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LA MAR DE NOMBRES LES FORMES DE L´UNIVERS Javier Barrallo

La Mar de Numeros

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Page 1: La Mar de Numeros

LA MAR DE NOMBRESLES FORMES DE L´UNIVERS

Javier Barrallo

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Page 3: La Mar de Numeros

LA MAR DE NOMBRESLES FORMES DE L´UNIVERS

Javier Barrallo

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RECTOR DE LA UNIVERSITAT

POLITÈCNICA DE VALÈNCIA

Juan Juliá Igual

VICERECTOR DE CULTURA

Joan Bta. Peiró

CAP D’ÀREA

DE GESTIÓ CULTURAL

Carlos Plasencia

COMISSARI

Javier Barrallo

COORDINACIÓ

Lola Gil

Asunción García

Carlos Ayats

CATÀLEG

Textos.

Joan Bta. Peiró

Vicent Estruch

Lucía Agud

Javier Barrallo

TRADUCCIONS

Àrea de Promoció

i Normalització Lingüistica.

Lambe & Nieto.

DISSENY I MAQUETACIÓ

Tassen Estudio

IMPRESSIÓ

La Imprenta CG

EDICIÓ

Editorial de la UPV / Ref: 2007.0000

ISBN: 00-0000-000-0

DEPÒSIT LEGAL: V-0000-0000

© de les imatges, els autors

© dels textos, els autors

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INDEX

INTRODUCCIÓ / Joan Bta. Peiró

IMAGINACIÓ POÈTICA I MATEMÀTIQUES / Vicent Estruch Fuster

COM PREGONAVA L’ESLÒGAN DE L’ANY 2000 / Lucía Agud

EXPOSICIONSLa mar de nombresLes formes de l´univers / Javier Barrallo

LA MAR DE NOMBRES

LES FORMES DE L´UNIVERS

ENGLISH TRANSLATIONS

004 006

008

010

012

044

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LA MAR DE NOMBRES

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ALBA SOBRE L´OCEÀ

JosLeys

L’alba sobre l’oceà és una de les imatges més belles que poden ser contemplades al nostre planeta. Per a la interpretació d’aquesta es-cena, l’autor féu emergir de l’aigua un artefacte denominat grup klei-nià, anomenat així en honor del matemàtic Felix Klein.Dues transformacions matemàtiques, en harmonia amb les seues inverses, produeixen la intricada forma composta per cercles i ones que embelleixen el Sol elevant-se sobre l’oceà.

AMANECER SOBRE EL OCÉANO

El amanecer sobre el océano es una de las imágenes más hermosas que pueden ser contempladas en nuestro planeta. Para la interpre-tación de esta escena su autor hizo emerger del agua un artefacto denominado grupo Kleiniano, llamado así en honor del matemático Felix Klein.Dos transformaciones matemáticas, en armonía con sus inversas, producen la intricada forma compuesta por círculos y ondas que em-bellecen al sol elevándose sobre el océano.

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OXIGENASCENDINT

Nicholas Rougeaux

L’oxigen arriba al mar procedent de l’atmosfera, d’on l’agafen animals i vegetals per a la seua respiració, els quals, al seu torn, contribueixen a incrementar-lo grà-cies al procés de fotosíntesi. La proporció d’oxigen dis-solt en l’aigua del mar és inversament proporcional a la profunditat, de manera que hi ha àrees sense aquest gas en els grans fons marins.L’autor de la imatge, un incansable explorador de for-mes diferents de mirar el nostre món, va emprar la fórmula ManPhoenix per a representar les bombolles ascendint des del fons marí. Les bombolles foren sub-tilment ombrejades per a simular la visió del sol des de la profunditat de l’oceà.

OXÍGENO ASCENDIENDO

El oxígeno llega al mar procedente de la atmósfera, de don-de lo toman animales y vegetales para su respiración y que, a su vez, contribuyen a incrementar gracias al proceso de fotosíntesis. La proporción de oxígeno di-suelto en el agua del mar es inversamente proporcio-nal a la profundidad, encontrándose áreas carentes de este gas en los grandes fondos marinos.El autor de la imagen, un incansable explorador de formas diferentes de mirar nuestro mundo, empleó la fórmula ManPhoenix para representar las burbujas ascendiendo desde el fondo marino. Las burbujas fue-ron sutilmente sombreadas para simular la visión del sol desde la profundidad del océano.

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Encara que el corall suposa la major part de la mas-sa d’un escull, els organismes responsables del seu creixement contra l’assetjament constant de les ones oceàniques són les algues calcàries. Els coralls no fan la fotosíntesi, però viuen en una relació simbiòtica amb aquestes algues microscòpiques que sí que la realit-zen.Per a presentar en una imatge matemàtica l’enorme biodiversitat de l’escull s’ha emprat la tècnica d’Orbit Traps (captura d’òrbites), que aplicada sobre una fór-mula fractal convencional produeix una gran varietat de formes. Amb una certa habilitat, la combinació har-mònica d’aquestes formes assoleix simular la com-plexitat i el colorit de l’escull.

ARRECIFE CORALINO

Aunque los corales suponen la mayor parte de la masa de un arrecife, los organismos responsables de su crecimiento contra el constante acoso de las olas oceá-nicas son las algas calcáreas. Los corales no realizan fotosíntesis, pero viven en una relación simbiótica con estas algas microscópicas que sí la realizan.Para presentar en una imagen matemática la enorme biodiversidad del arrecife se ha empleado la técnica de Orbit Traps (Captura de Órbitas) que aplicada sobre una fórmula fractal convencional produce una gran va-riedad de formas. Con cierta habilidad, la combinación armónica de estas formas logra simular la compleji-dad y colorido del arrecife.

ESCULLCORAL·LÍ

Linda Allison

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Les badies han tingut i tenen gran importància estratègica i econòmica ja que, normalment, són llocs ideals per a la construcció de ciutats i ports, perquè ofereixen refugi con-tra les tempestes i faciliten les activitats pesqueres i por-tuàries.Per a aquesta imatge l’autora va triar una variant d’una de les fórmules fractals més conegudes: el conjunt de Man-delbrot. Lluny de buscar artificis complexos per a realçar la imatge, li va bastar disposar un senzill gradient de color per a ressaltar les llums de la ciutat, el blau de la badia i el cel púrpura del capvespre.

EL ANOCHECER SOBRE LA BAHÍA

Las bahías han tenido y tienen gran importancia estratégica y económica ya que, normalmente, son lugares ideales para la construcción de ciudades y puertos, por ofrecer abrigo contra las tormentas y facilitar las actividades pesqueras y portuarias.Para esta imagen su autora eligió una variante de una de las formulas fractales más conocidas: el Conjunto de Man-delbrot. Lejos de buscar complejos artificios para realzar la imagen, le bastó disponer un sencillo gradiente de color para resaltar las luces de la ciudad, el azul de la bahía y el cielo púrpura del anochecer.

CAPVESTRE SOBRE LA BADIA

Sylvie Gallet

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TSUNAMIDave Makin

Les ones són ondes que es desplacen per la superfície de mars i oceans, engegades pel vent. Un tipus particular són els tsunamis, que són ones produïdes per un sisme submarí o una explosió vol-cànica. L’ona única produïda per un tsunami avança a gran velocitat, poden ser milers de quilòmetres per hora, i arriba a tenir una altura superior als 20 metres.La forma pseudoaleatòria que presenten aquestes ones, la simetria de les quals es trenca gradualment, inspirà el procés seguit per a aconseguir aquesta imatge, que consisteix en l’aplicació d’una trans-formació matricial per a cada punt representat.

TSUNAMI

Las olas son ondas que se desplazan por la superficie de mares y océanos, puestas en marcha por el viento. Un tipo particular son los tsunamis, que son olas producidas por un maremoto o una explosión volcánica. La ola única producida por un tsunami avanza a gran ve-locidad, pueden ser miles de kilómetros por hora, y llega a tener una altura superior a los 20 metros.La forma seudo aleatoria que presentan estas olas, cuya simetría se rompe gradualmente, inspiró el proceso seguido para conseguir esta imagen, consistente en la aplicación de una transformación matricial para cada punto representado.

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Rick Spix

FONSSUBMARÍ

Gairebé un 80 % del fons marí està per sota dels 1.000 metres, de manera que la pressió hidrostàtica és unes cent vegades superior a la pressió atmosfèrica a la qual estem habituats. Aquestes condicions adverses no impedeixen el desplegament d’una diversa vida animal fins i tot en els fons dels avencs més profunds.Per a compondre aquesta imatge s’utilitzaren fins a 170 fórmules, principalment del tipus Mandelbrot, Julia i Lyapunov. L’autor va pretendre representar les mutacions evolutives ocorregudes milions d’anys en-rere a partir de formes vivents primitives.

FONDO MARINO

Casi un 80% del fondo marino está por debajo de los 1.000 metros, por lo que la presión hidrostática es unas cien veces superior a la presión atmosférica a la que estamos habituados. Estas condiciones adversas no impiden el despliegue de una diversa vida animal in-cluso en los fondos de las simas más profundas.Para componer esta imagen se utilizaron nada menos que 170 fórmulas, principalmente del tipo Mandelbrot, Julia, y Lyapunov. Su autor pretendió representar las mutaciones evolutivas ocurridas millones de años atrás a partir de formas vivientes primitivas.

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Els nàutils són mol·luscos marins de petita grandària que habiten a l’oceà Índic. Curiosament, la construcció espiral de la seua conquilla manté una proporció matemàtica anomenada nombre d’or o propor-ció àuria, també coneguda com phi (1,61803…).Las espirals àuries constitueixen les estructures de creixement i de conservació d’energia més eficients, íntimament lligades amb altres objectes matemàtics com els sòlids platònics o el conjunt de Mandel-brot. El seu disseny tan elegant ha servit d’inspiració per a aquesta imatge, complementada amb altres fórmules per a produir la textura i la il·luminació de l’escena.

NAUTILUS

Los nautilus son moluscos marinos de pequeño tamaño que habitan en el Océano Índico. Curiosamente la construcción espiral de su con-cha mantiene una proporción matemática denominada Número de Oro ó Proporción Áurea, también conocida como Phi (1.61803…).Las espirales áureas constituyen las más eficientes estructuras de crecimiento y conservación de energía, íntimamente ligadas con otros objetos matemáticos como los sólidos platónicos o el Conjunto de Mandelbrot. Su elegante diseño ha servido de inspiración para esta imagen complementada con otras fórmulas para producir la textura y la iluminación de la escena.

NÀUTILSHeatherLamb

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NAUFRAGI

EtienneSaint-Amant

Cada any es produeixen milers de naufragis als mars del món, però no tots són famosos com els del Titanic o el Lusitania. La majoria són petits navilis o vaixells pesquers que en molts casos són abandonats.Per a la creació d’aquesta imatge, de gran complexi-tat, s’acoblaren i es refinaren diversos conjunts frac-tals per a tractar d’aconseguir un vaixell surrealista. L’efecte de l’onatge i la boirina present en l’escena es va aconseguir combinant tècniques de moviment brownià i l’aplicació de textures aquàtiques.

NAUFRAGIO

Cada año se producen miles de naufragios en los ma-res de mundo, pero no todos son famosos como los del Titanic o el Lusitania. La mayoría son pequeños navíos o barcos pesqueros que en muchos casos son abando-nados a su suerte.Para la creación de esta imagen, de gran complejidad, se ensamblaron y refinaron varios conjuntos fractales tratando de conseguir un barco surrealista. El efecto del oleaje y la neblina presente en la escena se consi-guió combinando técnicas de Movimiento Browniano y aplicación de texturas acuáticas.

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LES FORMES DE L´UNIVERS

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046

Samuel Monnier

JÚPITER & IÓ

Descoberta per Galileu Galilei el 1610, Ió és la lluna més propera a Júpiter i el cos del sistema solar de més activitat volcànica, amb erupcions que emeten mate-rial a més de 300 quilòmetres d’altura.En aquesta imatge es van combinar elements de geo-metria fractal per a representar la riquesa de color i textura produïda per l’intens vulcanisme d’Ió. L’objectiu de la imatge no era buscar el fotorealisme, sinó usar Júpiter i Ió com a font d’inspiració per a una imatge de naturalesa matemàtica i caràcter merament estètic.

JUPITER & IO

Descubierta por Galileo Galilei en 1610, Io es la luna más cercana a Jupiter y el cuerpo del Sistema Solar de mayor actividad volcánica con erupciones que emiten material a más de 300 Kilómetros de altura.En esta imagen se combinaron elementos de Geo-metría Fractal para representar la riqueza de color y textura producida por el intenso vulcanismo de Io. El objetivo de la imagen no era buscar el fotorealismo sino usar a Júpiter e Io como fuente de inspiración para una imagen de naturaleza matemática y carácter meramente estético.

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JosephSamuel Monnier

ANDRÒMEDA La galàxia d’Andròmeda ja va ser observada l’any 964 per astrònoms perses. Andròmeda pot observar-se a primera vista sota un cel suficientment fosc i lluny de fonts de contaminació lumínica.Dibuixar una galàxia de forma realista emprant úni-cament fórmules matemàtiques és més difícil del que sembla. Hi ha una interacció entre la forma regular global de la galàxia, generalment una el·lipse o espi-ral, i els detalls més complexos, basats en els núvols de pols i gas que la componen.

ANDRÓMEDA

La Galaxia de Andrómeda ya fue observada en el año 964 por astrónomos persas. Andrómeda puede obser-varse a simple vista bajo un cielo suficientemente os-curo y lejos de fuentes de contaminación lumínica.Dibujar una galaxia de forma realista empleando úni-camente fórmulas matemáticas es más difícil de lo que parece. Existe una interacción entre la forma re-gular global de la galaxia, generalmente una elipse o espiral, y los detalles más complejos, basadas en las nubes de polvo y gas que la componen.

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El nombre d’estrelles visibles a primera vista des de la Terra s’ha calculat en 8.000. Les estrelles no estan distribuïdes uniformement en l’univers, sinó agru-pades en galàxies. La seua quantitat, a la Via Làctia, ascendeix a centenars de milers de milions i, al seu torn, només és una dels diversos centenars de milions de galàxies visibles mitjançant els potents telescopis moderns.L’ésser humà, des de l’origen, ha donat nom i ha creat històries sobre les estrelles, que es transmeten a tra-vés de generacions. Per mitjà d’aquesta imatge, l’autor pretén homenatjar les estrelles, que han actuat de guia i inspiració per a la humanitat. Per a compondre-la, s’han utilitzant parts del clàssic conjunt de Mandelbrot disposats a partir d’una reixeta de discos circulars.

NOCHE ESTRELLADA

El número de estrellas visibles a simple vista desde la Tierra se ha calculado en 8.000. Las estrellas no están distribuidas uniformemente en el Universo, sino agru-padas en galaxias. Su número, en la Vía Láctea, as-ciende a cientos de miles de millones y, a su vez, sólo es una de los varios cientos de millones de galaxias visibles mediante los potentes telescopios modernos.El hombre, desde su origen, ha dado nombre y creado historias sobre las estrellas, que se transmiten a tra-vés de generaciones. Mediante esta imagen, su autor pretende homenajear a las estrellas, que han actuado de guía e inspiración para la humanidad. Para compo-nerla, se han utilizando partes del clásico Conjunto de Mandelbrot dispuestos a partir de una rejilla de discos circulares.

NITESTRELLADA

Nicholas Rougeux

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UNIVERS ENEXPANSIÓ

Heather Lamb

Totes les cosmologies antigues assumeixen un Univers etern, estàtic i immutable. Però el 1929, l’astrofísic nord-americà Edwin Hubble, després de lo-calitzar i identificar diferents galàxies, va observar que s’allunyaven de nosaltres. Això implica que l’univers no és estàtic, s’està expandint i hi hagué un temps en què degué estar totalment concentrat.La interpretació personal d’aquesta teoria se sustenta en la imatge amb una fórmula que representa núvols de pols i gas distorsionats per mitjà d’una espiral. Els planetes s’hi afegiren per a proporcionar una sensació de flotació i moviment suau enfront de l’intens i dra-màtic fons.

UNIVERSO EXPANDIÉNDOSE

Todas las cosmologías antiguas asumen un universo eterno, estático e inmutable. Pero en 1929, el astrofísi-co norteamericano Edwin Hubble, tras localizar e iden-tificar diferentes galaxias, observó que se alejaban de nosotros. Esto implica que universo no es estático, se está expandiendo y que hubo un tiempo en el que de-bió estar totalmente concentrado.La interpretación personal de esta teoría se sustenta en la imagen mediante una fórmula que representa nubes de polvo y gas distorsionadas mediante una espiral. Los planetas se añadieron para proporcionar una sensación de flotación y movimiento suave frente al intenso y dramático fondo.

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NEBULOSA TARÀNTULA

Sylvie Gallet

La nebulosa Taràntula està situada al Gran Núvol de Magallanes, una de les galàxies satèl·lits de la nostra Via Làctia. Dintre d’aquest aràc-nid còsmic jau un cúmul jove d’estrelles pesades, la intensa radiació i els forts vents del qual han contribuït a energitzar la lluentor nebu-lar i donen la forma als filaments laterals.L’autora d’aquesta imatge va necessitar una exploració pacient de diverses fórmules fractals fins a aconseguir el resultat perseguit, que va realçar emprant una suau paleta de color que proporciona equilibri i harmonia al conjunt.

NEBULOSA TARÁNTULA

La Nebulosa Tarántula está ubicada en la Gran Nube de Magallanes, una de las galaxias satélites de nuestra propia Vía Láctea. Dentro de este arácnido cósmico yace un cúmulo joven de estrella pesadas, cuya intensa radiación y fuertes vientos han contribuido a energizar el brillo nebular y le dan la forma a los filamentos laterales.La autora de esta imagen necesitó una paciente exploración de di-versas formulas fractales hasta conseguir el resultado perseguido, que realzó empleando una suave paleta de color que proporciona equilibrio y armonía al conjunto.

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PEGÀS 51

Samuel Monnier

Pegàs 51 és el nom d’una estrella de tipus solar de la constel·lació de Pegàs, situada a 47,9 anys llum del sistema solar. El 6 d’octubre de 1995 es va descobrir al costat d’aquesta estrella el primer planeta extrasolar, Pegàs 51B. L’autor va concebre l’escena com un tribut a aquest primer planeta descobert al voltant d’una es-trella diferent del Sol. La imatge suggereix la presèn-cia d’un hipotètic satèl·lit rocós girant, al seu torn, al voltant de Pegàs 51B. Aquesta interpretació és una lli-cència artística, ja que no té rigor científic.

PEGASO 51

Pegaso 51 es el nombre de una estrella de tipo solar en la constelación de Pegaso, situada a 47.9 años luz del Sistema Solar. El 6 de octubre de 1995 se descubrió junto a esta estrella el primer planeta extrasolar, Pe-gaso 51B.El autor concibió la escena como un tributo a este pri-mer planeta descubierto alrededor de una estrella di-ferente del Sol. La imagen sugiere la presencia de un hipotético satélite rocoso girando, a su vez, alrededor de Pegaso 51B. Esta interpretación es una licencia ar-tística, ya que carece de rigor científico.

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Mathematics require a high capacity for abstraction to catch a glimpse of its ultimate meaning in the infinite darkness and in the dazzling brightness of thought.

Artistic abstraction was yet another of the many revolutions that swept the world during the early decades of the 20th century. The complete break with the figurative represen-tation reigning in our Western culture also brought about a rupture between the work and the spectator that remains open to this day, with the latter still not understanding abstract forms divested of representational interpretations.

Perhaps it is this double abstract conjunction that underpins this especially interesting and attractive exhibition openly grounded in mathematics and offering us images of powerful formal beauty coupled with an equal capacity to relate with the natural spaces of oceans and skies.

Over and beyond the evident first common denominator of mathematics, the sea and the firmament, is humanity.That said, it strikes me as extraordinarily significant that a second common denomina-tor has arisen, less obvious yet for all that much more fascinating, and that is beauty. Yet not so much a conceptual beauty, though that too, as visual.

I am absolutely convinced that whoever sees this exhibition will be initially fascinated, then left pensive and finally enjoy it to the full.

INTRODUCTION / Joan Bta. Peiró

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Godfrey H. Hardy maintained that a mathematician, not unlike a painter or a poet, is an inventor of models and that if mathematic models are lasting, then that is owing to the fact that they are constructed with ideas. Besides, he claimed that a mathematician’s models, like those of a painter or a poet, should be beautiful and that beauty is simply the first test of validity; denying a permanent place in the world for ugly mathema-tics. Hardy was a poet of reason, whose poetry consisted in realizing beautiful abstract constructions based on impeccable reasoning. Benoît Mandelbrot, the father of fractal geometry, was conversant with Hardy’s work yet he objected to his aesthetic concep-tion of the mathematical work, rejecting any mathematics whose goal was not to con-trol reality in itself. It is highly probable that Hardy’s poetic mindset would have intuited, well in advance, that the beauty of mathematical constructions and the reality of the forms of nature would eventually be fused in a geometry easy to intuit yet difficult to formalise, a geometry that would lead to known yet unimaginable images, a complex yet beautiful geometry, observable yet limitless, posterior to man yet anterior to the existence of almost everything. Nonetheless, Hardy could never have imagined that it would be Mandelbrot, a mathematician attached to cold reality, who, guiding machines with memory but without memories, and a silicon soul, would finally construct, preci-sely with mathematics, those beautiful images that his poetry lacked.

POETIC IMAGINATION AND MATHEMATICS /Vicent Estruch Fuster

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As the slogan for the World Mathematical year in 2000 claimed, this is a discipline that impinges in our everyday life in any moment or situation in the form of geometry, cal-culus or computation. All we have to do is open our eyes to our surrounding reality and we will find any or all of the branches embraced by mathematics.

Mathematics helps to render what the eye sees, but also to artificially create, thanks to its numerous and powerful tools, new forms that resemble and sometimes go beyond natural phenomena and the grandiosity of nature.

We can discover part of this surprising world in this exhibition primarily based on frac-tal geometry. The word fractal comes from the Latin “fractus” which means “broken”, due to the fact that these figures are forms that can be divided into parts, each a smaller version of the whole. Using any piece, even at the smallest scale, we can recompose the initial structure. This branch of science has been developed thanks to the advances in technology and computation because, without these tools, it would not be possible to generate them in infinitely small scales that the eye cannot distinguish.

When the many disciplines of science are brought together, they complement one ano-ther, reaching unsuspected and wonderful heights, and this is true from many viewpo-ints: technological, aesthetic, functional, etc.

Let ourselves be surprised by everything this exhibition shows us, and let us open up our eyes and our understanding to the world that science wishes to reveal to us.

AS THE SLOGAN FOR THE WORLD MATHEMATICAL YEAR IN 2000 CLAIMED / Lucía Agud

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The secrets still concealed by the Universe and the Oceans have given free rein to the imagination of writers, painters, musicians and, in general, all kinds of artists.We remember, among others, Jules Verne’s novel From the Earth to the Moon or Stan-ley Kubrick’s film 2001: a Space Odyssey. On the other hand, scientists such as Jac-ques Cousteau, Edwin Hubble or Stephen Hawking have devoted their lives to unveil the mysteries of the ocean depths and the limits of the Universe. But combining Art and Science starting from images taken from the Oceans and from the Universe, only by means of mathematical formulas, is a challenge which had never been faced before this exhibition.This unusual creative challenge requires a combination of a delicate artistic sensitivity with a deep knowledge of Calculus and Mathematics. It is necessary to point out that all the images exhibited have been designed from formulas and algorithms without any subsequent artistic manipulation or computer retouching.Most of the formulas used belong to the branch of Mathematics known as Fractal Geo-metry, formulated by the scientist Benoit Mandelbrot in 1975. Unlike Euclidean Geome-try, developed to measure the objects designed by man: cubes, spheres, parabolas, straight lines,… Fractal Geometry studies the objects whose shapes are drawn from patterns present in nature such as clouds, galaxies, waves and coral reefs.The artistic interpretation of the impressive images coming from faraway nebulas and the sea bed by means of Fractal Geometry is amazing, and shows a much closer rela-tionship between Art and Science than we can imagine.

EXHIBITIONSA sea of numbersThe forms of the univers / Javier Barrallo

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DAWN OVER THE OCEAN / Jos Leys

A dawn over the ocean is one of the most beautiful images that can be contempla-ted on our planet. In order to capture this scene its author made an artefact known as Kleinian group, named after the ma-thematician Felix Klein, emerge from the water.Two mathematical transformations, in harmony with their inverses, produce the intricate shape made up of circles and waves that embellish the sun rising over the ocean.

OXYGEN RISING/ Nicholas Rougeaux

Oxygen is supplied to the sea from the atmosphere, and once there it is used by animals and vegetables for breathing, which in turn contribute to increase its levels thanks to the photosynthesis pro-cess. The proportion of oxygen dissolved in sea water is inversely proportional to water depth, so there are areas in the sea bed that lack this gas.The author of the image, a tireless ex-plorer of different ways of looking at our world, used the ManPhoenix formula so as to depict the bubbles coming up from the seabed. The bubbles were subtly sha-ded to simulate the sight of the sun from the ocean depths.

THE PERFECT STORM / David April

Japanese painter Hokusai (1760-1849) al-ready used in his works one of the most fascinating properties of fractal mathema-tics: self-resemblance. According to this property, objects display similar structu-res on a different scale, both regarding their geometry and their behaviour.Likewise, in this image, each wave is made up of other smaller waves, whi-ch in turn are made up of other smaller waves, and so on. Technically the image is generated by only using pseudo-random points arranged by means of an Iterated Function System (IFS).

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NAUTILUS / Heather Lamb

Nautilus are small marine molluscs that live in the Indian Ocean. Surprisingly, the spiral construction of their shells holds a mathematical proportion called Golden Ratio or Divine Proportion, also known as Phi (1.61803...). Golden spirals constitute the most effi-cient structures for the growth and pre-servation of energy, closely linked to other mathematical objects such as Platonic so-lids and the Mandelbrot set. Their elegant design has inspired this image, which has been complemented with other formulas to provide texture and lighting.

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BLACK TIDE / Janet Parke

Black tides such as those ones caused by the Gulf War, the Exxon Valdez or the Prestige are among the most serious eco-logical disasters on our planet, and their effects still persist years after the leaka-ge.The author of this image used the Popcorn formula to simulate the winding shapes adopted by the moving sea. The picture was completed with various layers that helped to both achieve the right colours and the oily stains and drops characteris-tic of a black tide.

EROSION / Heather Lamb

The coastline is constantly subject to the erosive action of water, which creates many different shapes depending on the action of waves, tides and sea currents. The author of this image was inspired by the intricate structure of the Scottish coa-sts. The rough texture of rocks is based on triangular geometric models created from the Sierpinski Triangle. The calm sea, the sun and the sky were created by using different algorithms so as to provide con-trast and perspective with the rocks.

THE SARGASSO SEA / Javier Barra-llo

The Sargasso Sea is a strange sea of calm waters not confined by land margins. Its superficial waters abound with the al-gae that give it its name, the sargassum seaweed, which forms large rafts brim-ming with marine organisms and with almost no animal life, except for the eels that choose it as their spawning place.In order to emphasise this image, its au-thor used one single formula repeated in different layers, applying a progressive atmospheric effect for each layer with the aim of adding a sense of sub-aquatic depth.

CORAL / Kerry Mitchell

Corals are small animals living in colonies and which are joined together by their cal-careous skeletons, forming red or pinkish reefs and branched structures.This interpretation of coral is an enlarge-ment of the Mandelbrot set, where every calculation is repeated six times with slig-htly different parameters. The final result is the arithmetic mean of the six calcula-tions, which adds a sense of depth and fragility to the image.

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NIGHTFALL ON THE BAY/ Sylvie Gallet

Bays have had, and still have, a great strategic and economic importance since they are usually the perfect place to build cities and ports, as well as offering shel-ter against storms and facilitating fishing and port activities.For this image the author chose a variant of one of the most popular fractal formu-las: the Mandelbrot set. Far from looking for complex artifices to enhance the ima-ge, it was enough for her to use a simple colour gradient so as to emphasise the city lights, the blue of the bay, and the pur-ple sky of the sunset.

TSUNAMI / Dave Makin

Sea waves propagate over the surface of the seas and the oceans, set in motion by the wind. Tsunamis are a particular type of wave caused by a seaquake or a volca-nic explosion. The only wave generated by a tsunami is propelled at great speed, re-aching thousands of kilometres per hour, and can exceed 20 metres in height.The pseudo-random shape of these waves, whose symmetry is gradually broken, inspired the process followed to achieve this image, consisting of applying a matrix transformation to each point re-presented.

THE INFINITE OCEAN/ Samuel Monnier

Oceans cover 71% of the Earth’s surface. The deepest point is in the Marianas Tren-ch, which has a depth of 11.033 m. For some primitive cultures oceans symboli-sed the boundless: the infinite.The brilliant Dutch artist M.C. Escher used a mathematical model called hyperbolic plane that can represent the infinite as the objects become smaller in proportion to their distance to a given point, being con-tained inside a boundary circumference. This image is a homage to Escher, and uses two logarithmic spirals to shrink a regular mosaic to the central point of the image, in a similar way as Escher defor-med his fish.

SEA BED / Rick Spix

Almost 80% of the sea bed is under 1,000 metres, therefore the hydrostatic pres-sure is about one hundred times the at-mospheric pressure we are used to. The-se adverse conditions do not prevent the existence of a variety of animal life even in the deepest abysses. In order to create this image 170 formulas were used, mainly of the Mandelbot, Julia and Lyapunov types. The author intended to depict the evolutionary mutations occu-rred millions of years ago from primitive living forms.

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CORAL REEF / Linda Allison

Even though corals represent the ma-jor part of a reef mass, the organisms responsible for its growth against the constant pounding of oceanic waves are calciferous algae. Corals do not carry out photosynthesis, but they live in a symbio-tic relationship with those microscopic algae and rely on them to perform this function.In order to depict a mathematical image of the enormous biodiversity on the reef the author has used the Orbit Traps tech-nique, which applied on a conventional fractal formula produces a wide variety of forms. With some skill, the harmonic combination of those forms simulates the complexity and colours of the reef.

SHIPWRECK / Etienne Saint-Amant

There are thousands of shipwrecks in the world’s seas every year, but not all of them are as famous as those of the Titanic or the Lusitania. Most of them are small ships or fishing boats, which in many ca-ses are left to their own devices.To create this image of great complexity, different fractal sets were assembled and refined trying to produce a surrealist ship. The swell and the mist effects depicted in the image were achieved by combining Brownian Movement techniques and the application of aquatic textures.

WAVES IN THE NIGHT/ Damien Jones

Tides are caused by the gravitational pull on the seas of the Sun and, especially, of the Moon because as it is closer to the Earth, it exerts a stronger pull. Technically, the image has been created by combining Newton’s Method for the solution of complex roots with the intro-duction of small turbulences mathemati-cally produced. In this way, an unexpected effect known as Chaos Theory is obtained: whereas small areas of the image appear random, if we observe the whole image there is a feeling of regularity that is remi-niscent of the waves sequence produced by tides.

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BIG BANG / Kerry Mitchell

The Big Bang Theory is a model that des-cribes the development of the Universe and its evolution. The term is used to refer both to the moment in which the observa-ble expansion of the Universe began, and to the paradigm that explains its origin and its evolution.This image has been generated from a co-loured Julia set, according to the proximity of its iterations to the Gaussian integers. The colour is defined by the iteration angle and the lighting by the distance. The pearls chains in the background suggest the ele-mental particles that existed immediately after the Big Bang, and how they combined in order to form stars and galaxies.

OMEGA / Nicholas Rougeux

The Omega Nebula, also known as the Swan Nebula, is located at a distance of 5,000 to 6,000 light years, and it is about 15 light years in diameter.This image has been created with plen-ty of orbit trap algorithms and Brownian colouring, trying to show the magic of the immensity of the universe. For the author, nebulas stimulate the imagination and in-vite us to escape from reality.

BLUE PLANET / Heather Lamb

The Earth is the only planet in the Solar System where water exists in liquid form, covering three quarters of the Earth’s sur-face. The presence of water is essential for the existence of life, and our planet is the only known place where life can develop.This image is inspired by the photographs taken from the International Space Station. The Earth was designed by using two types of sphere: the first one is smooth, and con-tains lights and shadows, and the second one is rough, perfect for depicting clouds, land and oceans. Finally, it was embellis-hed by adding the aura of a circular coro-na with a bluish gradient. The Sun and the Moon where created in a similar way.

JUPITER & IO / Samuel Monnier

Io, discovered by Galileo Galilei in 1610, is the closest moon to Jupiter, and it has the highest volcanic activity in the solar system, with eruptions that eject material more than 300 kilometres high.Elements of Fractal Geometry were com-bined in this image to depict the richness of colour and texture produced by the in-tense volcanism of Io. The aim of this ima-ge was not to achieve photorealism but to use Jupiter and Io as inspiration for the creation of an image of a mathematical nature and a merely aesthetic character.

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PAPILLON NEBULA N159/ Joseph Presley

Nebulas are clouds of cosmic gas and dust, whose importance lies in the fact that they are regarded as the place whe-re solar systems similar to our own are born, develop and die.Nebulas are among the most photogenic objects in astronomy, and their beauty is a source of inspiration for the author of this image, who resorted to a spiral shape in order to depict it. Spirals are common in nature, from the smallest organisms to the colossal spirals of the grand cosmos.

SIRIUS / Samuel Monnier

Sirius is the brightest star that can be seen at night from the Earth. This hea-venly body is actually made of two stars. Sirius’ partner, called Sirius B, is smaller and extremely denser, since a litre of Si-rius B’s matter would weigh 60,000 kg on the Earth.The image is made up of a grid of squares containing circles. Each space between four circles shapes up a star. This struc-ture is repeated in different scales produ-cing stars of various sizes.

GAS STORM / Íñigo Quílez

The hydrogen, oxygen and sulphurous gases storms illuminated by floods of ultraviolet radiation coming from young stars nearby, create incredibly beautiful scenes.Inspired by these scenes, the image shows the Mandelbrot set represented with the Buddhabrot algorithm. A colour algorithm similar to the one used by the NASA to ar-tificially colour the images received by the Hubble telescope has been applied. The image was created through the iteration of seven thousand million dots and it took several days to be generated.

ECLIPSE / Heather Lamb

An eclipse (from the Greek ekleipsis, “abandonment”) is an event in which the light coming from a heavenly body is hidden by another one. Solar eclipses oc-cur when the Moon blocks the sunlight. In this image, the Moon passes between the Sun and the Earth creating a partial eclipse. The energy and intensity of the eclipse is obtained by projecting a clear-dark gradient on the Moon which is sha-ded in its lower part in contrast with the intense brightness of the upper part, di-rectly illuminated by the Sun.

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PEGASUS 51 / Samuel Monnier

Pegasus 51 is the name of a solar-type star within the Pegasus constellation, lo-cated 47.9 light years away from the Solar System. Pegasus 51B, the first extra-so-lar planet, was discovered next to this star on 6th October 1995. The author conceived the scene as a tri-bute to the first planet discovered around a star other than the Sun. The image sug-gests the presence of a hypothetical rocky satellite rotating in turn around Pegasus 51B. This interpretation is an artistic li-cence as it lacks scientific rigour.

THE SUN / Javier Barrallo

The Sun is the star of the planet system in which the Earth is located, and its pre-sence in the sky marks the days and the nights. The heat irradiated by the Sun provides the energy that drives weather processes and it is used by the photosyn-thetic beings to start their life cycle.To create this image a Julia set was cha-racterised with a deliberately simple de-sign, both geometrically and chromatica-lly, enhancing the exuberant corona. The heavenly bodies in the background are embellished as they adopt spiral shapes.

ANDROMEDA / Samuel Monnier

The Andromeda Galaxy was already ob-served by Persian astronomers in the year 964. Andromeda can be seen with the naked eye under a dark enough sky and far from light pollution sources.Drawing a galaxy in a realistic way by using only mathematical formulas is more difficult than it seems. There is an interac-tion between the global regular shape of the galaxy, generally an ellipse or a spiral, and the more complex details, based on dust and gas clouds that form it.

MARS / Paul DeCelle

The Mars Exploration Rover mission is an unmanned NASA operation sent to ex-plore Mars. The main scientific aim of the mission is the search and analysis of soil that may prove the existence of water on the surface of Mars at any time in the past. Mars is inspired by the first photograph sent back from the surface of the red pla-net by the Spirit Rover in January 2004. When looking at the picture, the author discovered that the image could be nearly represented with a combination of fractal formulas, colour algorithms and trans-formations. As a slight artistic licence, the author included a rising sun which fitted in with the shadows cast by the rocks.

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STARRY NIGHT / Nicholas Rougeux

It has been estimated that the number of visible stars from the Earth, with the naked eye, are 8,000. Stars are not evenly distributed in the Universe, but grouped in galaxies. Their number, in the Milky Way, rises to hundreds of thousands of millions and, in turn this is just one of the various hundreds of millions of galaxies that can be seen with the powerful modern tele-scopes. From the beginning, human be-ings have given name and created stories about stars which have been passed from generation to generation. With this image, its author wishes to pay tribute to the stars, which have acted as a guide and inspira-tion for mankind. To create it the author has used parts of the classic Mandelbrot set, which have been arranged from a grid of circular disks.

EXPANDING UNIVERSE / Heather Lamb

All the ancient cosmologies accept an eternal, ecstatic and unchangeable univer-se. However, in 1929, the North-American astrophysicist Edwin Hubble, after locating and identifying different galaxies, noticed that they moved away from us. This im-plies both that the universe is not ecstatic but it is expanding, and that at some po-int it must have been totally concentrated. The personal interpretation of this theory is developed in the image through a for-mula that depicts dust and gas clouds dis-torted by a spiral. The planets were added to provide a sense of floating and smooth movement as opposed to the intense and dramatic background.

TARANTULA NEBULA / Sylvie Gallet

The Tarantula Nebula is located in the Lar-ge Magellanic Cloud, one of the satellite galaxies of our own Milky Way. Inside this cosmic arachnid lies a young accumula-tion of heavy stars, whose intense radia-tion and hard winds have contributed to energise the nebular brightness, and give the lateral filaments their shape.The author of this image performed a patient exploration of several fractal for-mulas until she found the desired result, which she enhanced by using a smooth colour palette that brings balance and harmony to the whole image.

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