye to the cerebral

cortex and the elementary laws of Optics, several

devices using these properties have been built.

photo by M.G. Lorenzi

The spectrum of irradiating energy is usually divided into families that overlap at the


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The spectrum of irradiating energy is usually divided into families, that overlap at the

extremes, called microwaves, ultraviolet, infrared, etc.; Human retina is sensibile to

electromagnetic waves in a short interval: the term light is usually referred to

electromagnetic wave-frequencies within it. Thus visibile light is a very small region of the

electromagnetic spectrum.

A light wave, irradiating into space at the constant velocity of about 300.000 kilometers per

second, can therefore energetically interact with a detector, that could be a film, a retina

or a photoelectric device.

Photography: its all about light


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g p y g

Many people do not realize and even

experienced photographers can forget, that

Photography is all about light.

The word Photography comes from two

Greek words, Photos (light) and Graphos

(writing, painting), so drawing with the light.

Photography is not about objects or people orscenery, rather it is about how the light reveals

those things. The action of light on a light-

sensitive material (film or electronic devices)

creates the image. An object can be lighted so

that it almost disappears or so that it is virtually

three -dimensional.

I often think the night is

more alive and more

richly coloured than the

day."

Vincent van Gogh, 1888


Color photography is a relatively recent invention,

so it is not only the intensity of the light but also

Digital photography could

be a major art form in the


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so it is not only the intensity of the light, but also

the color of the light that creates the image. This

complicates things: for example, a scene may

include different light sources which have their

own particular color (color temperatures and wavelengths) and subtleties of color that may be seen

differently by the camera than by the human eye.

1950s: single lens reflex (SLR) camera

For the first time the photographer could see

exactly what the lens saw.

1990s: Digital Photography

artist see what the camera is seeing in "real time"

on a LCD screen (approximation).

be a major art form in the

next century. It may be the

culmination of the

development of

photography. Digitalcameras may give us the

power to set photography

loose.

Rick Doble


The photograph can be used to record a

passage though time, a movement

"I'm an eye. A mechanical eye. I, the

machine, show you a world the way only I

can see it I free myself for today and


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passage though time, a movement

through a space, even an emotion

characterized by the movement of the

camera.

The elusive change of light, the fleeting

expression, the ephemeral form all

could be captured by the camera, an

instrument for working both in time and

space as the English painter John Piperpointed out.

Indeed taking pictures needs some

devices and a particular process in Space

and Time.

can see it. I free myself for today and

forever from human immobility. I'm in

constant movement. I approach and pull

away from objectsThis is I, the machine,

manoeuvring in the chaotic movements,recording one movement after another in the

most complex combinations.

Freed from the boundaries of time and

space, I co-ordinate any and all points of

the universe, wherever I want them to be.

My way leads towards the creation of a

fresh perception of the world. Thus I explain

in a new way the world unkown to you."

Dziga Vertov, 1923

Today scientists no longer limit themselves

to the three dimensions of Euclid TheAccording to Albert Einstein the basic


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to the three dimensions of Euclid. The

painters have been led quite naturally, one

might say by intuition, to preoccupy

themselves with the new possibilities of

spatial measurement which, in the languageof modern studios, are designated by the

term: the fourth dimension.

Regarded from the plastic point of view, the

fourth dimension appears to spring from the

three known dimensions: it represents the

immensity of space eternalizing itself in all

directions at any given moment. It is space

itself, the dimension of the infinite.

Guillaume Apollinaire, 1913

structure of our world is SpaceTime and

things exist in a spacetime continuum, a

world of four dimensions: height, width,

depth and time.

A generative process is usually referred to

as setting in motion. Motion is the

essence of Life. To be alive is to move.

It seems that both Cubism and Futurism

were deeply affected by Einstein's Special

Theory of Relativity, which was published

in 1905. The goal of the Futurists was to

include motion (and therefore Time) in a

painted image, much like the cubist

wanted to include multidimensions in a

portrait.

Photography may be the visual Art

best suited to creating still images

f bj i i Thi i b

Photography, and Digital Photography in

particular is uniquely capable of


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of subjects in time. This is because a

photograph is made by recording an

object (via the lens) over time (by

opening the shutter for a specific

d u r a t i o n ) . T h e r e f o r e , a

photographic exposure is a

combination of space and time, a

recording of space and time Rick

Doble

particular, is uniquely capable of

recording a space/time image.

Yet to photograph a space/time image is

quite complex. For example, the correct

shutter speed to depict motion varies

considerably depending on the motion of

the subject and the artistic intentions of

the photographer. In addition there are

many other variables to movement. And

to record this kind of imagery

successfully, the photographer must have

a tool that allows instant display of the

imagery just taken so that adjustments

can be made based on that feedback -

which is the very powerful capability

provided by Digital Photography.


Digital photography and

Futurism


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The artist and the scientist

each substitute a self-created

world for the experiential one,

with the goal of transcendence

Albert Einstein

Futurism

Digital photographers who are interested in

the depiction of movement have tried to addto the Futurist idea of an algebra of

movement. The craft of Photography

defines two fundamental kinds of movement:

subject movement and camera

movement. It also adds a third kind of

movement which is the combination ofsubject and camera movement such as

panning a camera with a moving subject.

The Futurist's notion of absolute and relative

movement is part of subject movement and

works very nicely with other photographic

considerations when it comes to taking

pictures of a subject in motion.


The eyes within the camera

shift and capture each moment.Some digital photographers have added other


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shift and capture each moment.

This act of perception trasmutes

and trasfigures each object

perceived creating a montage of

unique and pivotal images.

Yulla

aspects of subject movement to their ideas.

For example, one of us (RD) defined these

different types of subject movement:

Regular movement: Some movement is

unchanging, like that of a train; it moves at a fairly

steady pace in a predetermined direction. A cars

movement is also regular but with some variables,

such as swerving a bit to the left or right and

slowing down or speeding up.

Predictable movement: Less precise than regular

movement is predictable movement. A car heading

down the road will continue to head in that

direction; a car with its right turn signal blinking

will turn right. A dancer doing a traditional dance

will repeat the same steps but not in exactly thesame spot.



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Left: Le mani del Violinista, by G. Balla

Right: Violinist, subject movement, photo byRick Doble

Irregular movement: Some movement repeats but in an irregular fashion, such

as a dancer who moves in a free-form manner. Nevertheless, this dancer will


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repeat many of the same motions and, after a while, a photographer might gain a

sense of how that particular dancer is likely to move.

Erratic movement: The movement of a singer on a stage or a child playingwith a dog can be hard to predict, however, scenes such as these can yield

exciting and unusual imagery.

Left/right: Photodynamic portraits of Dee Dee Bridgewater singing and dancing with a

green fan during a Jazz concert. Photos by M.G. Lorenzi

Can an art form, which has

been committed to creatingCamera Movement


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g

high resolution images of the

real world, find happiness as a

contemporary art form that

includes things thatphotography has been avoiding

up to now, such as blurriness,

overexposure, underexposure,

camera movement, subject

movement, graininess and long

exposures in which the

unexpected happens?

Rick Doble

Camera movement, by itself, depicts

motion from the photographers point-of-

view. This type of imagery has been

called Painting with Light and also

camera painting.

It is essentially a new Art form that has

only been made practical with the advent

of Digital Photography and leads to

dynamic abstract imagery much like theabstract work of the Futurists.

NOTE: while this imagery was

technically possible with film

photography, not much work was done in

this area due to the high cost and largeamount of effort that was required.


I was by myself


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In addition camera and subject

movement together add

powerful techniques in thedepiction of movement as well

as providing considerable

individual artistic control.

Combined camera and subjectmovement can record the most

dynamic imagery where the

world seems to be rushing to fill

the picture.

My Wife Driving, subject and camera movement combined, photo by Rick Doble

I was by myself

action painting

with a camera

Rick Doble

The vital instant is the

decisive moment of

The New Digital Photographic

Capabilities


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decisive moment of

snapping the shutter

Henri Cartier-Bresson

Capabilities

The capabilities of Digital Photography can make

use of all of these ideas about movement in afashion that the Futurists could only dream about.

LCD Monitor: To begin, digital photographers can

see a rapid thumbnail photograph immediately

after taking a photograph. The instant image on the

LCD monitor gives digital photographers theessential tool they need - since photographing

motion is so complex and requires a good deal of

trial and error. Yet the LCD monitor allows the

accomplished photographer to hone his or her

imagery in real time and to take pictures both in the

studio and in the streets to record the vital pulse ofLife.


Low Cost: prohibitive cost of film and

pr c ssing in th past Taking pict r s f


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Camera movement

combined with a slow

shutter speed can

create fluid images

full of energy.

processing in the past. Taking pictures of

movement, by necessity, requires a lot of

test shots and shots that are not the best.

Digital photographers can now shoothundreds of pictures without worrying

about the cost.

Stabilizing Control: The new stabilizing

feature on most digital cameras allows

photographers to handhold shots at verylow shutter speed such as 1/2 second with

no camera shake. This means that

photographs of subject movement by

itself, for example, can be accurately shot

without any camera shake or with minimal

camera shake. And this is a new featureonly now available with digital

technology.


EXIF Data: Also important is the EXIF

d t th t i d d b t

I suppose I am interested,

above all in investigating the


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exposures data that is recorded by most

digital cameras and embedded in the

photographic image. This invaluable new

digital tool lets a photographer go back andreview the settings, such as shutter speed, that

were used with different photographs and

then allows the photographer to learn from

and to build on that information.

Expressive Control: And while DigitalPhotography is a technical craft, it is also an

expressive medium. Different photographers

can make very different images that reflect

their personalities and their artistic visions.

As a result the world in motion can be both

accurately recorded and also depicted in anexpressive and individual manner.

above all, in investigating the

golden ability of the artist to

achieve a metamorphosis of

quite ordinary things into

something wonderful and

extraordinary

Eduardo Paolozzi, 1959

photo by Rick Doble

Selecting particular initial conditions,

adding a fourth dimension andThe artist and the scientist



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photographing motion by means of

randomised generative processes can

give rise to very expressionistic results,

in full agreement with Galantersdefinition of Generative Art.

Rick Doble proposed a new term for

this Photography, namely photo

expressionism. Generally speaking the

most interesting effects will occur atextremes, because it is at these "edges"

that the normal relationship between

light and film/CCD breaks down and

something unusual happens.



with the goal of transcendence

Albert Einstein


Real context Light painting


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photos by M.G. Lorenzi

Real context Light paintingphotos by M.G. Lorenzi


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utu e o utu s

Photo Lorenzi

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Photo Lorenzi

Thank you for your attention!


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Contacts:

[email protected]

[email protected]

[email protected]

The artist and the scientist




mathematics, arts and photography by dr marcella lorenzi, university of calabria

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