"También se esperó entonces la aclaración de los misterios básicos de la humanidad: el origen de la Biblioteca y del tiempo. Es verosímil que esos graves misterios puedan explicarse en palabras: si no basta el lenguaje de los filósofos, la multiforme Biblioteca habrá producido el idioma inaudito que se requiere y los vocabularios y gramáticas de ese idioma."
(Jorge Luis Borges, La Biblioteca de Babel)
A Simple Algebraic Game
How to play. The aim of the game is to produce a fully colored picture by clicking onto the cells composing the 4x4 array. Each click will switch the color, from grayscale to colored or viceversa, of those cells belonging to a neighborhood whose shape is shown bottom right. It is possible to change the shape of the cell neighborhood by clicking onto "prev shape" and "next shape". The neighborhood shape could be modified also during a single game, even though you oughta keep it constant, according to the spirit of the game. Before clicking onto any cell, a preview of the modified array is shown. The array is supposed to have a toroidal topology, so that the top side is identified with the bottom one and the right side is identified with the left one, according to the standard 2D model of a torus. A move counter is displayed top right. Any time, you can reset the game and restart by clicking onto "Reset & restart" (I beg your pardon... this short account is intended to be idiot proof). You can start from the scratch, that means, from a fully grayscale picture, or you can also start from a random configuration. Random configuration are obtained by pressing onto "Random configuration". Since there are 16 cells, each of which may have 2 states, there are 216=65536 total possible states, one of which is the final goal. Of course, you may get lost and be looping in the graph of possible states, so, if you give up, an automatic solver is started by clicking onto "Automatic solution". The algorithm will pause after any move, allowing you to watch how to proceed step-by-step. Finally, in case your network connection is slow, you should wait for a little while until everything is loaded, before starting to play. If some problem still persists, please reset & restart.
Vincent van Gogh, L'eglise d'Auvers-sur-Oise 1890, oil on canvas, Musée d'Orsay, Paris |
Moves used
Reset & restartRandom configurationAutomatic solutionMathematical theoryChange the shapeof the transformation
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My Photos
Canary Islands, Summer 2006
The volcanic landscape of Lanzarote island.
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Son take a good look around, this is my hometown
"I was eight years old and running with a dime in my handInto the bus stop to pick up a paper for my old man
I'd sit on his lap in that big old Buick and steer as we drove thru town
He'd tousle my hair and say son take a good look around this is your hometown"
(Bruce Springsteen, My Hometown)
"I will stare the sundown until my eyes go blind"
(Pearl Jam, Indifference)
Thru the months of the year
The colors of early fall (Gallarate, October 2008).
One foggy day of fall (Gallarate, November 2008).
The great snow fall of the Epiphany day (Gallarate, January 2009).
Thawing (Gallarate, January 2009).
Springtime (Gallarate, April 2010).
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Fog
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Snow
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A trip to Brunate's lighthouse
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The Fractal Geometry of Nature
The Fractal Geometry of Nature is the title of a popular book published in 1975 by Canadian mathematician Benoît Mandelbrot, the father of fractal sets theory. Fractals are glamour examples of topological spaces with real, but not integer, Hausdorff(-Besicovitch) dimension.
Here's Brassica oleracea, exhibiting a stunning self-similarity property! In italian, this vegetable is called broccolo romano, because it is typical of the region of Rome, and in english language... romanesco broccoli... well, for an italian speaker this expression is really ridicolous!!! Sounds like the absurd names given to (non-existent) italian foods in (supposed) italian restourants for non-italians.
The artificial lake produced by the Assuan dam on the Nile river in Egypt (left) and the Grand Canyon in the USA (right). These two images have been produced by surfing thru the web site of the Jet Propulsion Laboratory (JPL) at the California Institute of Technology (Caltech). Probably, Google Maps could deliver them as well.
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Family reunion, December 2008 & December 2010
Images from two weekends by my uncles and cousins in Pisa. From left to right, top to bottom: piazza dei Miracoli in the afternoon; Lungarno in the morning; piazza dei Cavalieri as the evening light embraces the city; the famous tower; piazza dei Miracoli again; the tower top seen from inside the Arcivescovado; the internal court of the Arcivescovado; reflections of the tower into a puddle and piazza dei Miracoli from above.
A trip to the city of Lucca: the elliptic piazza Anfiteatro; the church of San Michele; Lucca from above (from top of Guinigi tower, with trees above), with the shape of piazza Anfiteatro clearly visible (on the background, the big and beautiful romanic church of San Frediano).
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Amsterdam, the Netherlands, October 2010
Images from the weekend before my doctoral defense in Holland. The streets, the canals and the houses of Amsterdam. A couple of shots of Ajax Arena close the collection.
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Delft, the Netherlands, October 2010
Here and later, left to right: the leaning tower of the Oude Kerk (old church); a canal; the inside of the Oude Kerk, diffeomorphically mapped onto the surface of a big chandelier; the Stadhuis (city hall) facing Markt square.
Blinded by the light: the Stadhuis; the inside of the Oude Kerk diffeomorphically mapped onto the base of a pedestal; a view from top of the tower of the Nieuwe Kerk (new church) with the twin towers of the Catholic curch, then TU Delft and Rotterdam faraway on the background.
Blinded by the light: sunset on a canal; the Oostport (eastern gate); a canal surrounded by trees in fall, in front of the Oude Kerk main door; the Oude Kerk from the Nieuwe Kerk tower.
Two views of the old houses surrounding the Markt square, and then the Oude Kerk, all seen from top of the Nieuwe Kerk tower.
The peaceful garden of the Prinsenhof; the quiet of a Monday morning in the Beestenmarkt (former cattle market place), from the inside of the old dress shop where i rented the tux for the doctoral defense; the Beestenmarkt again in the other two shots.
The Technische Universiteit Delft (Delft University of Technology): the tall building is the Faculty of Electrical Engineering, Mathematics and Computer Sciences; the wide building over the pond is the Faculty of Maritime, Mechanical and Material Engineering.
I also took some color shots from the top of the Nieuwe Kerk: canals, banks and bridges; the green square of the Beestenmarkt; the statue of the famous jurist Hugo de Groot (Grotius), born in Delft.
The organ pipes of the Oude Kerk; a look thru a window of the Oude Kerk; Grotius statue again.
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Liquid Shapes
Reflections of the city of Delft, the Netherlands, in the water of its canals. All shots taken the eve of my doctoral dissertation, in a strangely warm and sunny October afternoon.
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Trees in Luminy
Monumental trees in the garden of CIRM (Centre International de Recontres Mathématiques), Luminy (Marseille), France.
The garden and building remind a bit of the psichiatric hospital (Saint-Paul Asylum, Saint-Rémy) where Vincent van Gogh was interned...
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Marseille, France, July 2011
Images from Marseille, France, and its surroundings; particularly the calanques de Sugiton in Luminy, the small village and harbour of Cassis, the Basilique de Notre-Dame de la Garde and the views onto the city from above.
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Xi'an, China, October 2011
Pictures from the city of Xi'an, in Shaanxi province: the big wild goose pagoda, the small wild goose pagoda, the city walls and the impressive tomb of Qin Shi Huang, the first emperor to unite China in a single nation, with the world famous terracotta army.
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Beijing, China, October 2011
Scattered memories from a long weekend in Beijing, the apocaliptic city where poison is in the air to breath, in the water to drink and the dust of death covers the earth, "falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead" (James Joyce, The Dead, Dubliners). The environmental disaster foreseen in many sci-fi movies is already began, and China is a post-catastrophe waste land. I have never seen the sky during more than 2 weeks, and also the sun disk was hard to detect clearly. All the colors are lost in the white-brown dirt fog. Sometimes, a piercing smell of burnt plastic makes you throat sick, and makes your eyes weep.
I knew it was bad, not to such an extent. How to slow down such a large and an inertial system (not only China, the whole world)? The human civilization as a system evolves according to its own rules, which are well above the will and possibilities to control of individuals and also of nations. How long will it take for this sad scenery of death to enlarge and cover the islands of wealth, where people live happy in the ignorance of the apocalypse which shall invest and erase them? It really seems that our planet is definitely lost, and the destruction of humanity is fast to come. We only have one or two decades remaining, and the point of no return is already over. After this travel, the feeling is that life shall turn to a struggle to resist, with no hope for the future generations.
Still, Beijing is an incredibly wonderful city, and also China offers treasures of stunning beauty. The photos are about the Temple of Heaven, the (endless!) Forbidden City, the Bird's Nest and the Watercube (olympic stadium and swimming pool, resp.), the National Theater, and the financial district with its skyscrapers, particularly the building of China Television. Dragons and the phoenix, lions guarding the staircase of a pagoda as the gate of a next universe, embedded into another one in an endless sequence of nested, self-similar fractals, kites drawing lines in the sky and vanishing in the gray cloud of oblivation.
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Great Wall & Ming's Tombs, China, October 2011
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El jardín de senderos que se bifurcan
Tlön. Hay que haber leído Jorge Luis Borges para entender el sentido general de esa seccion. dhcmrlchtdj. Gotta have seen M.C. Escher's "Relativity" to feel familiar with the Bird's Nest. Uqbar. (Tú, que me lees, ¿estás seguro de entender mi lenguaje?) Conoscendo il teorema di Gödel si può forse immaginare un labirinto immateriale ma non meno intricato. Orbis tertius. Oh tiempo tus pirámides. Acaso existe o puede existir un lenguaje en el que eso texto sea llano y comprensible. Axaxaxas mlö.
M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V M C V
(Homenaje al genio de Jorge Luis Borges, Buenos Aires, 24 de agosto de 1899 - Ginebra, 14 de junio 1986)
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Doors and gates
Doors #1,2, 3 are in Pisa, Italy. Door #4 is in Gallarate, Italy.
Door #1 is in Alberobello, Italy. Doors #2,3,4 are in Gallipoli, Italy.
Door #1 is in Gallipoli, Italy. Door #2 is in Taranto, Italy. Doors #3,4 are in Polignano a mare, Italy.
All doors are in Ostuni, Italy.
Door #1 is in Ostuni, Italy. Door #2 is in Perugia, Italy. Doors #3,4 are in Otranto, Italy.
Door #1 is in Ascoli Piceno, Italy. All other doors are in Lecce, Italy.
All doors are in Lecce, Italy.
Side doors are in Amsterdam and in Delft, the Netherlands. The central door is in Lecce, Italy.
All doors are in Delft, the Netherlands.
All doors are in Amsterdam, the Netherlands.
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Italian landscapes
Winter dawn in the countryside by Asmonte di Ossona, NW of Milan. I took this shot on my way to work, on 30.DEC.2008, with many thanks to light refraction in the earth's atmosphere for the pink shift of the spectrum down by the horizon (thx to the electronic filter of the digital camera as well!).
Summer morning in Todi, in the central region of Umbria, on 07.AUG.2009. A nice place to see when having breakfast!
The city of Orvieto, in the central region of Umbria, rising behind a sunflower field, on 09.AUG.2010.
Sunset over my hometown Gallarate, on 27.DEC.2011, during a week of stunning clear sky in that strangely calm period between Christmas and the new years's eve.
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Madrid, March 2009
(Left to right, top to botttom.) Plaza del la Puerta del Sol, with the bear statue, symbol of Madrid [metro: Puerta del Sol]. Estacion de Atocha, with the palmhouse [metro: Atocha or Atocha Renfe]. Plaza de los Carros [metro: la Latina]. Cuatro torres business area [metro: Begoña]. Puerta de Alcala and the boat lake (el embarque) in the Parque del Retiro [metro: Retiro]. The monument to Miguel de Cervantes Saavedra, the father of Don Quijote and Sancho Panza in Plaza de España [metro: plaza de España]. Plaza Mayor and its surroundings [metro: Puerta del Sol]. Finally three other shots taken in the Parque del Retiro [metro: Retiro] and the glass house found therein.
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Villa d'Este, August 2009
"Water, water, every where,
And all the boards did shrink;
Water, water, every where,
Nor any drop to drink."
(Samuel Taylor Coleridge, The Rime of the Ancient Mariner)
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On the road thru Italy, Summer 2009
Memories from a wonderful travel thru central and southern Italy. Found a secret treasure of joy and beauty exceeding all hopes. First row: lake Trasimeno's shore in Tuoro; palazzo dei Priori and fontana maggiore in Perugia; Todi. Second row: villa di Adriano in Tivoli; light frames in Otranto; the triumph of baroque in the basilica di Santa Croce (Holy Cross), Lecce; a street in Alberobello. Third row: Polignano a mare; the magic city of Otranto again, with its cathedral; Ostuni's cathedral. Fourth row: via Appia in Perugia; the water sprawl at the feet of Marmore waterfalls; a narrow road leading to the terrace over Otranto's harbour; Polignano a mare again. Fifth row: piazza del Popolo and piazza Arringo in Ascoli Piceno; Marmore waterfalls.
"A thing of beauty is a joy for ever" (John Keats)
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My Documents
Soluzione dell'equazione di Fourier per per piastra sottile con sorgente termica in moto uniforme. The solution of Fourier equation for transient, conductive heat transfer, that is, a parabolic, linear, second order PDE, for a uniformly moving heat source. By using a moving reference frame the solution is trivial in a 3D space. In a 2D plane Bessel functions are needed, the basic idea and method being the same. In italian. [download]
Second Order Nurbs Interpolation of Real Affine and Projective Plane Curves. Usually third order (or higher) Nurbs are used as a standard in computer graphics. Obviously, no one forbids to use a simpler approximation. The outcome is particularly beautiful when interpreted in the sense of projective geometry. Unfortunately I was too long. I hope this could serve for those new to Nurbs theory. [download]
>> topMy Favorite Books
What Is Mathematics?
Richard Courant and Herbert Robbins, What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, Oxford, 1941.I read it (actually the italian translation, "Che cos'è la matematica? Introduzione elementare ai suoi concetti e metodi", by Boringhieri) in the summer following my diploma and immediately thereafter I understood what to do for a living. Richard Courant, one of the greatest mathematician of the XX century, drives the reader in a trip thru the basic and appealing topics of mathematics. Also relatively advanced topics are seen from an elementary - though rigorous - viewpoint, providing a strong intuitive insight and an often amusing approach, useful also to the trained mathematician. Really a pretty good book!
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Gödel, Escher, Bach
Douglas R. Hoftstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, Inc., 1979.Also in this case I read the italian translation ("Gödel, Escher, Bach: un'Eterna Ghirlanda Brillante - Una fuga metaforica su menti e macchine nello spirito di Lewis Carroll", by Adelphi). It was a pleasant summer reading, showing interconnections between mathematical logic (Kurt Gödel), art (M.C. Escher) and music (Johann Sebastian Bach). I am an ignorant in musical theory, so I didn't get the point for Bach (the concept is simple and clear, it's a mistery how this should sound to the ear). The paradox of Epimenides from Knossos (the liar paradox), the paradox of Zeno from Elea about Achilles and the tortois, the famous incompleteness theorem for any axiomatic system capable to code arithmetic, theorems, meta-theorems, meta-meta-theorems, ... , (meta-)^k theorems, wishes, meta-wishes, meta-meta-whishes, ... , (meta-)^k wishes, artificial intelligence, self knowledge as a consequence of complexity, etc.
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Flatland
Edwin A. Abbott, Flatland - A Romance of Many Dimensions.Very quick and amusing to read. Wonderful idea. It's the story of an inabitant of a 2D world that experiences the orthogonal crossing of a 3D sphere (seeing its size varying 2D section, of course!). The poor triangle is believed mad and banished from the community. The book inspired a series of similar novels with a bit more topological flavor. When I studied the topology of surfaces I read the story about 2D explorers living on a non orientable 2-manifold, leaving for a journey to the unknown lands where no one has ever dared to venture and coming back reversed from the opposite side. Italian translation: Flatlandia - Racconto fantastico a più dimensioni, by Adelphi.
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Number Theory
Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer, London, 1998.A nice introduction to the fascinating realm of number theory. A valuable merit of this book is being extremely simple and clear while being at the same time rigorous and complete (as regards the issues covered, obviously not the whole Number Theory). Elementary number theory and a little analytic number theory are considered. The beginning of the book is centered on modular arithmetic and related classical topics (primes and the fundamental theorem of arithmetic, congruences, Diophantine equations, the Chinese remainder theorem, Euler phi function, public key cryptography). Then a little more algebra is introduced, with groups of units and subgroups of quadratic residues (and their algebraic decomposition), Legendre symbol and the law quadratic reciprocity. The best part is related to Riemann zeta function, Möbius inversion formula, Dirichlet series and Euler products. A little boring chapter on the sums of squares and one on Fermat's last theorem close the book. I had fun reading it and I suggest it to whoever wants to explore the field by himself and starting from the scratch.
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[to be continued...]
My Favorite Quotations
- "Life is a pendolum oscillating in between boredom and sorrow"
(A. Schopenauer) - "Is a dream a lie if it don't come true, or is it something worse?"
(B. Springsteen, The River) - "Only the strong survive"
(C. Darwin) - "He who takes the coast will never discover new oceans"
(F. de Magallanes) - "Qui auget scientiam auget dolorem"
(Qoelet 1,18) - "The shortest path in between two true statements in the real field passes thru the complex field"
(J. Hadamard) - "Forgive your enemies, but remember their names"
(J.F. Kennedy) - "Fácilmente aceptamos la realidad, acaso porque intuimos que nada es real"
(J.L. Borges, in "El inmortal") - "I know, I know, a dirty world"
(K.D. Cobain, Smells Like Teen Spirit) - "We adore chaos because we love to produce order"
(M.C. Escher)
I can die happy because...
1. Math is the joy of endlessly discovering
Mandelbrot set.
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2. I've seen the greatest of all times playing
Here's MJ bringing number 6 to the windy city.
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3. I was 17 when the Grunge revolution exploded
Some term Generation X all those people born from 1965 to 1980. To cut a long story short, we are those come too late. Sons of the economic wealth in the western world after WW2, but destined to end our life in the inescapable decadence of humanity, and to assist in powerlessness resignation to the death of our planet, the wars for survival, the ecological catastrophe, the death of multitudes. Life at the times of the Roman Empire's fall, in a climate of generalized mistrust in the future. Maybe our fathers, the baby boomers, won't ever perceive or share the pessimism, the sense for death, for the apocalypse short too come, the nichilism, the despair, the sickness, the disillusion, the rebellion and the self distructive attitude of the music we loved. Nor our younger friends, the Generation Yers, plastified in the big emptyness come thereafter. Pearl Jam, Nirvana, Alice in Chains, Soundgarden and the multitude of other alternative and indie rock bands are the soundtrack of our lives. They made Seattle the capital of music, it was the revenge of underground over commercial mass culture. It was all a breath, a very short while. The movement singing self destruction finally, genuinely, truthfully ended in self destruction. In the end, it could not be but so.
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4. I saw the Boss live in concert
Bruce Springsteen and the Seeger Session Band, Milano 12.MAY.2006.
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5. The greatest happiness lies in the purest moments
"Two years he walks the earth. No phone, no pool, no pets, no cigarettes. Ultimate freedom. An extremist. An aesthetic voyager whose home is the road. Escaped from Atlanta. Thou shalt not return, because «the West is the best». And now after two rambling years comes the final and greatest adventure. The climactic battle to kill the false being within and victoriously conclude the spiritual pilgrimage. Ten days and nights of freight trains and hitchhiking bring him to the Great White North. No longer to be poisoned by civilization he flees, and walks alone upon the land to become lost in the wild."
Christopher McCandless, May 1992.
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Important warning!
