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LA MAR DE NOMBRES LES FORMES DE L´UNIVERS

Javier Barrallo


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


RECTOR DE LA UNIVERSITAT

CATÀLEG

POLITÈCNICA DE VALÈNCIA

Textos.

Juan Juliá Igual

Joan Bta. Peiró Vicent Estruch

VICERECTOR DE CULTURA

Lucía Agud

Joan Bta. Peiró

Javier Barrallo

CAP D’ÀREA

TRADUCCIONS

DE GESTIÓ CULTURAL

Àrea de Promoció

Carlos Plasencia

i Normalització Lingüistica. Lambe & Nieto.

COMISSARI Javier Barrallo

DISSENY I MAQUETACIÓ Tassen Estudio

COORDINACIÓ Lola Gil

IMPRESSIÓ

Asunción García

La Imprenta CG

Carlos Ayats 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


INDEX 004

INTRODUCCIÓ / Joan Bta. Peiró

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IMAGINACIÓ POÈTICA I MATEMÀTIQUES / Vicent Estruch Fuster

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

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EXPOSICIONS La mar de nombres Les formes de l´univers / Javier Barrallo

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LA MAR DE NOMBRES

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LES FORMES DE L´UNIVERS

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ENGLISH TRANSLATIONS


LA MAR DE NOMBRES


ALBA SOBRE L´OCEÀ

Jos Leys

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 escena, l’autor féu emergir de l’aigua un artefacte denominat grup kleinià, 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 interpretación 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 embellecen al sol elevándose sobre el océano.

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OXIGEN ASCENDINT 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 gràcies al procés de fotosíntesi. La proporció d’oxigen dissolt 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 formes diferents de mirar el nostre món, va emprar la fórmula ManPhoenix per a representar les bombolles ascendint des del fons marí. Les bombolles foren subtilment 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 donde 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 disuelto en el agua del mar es inversamente proporcional 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 fueron sutilmente sombreadas para simular la visión del sol desde la profundidad del océano.

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ESCULL CORAL·LÍ Linda Allison

Encara que el corall suposa la major part de la massa 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 realitzen. 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 fórmula fractal convencional produeix una gran varietat de formes. Amb una certa habilitat, la combinació harmònica d’aquestes formes assoleix simular la complexitat 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 oceánicas 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 variedad de formas. Con cierta habilidad, la combinación armónica de estas formas logra simular la complejidad y colorido del arrecife.

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Sylvie CAPVESTRE SOBRE LA BADIA Gallet 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 contra les tempestes i faciliten les activitats pesqueres i portuàries. Per a aquesta imatge l’autora va triar una variant d’una de les fórmules fractals més conegudes: el conjunt de Mandelbrot. 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 Mandelbrot. 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.

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TSUNAMI

Dave 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 volcànica. 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 transformació 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 velocidad, 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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FONS SUBMARÍ Rick Spix

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 enrere 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 incluso 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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NÀUTILS

Heather Lamb

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 proporció 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 Mandelbrot. 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 concha 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.

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NAUFRAGI Etienne Saint-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 complexitat, s’acoblaren i es refinaren diversos conjunts fractals 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 mares 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 abandonados 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 consiguió combinando técnicas de Movimiento Browniano y aplicación de texturas acuáticas.

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LES FORMES DE L´UNIVERS


JÚPITER & IÓ Samuel Monnier

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 material a més de 300 quilòmetres d’altura. En aquesta imatge es van combinar elements de geometria 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 Geometría 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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ANDRÒMEDA Joseph Samuel Monnier

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 únicament 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 espiral, 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 observarse a simple vista bajo un cielo suficientemente oscuro y lejos de fuentes de contaminación lumínica. Dibujar una galaxia de forma realista empleando únicamente fórmulas matemáticas es más difícil de lo que parece. Existe una interacción entre la forma regular 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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NIT ESTRELLADA Nicholas Rougeux

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 agrupades 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 través 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 agrupadas en galaxias. Su número, en la Vía Láctea, asciende 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 través de generaciones. Mediante esta imagen, su autor pretende homenajear a las estrellas, que han actuado de guía e inspiración para la humanidad. Para componerla, se han utilizando partes del clásico Conjunto de Mandelbrot dispuestos a partir de una rejilla de discos circulares. 068


UNIVERS EN EXPANSIÓ Heather Lamb

Totes les cosmologies antigues assumeixen un Univers etern, estàtic i immutable. Però el 1929, l’astrofísic nord-americà Edwin Hubble, després de localitzar 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 dramàtic fons.

UNIVERSO EXPANDIÉNDOSE Todas las cosmologías antiguas asumen un universo eterno, estático e inmutable. Pero en 1929, el astrofísico norteamericano Edwin Hubble, tras localizar e identificar 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 debió 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 aràcnid còsmic jau un cúmul jove d’estrelles pesades, la intensa radiació i els forts vents del qual han contribuït a energitzar la lluentor nebular 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 diversas 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 estrella diferent del Sol. La imatge suggereix la presència d’un hipotètic satèl·lit rocós girant, al seu torn, al voltant de Pegàs 51B. Aquesta interpretació és una llicència 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, Pegaso 51B. El autor concibió la escena como un tributo a este primer planeta descubierto alrededor de una estrella diferente 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 artística, ya que carece de rigor científico.

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INTRODUCTION / Joan Bta. Peir贸 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 representation 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 denominator 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.

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POETIC IMAGINATION AND MATHEMATICS / Vicent Estruch Fuster 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 mathematics. 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 conception of the mathematical work, rejecting any mathematics whose goal was not to control 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, precisely with mathematics, those beautiful images that his poetry lacked.


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AS THE SLOGAN FOR THE WORLD MATHEMATICAL YEAR IN 2000 CLAIMED / Lucía Agud 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, calculus 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 fractal 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 another, reaching unsuspected and wonderful heights, and this is true from many viewpoints: 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.

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EXHIBITIONS A sea of numbers The forms of the univers / Javier Barrallo 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 Stanley Kubrick’s film 2001: a Space Odyssey. On the other hand, scientists such as Jacques 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 Geometry, formulated by the scientist Benoit Mandelbrot in 1975. Unlike Euclidean Geometry, 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 relationship between Art and Science than we can imagine.


A SEA OF NUMBERS

DAWN OVER THE OCEAN / Jos Leys

NAUTILUS / Heather Lamb

A dawn over the ocean is one of the most beautiful images that can be contemplated on our planet. In order to capture this scene its author made an artefact known as Kleinian group, named after the mathematician 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.

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 efficient structures for the growth and preservation of energy, closely linked to other mathematical objects such as Platonic solids and the Mandelbrot set. Their elegant design has inspired this image, which has been complemented with other formulas to provide texture and lighting.

OXYGEN RISING / Nicholas Rougeaux

THE PERFECT STORM / David April

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 process. 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 explorer 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 shaded to simulate the sight of the sun from the ocean depths. 080

Japanese painter Hokusai (1760-1849) already used in his works one of the most fascinating properties of fractal mathematics: self-resemblance. According to this property, objects display similar structures on a different scale, both regarding their geometry and their behaviour. Likewise, in this image, each wave is made up of other smaller waves, which 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).


A SEA OF NUMBERS

THE SARGASSO SEA / Javier Barrallo The Sargasso Sea is a strange sea of calm waters not confined by land margins. Its superficial waters abound with the algae that give it its name, the sargassum seaweed, which forms large rafts brimming 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 author 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.

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 ecological disasters on our planet, and their effects still persist years after the leakage. 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 characteristic of a black tide.

CORAL / Kerry Mitchell

EROSION / Heather Lamb

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

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 coasts. 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 contrast and perspective with the rocks.


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THE INFINITE OCEAN / Samuel Monnier

NIGHTFALL ON THE BAY / Sylvie Gallet

Oceans cover 71% of the Earth’s surface. The deepest point is in the Marianas Trench, which has a depth of 11.033 m. For some primitive cultures oceans symbolised 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 contained 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 deformed his fish.

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 shelter against storms and facilitating fishing and port activities. For this image the author chose a variant of one of the most popular fractal formulas: the Mandelbrot set. Far from looking for complex artifices to enhance the image, 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 purple sky of the sunset.

TSUNAMI / Dave Makin

SEA BED / Rick Spix

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 volcanic explosion. The only wave generated by a tsunami is propelled at great speed, reaching 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 represented.

Almost 80% of the sea bed is under 1,000 metres, therefore the hydrostatic pressure is about one hundred times the atmospheric pressure we are used to. These 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 occurred millions of years ago from primitive living forms.


A SEA OF NUMBERS

CORAL REEF / Linda Allison Even though corals represent the major 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 symbiotic 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 technique, 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 cases 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 introduction of small turbulences mathematically 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 reminiscent of the waves sequence produced by tides.


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JUPITER & IO / Samuel Monnier

OMEGA / Nicholas Rougeux

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 combined in this image to depict the richness of colour and texture produced by the intense volcanism of Io. The aim of this image 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.

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 plenty 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 invite us to escape from reality.

BIG BANG / Kerry Mitchell

BLUE PLANET / Heather Lamb

The Big Bang Theory is a model that describes the development of the Universe and its evolution. The term is used to refer both to the moment in which the observable expansion of the Universe began, and to the paradigm that explains its origin and its evolution. This image has been generated from a coloured 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 elemental particles that existed immediately after the Big Bang, and how they combined in order to form stars and galaxies.

The Earth is the only planet in the Solar System where water exists in liquid form, covering three quarters of the Earth’s surface. 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 contains lights and shadows, and the second one is rough, perfect for depicting clouds, land and oceans. Finally, it was embellished by adding the aura of a circular corona with a bluish gradient. The Sun and the Moon where created in a similar way.


THE FORMS OF THE UNIVERSE

PAPILLON NEBULA N159 / Joseph Presley

GAS STORM / Íñigo Quílez

Nebulas are clouds of cosmic gas and dust, whose importance lies in the fact that they are regarded as the place where 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.

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 artificially 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.

SIRIUS / Samuel Monnier

ECLIPSE / Heather Lamb

Sirius is the brightest star that can be seen at night from the Earth. This heavenly body is actually made of two stars. Sirius’ partner, called Sirius B, is smaller and extremely denser, since a litre of Sirius 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 structure is repeated in different scales producing stars of various sizes.

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 occur 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 cleardark gradient on the Moon which is shaded in its lower part in contrast with the intense brightness of the upper part, directly illuminated by the Sun.


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MARS / Paul DeCelle

ANDROMEDA / Samuel Monnier

The Mars Exploration Rover mission is an unmanned NASA operation sent to explore 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 planet 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 transformations. As a slight artistic licence, the author included a rising sun which fitted in with the shadows cast by the rocks.

The Andromeda Galaxy was already observed 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 interaction 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.

THE SUN / Javier Barrallo

PEGASUS 51 / Samuel Monnier

The Sun is the star of the planet system in which the Earth is located, and its presence 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 photosynthetic beings to start their life cycle. To create this image a Julia set was characterised with a deliberately simple design, both geometrically and chromatically, enhancing the exuberant corona. The heavenly bodies in the background are embellished as they adopt spiral shapes.

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


THE FORMS OF THE UNIVERSE

EXPANDING UNIVERSE / Heather Lamb All the ancient cosmologies accept an eternal, ecstatic and unchangeable universe. However, in 1929, the North-American astrophysicist Edwin Hubble, after locating and identifying different galaxies, noticed that they moved away from us. This implies both that the universe is not ecstatic but it is expanding, and that at some point it must have been totally concentrated. The personal interpretation of this theory is developed in the image through a formula that depicts dust and gas clouds distorted 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 Large Magellanic Cloud, one of the satellite galaxies of our own Milky Way. Inside this cosmic arachnid lies a young accumulation of heavy stars, whose intense radiation 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 formulas until she found the desired result, which she enhanced by using a smooth colour palette that brings balance and harmony to the whole image.

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 telescopes. From the beginning, human beings 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 inspiration 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.



La Mar de Numeros