#thomas banchoff

[siryl]

“Klein Bottle” and “True Banchoff Klein Bottle” by Clifford Pickover.

[wozziebear]

These and other surprising examples made it clear that mathematicians needed to prove that dimension is a real notion and that, for instance, n- and m-dimension Euclidean spaces are different in some fundamental way when n ≠ m. This objective became known as the “invariance of dimension” problem.

Finally, in 1912, almost half a century after Cantor’s discovery, and after many failed attempts to prove the invariance of dimension, L.E.J. Brouwer succeeded by employing some methods of his own creation. In essence, he proved that it is impossible to put a higher-dimensional object inside one of smaller dimension, or to place one of smaller dimension into one of larger dimension and fill the entire space, without breaking the object into many pieces, as Cantor did, or allowing it to intersect itself, as Peano did. Moreover, around this time Brouwer and others gave a variety of rigorous definitions, which, for example, could assign dimension inductively based on the fact that the boundaries of balls in n-dimensional space are (n − 1)-dimensional.

Although Brouwer’s work put the notion of dimension on strong mathematical footing, it did not help with our intuition regarding higher-dimensional spaces: Our familiarity with three-dimensional space too easily leads us astray. As Thomas Banchoff wrote, “All of us are slaves to the prejudices of our own dimension.”

Suppose, for instance, we place 2^n spheres of radius 1 inside an n-dimensional cube with side length 4, and then put another one in the center tangent to them all. As n grows, so does the size of the central sphere — it has a radius of √n − 1. Thus, shockingly, when n ≥ 10 this sphere protrudes beyond the sides of the cube.

The surprising realities of high-dimensional space cause problems in statistics and data analysis, known collectively as the “curse of dimensionality.” The number of sample points required for many statistical techniques goes up exponentially with the dimension. Also, as dimensions increase, points will cluster together less often. Thus, it’s often important to find ways to reduce the dimension of high-dimensional data.

The story of dimension didn’t end with Brouwer. Just a few years afterward, Felix Hausdorff developed a definition of dimension that — generations later — proved essential for modern math. An intuitive way to think about Hausdorff dimension is that if we scale, or magnify, a d-dimensional object uniformly by a factor of k, the size of the object increases by a factor of k^d. Suppose we scale a point, a line segment, a square and a cube by a factor of 3. The point does not change size (3^0 = 1), the segment becomes three times as large (3^1 = 3), the square becomes nine times as large (3^2 = 9) and the cube becomes 27 times as large (3^3 = 27).

One surprising consequence of Hausdorff’s definition is that objects could have non-integer dimensions. Decades later, this turned out to be just what Benoit B. Mandelbrot needed when he asked, “How long is the coast of Britain?” A coastline can be so jagged that it cannot be measured precisely with any ruler — the shorter the ruler, the larger and more precise the measurement. Mandelbrot argued that the Hausdorff dimension provides a way to quantify this jaggedness, and in 1975 he coined the term “fractal” to describe such infinitely complex shapes.

To understand what a non-integer dimension might look like, let’s consider the Koch curve, which is produced iteratively. We begin with a line segment. At each stage we remove the middle third of each segment and replace it with two segments equal in length to the removed segment. Repeat this procedure indefinitely to obtain the Koch curve. Study it closely, and you’ll see it contains four sections that are identical to the whole curve but are one-third the size. So if we scale this curve by a factor of 3, we obtain four copies of the original. This means its Hausdorff dimension, d, satisfies 3^d = 4. So, d = log3(4) ≈ 1.26. The curve isn’t entirely space-filling, like Peano’s, so it isn’t quite two-dimensional, but it is more than a single one-dimensional line.

A Mathematician's Guided Tour Through Higher Dimensions

https://www.quantamagazine.org/a-mathematicians-guided-tour-through-high-dimensions-20210913/ Comments

[hu-and-al]

PICASSO, DALÍ, DUCHAMP E O HIPERESPAÇO

Era improvável que o tuberculoso, penosamente pobre e patologicamente tímido Georg Bernhard Riemann, um homem que sofria de repetidos colapsos nervosos, pudesse desencadear uma completa revolução no pensamento científico e cultural. Mas em junho de 1854, na Universidade de Gottingen na Alemanha, Riemann fez uma célebre apresentação de seu ensaio "Sobre as hipóteses que residem nos fundamentos da geometria" expondo ao mundo as propriedades do espaço de múltiplas dimensões e fazendo desmoronar a geometria euclidiana que havia vigorado por dois mil anos.

A revolução riemanniana influenciaria fortemente as artes, a literatura e a filosofia na Europa pelas décadas que se seguiram. Sessenta anos após a conferência, Einstein usaria a geometria de Riemann para explicar a evolução do universo e 130 anos depois, os físicos usariam a geometria de 10 dimensões como base para uma das mais fortes candidatas à teoria de tudo. O cerne do trabalho do matemático foi o entendimento de que as leis da física se tornam mais simples num espaço de múltiplas dimensões.

Nossos sentidos são capazes de perceber apenas 3 dimensões: comprimento, largura e altura. Sendo assim, um objeto no espaço pode se mover apenas para frente e para trás, para cima e para baixo e para os lados. As ideias de Riemann fizeram com que os físicos passassem a considerar que o espaço pode sofrer mudanças na sua densidade, distorções e ondas que sejam provocadas por eventos que ocorrem em dimensões imperceptíveis aos nossos sentidos. Numa analogia simples, assim como os peixes em um lago estão limitados a perceber o lago como sendo todo o seu universo, nossos cérebros estão limitados a perceber o espaço tridimensional como sendo tudo que há.

Entre 1870 e 1920, o interesse por dimensões adicionais, em especial pela "quarta dimensão", atingiu o ápice e conquistou a imaginação popular. Oscar Wilde, Dostoiévski e Marcel Proust foram alguns dos escritores que trouxeram a ideia de uma quarta dimensão para dentro de suas obras. O matemático Charles L. Dodgson, que era professor na Universidade de Oxford, se eternizou na literatura e no folclore infantis com o pseudônimo de Lewis Carroll, incorporando as ideias matemáticas de Riemann em Alice no país das maravilhas. O buraco do coelho de Lewis e o espelho de Alice são interpretações de buracos de minhoca, portais para outro universo adaptados para o público infantil.

A quarta dimensão também inspirou enormemente as obras de Picasso, Marcel Duchamp, Miró, Kandinsky e Dalí e teve extrema influencia no desenvolvimento do Cubismo e do Expressionismo, dois dos mais atuantes movimentos artísticos do século XX. Na época, os artistas interpretavam a quarta dimensão como uma dimensão espacial (diferente da dimensão temporal que conhecemos pós teoria da relatividade). Acreditava-se que, se alguém pudesse se transportar para a quarta dimensão, seria capaz de ver todas as perspectivas de um evento ao mesmo tempo. Mas como projetar essas perspectivas na a tela? A resposta de Picasso para o dilema foi, obviamente, o Cubismo.

Dalí passou a procurar o matemático Thomas Banchoff para se aconselhar e buscou inspiração na física teórica sobre a quarta dimensão até a sua morte, em 1989. Em 1958, ele escreveu em seu manifesto: "no período surrealista, eu queria criar a iconografia do maravilhoso mundo interior do meu pai Freud...Hoje, o mundo exterior da física transcendeu o da psicologia. Meu pai hoje é o Dr. Heisenberg."

[primainfusione-blog]

Io non bevo tè, il tè sono io: il tea time con Salvador Dalí

Dalí e il mondo del tè? A prima vista, sul web, non sembrano avere molto in comune. Mentre i suoi baffi all’insù popolano i feed di Pinterest e Tumblr, di immagini della serie Celebrities che amano bere tè ne trovi a malapena una. Ma nella rete come nella vita reale il genio catalano è unico e fa specie a sé. Come recita uno dei suoi aforismi più celebri: “Io non mi drogo, la droga sono io”, che abbiamo parafrasato con: “Io non bevo tè, il tè sono io”. Perché sì, Dalí e il mondo del tè hanno un legame, ed è un legame stravagante, onirico e visionario come l’arte stessa del Maestro.

“50 dollari per aver interrotto il mio tea time. In contanti”

L’aneddoto lo racconta Alan Klevit, critico d’arte e colonnista americano, nel suo libro The Art Beat. Ecco il brano intero (la traduzione dall’inglese è mia):

Quel giorno indossava una camicia arruffata in pizzo bianco con uno spillo di diamanti e un mantello nero foderato di bianco che drappeggiava intorno alle spalle. Aveva un bastone da passeggio in lacca nera con un enorme manico in argento intagliato. Mentre sorseggiava il tè, teneva il mignolo rivolto verso il cielo mettendo così in mostra un anello ornato grande come una noce.

Un uomo si avvicinò al tavolo di Dalí e chiese: “Signor Dalí, posso avere il suo autografo?” Non era per nulla preparato a quello che accadde dopo.

“Non sono un ‘signore’. Sono il Maestro. Il prezzo per aver interrotto il mio tè e per ricevere il mio autografo è di cinquanta dollari. Cash.” Detto questo, Dalí posò la tazza e stese la mano con il palmo all’insù. L'uomo si girò per guardare, da sopra la sua spalla, la moglie. Lei annuì e lui posò tre banconote da venti sul palmo della mano di Dalí. Dalí le mise sul tavolo, tirò fuori una penna dal taschino della camicia e firmò su un tovagliolo che aveva di fronte. Rendendosi conto che Dalí non ha alcuna intenzione di dargli il resto l’uomo si allontanò, borbottando un grazie. Dalí fece un cenno al cameriere e gli diede le tre banconote da venti.

12 eliografie per Alice in Wonderland, edizione limitata e autografata del ‘69

Nel 1969, la casa editrice statunitense Random House commissiona a Salvador Dalí la realizzazione delle illustrazioni per una piccola ed esclusiva edizione di Alice nel Paese delle Meraviglie. Il volume viene presentato come “libro del mese”. Contiene 12 incisioni fotografiche (eliografie), una per ogni capitolo, e un disegno in acquaforte per il frontespizio firmato dallo stesso Dalí.

L’edizione ’69 è una piccola perla per i collezionisti, ricercatissima. Attualmente il suo prezzo online si aggira intorno ai 12 mila euro... Gulp. Ma abbiamo buone notizie per chi ha un budget più modesto. Dopo più di un mezzo secolo, le preziose illustrazioni di Dalí tornano con l’edizione speciale del 2015 che la Princeton University Press ha pubblicato in occasione dei 150 anni del libro di Carroll.

“/.../ quale tributo migliore se non quello di prendere il testo finale che si è sviluppato nel corso della vita di Carroll e che è generalmente considerato, dai carrolliniani e dallo stesso Carroll, la versione più autentica e corretta, e abbinarlo alle splendide illustrazioni contemporanee di Salvador Dali, finora disponibili solo in una edizione straordinariamente rara e costosa? – scrive il presidente della Lewis Carroll Society of North America Mark Burston nell’introduzione che precede il libro (la traduzione dall’inglese è mia) – In questo modo riusciamo mantenere intatto il sapore ottocentesco, mentre voi, lettori del ventunesimo secolo, potete godervi le immagini di uno dei più significativi artisti del Novecento.”

Oltre all’introduzione di Burston, il volume contiene l’articolo del matematico Thomas Banchoff, che ha avuto l’occasione di conoscere e collaborare con il Maestro.

L’omaggio a Dalí: le teiere d’arte di Noi Volkov

E per concludere, ecco due immagini delle teiere con i baffi daliniani realizzate dalla mano di Noi Volkov, ceramista e pittore russo noto per le sue ceramiche surrealiste.

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[integrrl]

“...Both Carroll and Dalí also had a connection in their interest in math. Carroll was a lecturer in mathematics at Oxford’s Christ Church, and Dalí often embedded mathematic principles in his work, such as the golden ratio in “The Sacrament of the Last Supper” (1955) and the hypercube in “Crucifixion (Corpus Hypercubus)” (1954). In an essay for the new edition, Thomas Banchoff, professor emeritus of mathematics at Brown University, describes his personal interactions with Dalí, who once told him his paintings were inspired by metaphysics and medieval polymath Ramon Llull...”

Read more

Buy the new edition

[deonstri05-blog]

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[lsakclifhypercube-blog]

The hypercube became a recurring theme within many artworks and science fiction literature within the 20th century (Robert A. Heinlein’s "And He Built a Crooked House" being the most influential during this time).However the most relevant pieces of art and one of the most fascinating artistic styles that reshaped the ways in which the eight cube tesseract was perceived was through Salvador Dali’s ‘Crucifixion (Corpus Hypercubus)’.Dali’s initial inspiration for Corpus Hypercubus came from his change in artistic style during the 1940s and 1950s. During this time, Dali’s interest in traditional surrealism diminished and he became fascinated with nuclear science. His interest grew from the bombing of Hiroshima at the end of World War II, which left a lasting impression on him. In his 1951 essay "Mystical Manifesto", he introduced an art theory he called "nuclear mysticism" that combined Dali’s interests in Catholicism, mathematics, science, and Catalan culture in an effort to re-establish classical values and techniques, which he extensively exhibits within in Corpus Hypercubus. Dali is no foreigner when it comes abstract thinking and illusive thought. Dali spent an intriguing about of time collaborating with mathematics and scientists alike in order to influence and shape the context behind his work. The effects of ‘Crucifixion’ became notably apparent within mathematics and had a lasting effect on mathematicians who worked directly with Dali during his time. Thomas Banchoff was one of Dali’s many partners in the scientific and mathematical communities. These influences appear in much of his work, ranging from themes of quantum mysticism and DNA to his use of stereoscopic painting techniques.When Dali died, he was collaborating with Banchoff on plans for a highly mathematical sculpture that, had it come to fruition, would have placed a horse’s rear end on the moon. By the end of his career Dali was one considered, and still to this day, one of the most celebrated painters of the 20th century. Spending lifetime of pursuing what some people consider to be opposing intellectual territories, led him to create some of his most renowned works

[scenestertv]

(Los Angeles—May 24, 2018) The Los Angeles County Museum of Art (LACMA) presents 3D: Double Vision, the first North American survey of 3D objects and practices, tracing cycles of optical investigation, creative expression, and commercial popularity over the past 175 years. Featuring artifacts of mass culture alongside historic and contemporary art, 3D: Double Vision addresses the nature of perception, the allure of illusionism, and our relationship to accompanying technologies and apparatuses. The more than 60 artworks featured in the exhibition activate this process by means of mirrors, lenses, filters, or movement—requiring active participation on the part of spectators to complete the illusion.

Many 3D media are included in the exhibition—from stereoscopic photography, film, video, anaglyph printing, and computer animation, to the glasses-free formats of holography and lenticular—alongside 2D works. The exhibition is curated by Britt Salvesen, Head of the Wallis Annenberg Photography Department and the Prints and Drawings Department at LACMA.

3D: Double Vision is organized in five thematic sections, which trace the generational cycles of 3D from the 19th century to the present day.

Some 3D formats, such as lenticular and holography, do not require glasses. The exhibition also includes 2D works that generate depth effects through motion.

More than 60 artists are featured in these exhibition, including Los Angeles artist Simone Forti, who created animated integral (or multiplex) holograms of herself in the mid-1970s in collaboration with the technique’s inventor, Lloyd Cross. Simone’s Striding (1975–78) is one of eight multiplex holograms recently rediscovered in the artist’s studio.

The exhibition is accompanied by a catatogue, with an essay by Britt Salvesen and contributions by Thomas F. Banchoff, Eric Drysdale, Erkki Huhtamo, Zach Rottman, and Gloria Sutton. Co-published by LACMA and DelMonico Books • Prestel. Each book includes a pair of anaglyph glasses and a stereo card viewer, allowing the reader to conjure captivating virtual images.

The exhibition presents a variety of stereoscopic formats, delivering slightly different images to the two eyes by means of mirrors, lenses, polarized filters, or colored filters, as in the familiar red-blue anaglyph glasses.

The optical principle underlying all 3D media is binocular vision—the process by which our brains synthesize the information received by our two eyes into a single, volumetric image. Visitors are invited to use 3D glasses and devices throughout the exhibition. The exhibition does not include any VR or 3D printed works.

LACMA Presents 3D: Double Vision #ArtScene #LACMA (Los Angeles—May 24, 2018) The Los Angeles County Museum of Art (LACMA) presents 3D: Double Vision…

[spdfilmography]

Harald Johnson, Mastering Digital Printing: The Photographer's and Artist's Guide to High-Quality Digital Output (2003)

Any compartmentalization of my life begins to break down here (1989).

Before I left Goldsholl Design and Film (1987), Mort had given me a copy of an article announcing the invention of Stereolithography.  I had attended a 1984 Conference on the study of Polyhedra at Smith College called “Shaping Space”.  I met Thomas Banchoff there, who was famous for making computer-generated movies of the Klein Bottle rotating in four dimensions.

I was working at the Graphics department of The Post Group in Hollywood.  We shared a building with Electric Paint, a commercial print graphics division.  I developed a proposal for an exhibition of mathematical sculpture -- Artifacts from Cyberspace -- made via rapid prototyping (3D printing).  Linda Rheinstein allowed me to print my digital images on the Iris Graphics ink-jet printers at Electric Paint.  Color 2D printing was still a big deal in those days.  Large-format 2D printing still is.

I sent a copy of my exhibition proposal to Stephen Wolfram, who invited me to Champaign, IL for several weeks to develop illustrations for the Second Edition of the Mathematica Book, which coincided with release 2.0 of the software.   One of these images wound up in Illustration: Judges' Choices, Print Magazine / IDEA Computer Art and Design Annual, Number 1, Print, Volume 45, Number 7, November, 1991.  

So, there was interest in digital art at that time.  I got connected with Mary Lou Bock at the Williams Gallery in Princeton, NJ.  I printed a bunch of large-format prints on Arches archival watercolor paper -- Not at Nash Editions, but at a competitor’s in Anaheim Hills, I think it was.  Later, as color 3D printing came on-line at Z-Corp in 1999, it became obvious that the inks they were using were not colorfast and were fading  rapidly.  So, I started bugging Mac Holbert at Nash Editions about the possibility of hybridizing their mineral pigment inks with 3D printers.

[georgetownuniversitypress]

The Jesuits and Globalization: Historical Legacies and Contemporary Challenges by Thomas Banchoff and Jose Casanova, Editors 

[sarinamitchel]

I made a lot of shapes for my AMAZING math class, taught by Prof. Thomas Banchoff. 

Cutting styrofoam shapes was extremely helpful in visualizing many of the ideas we're learning about, like how you can fit a tetrahedron inside of a cube and have the vertices overlap. 

[georgetownuniversitypress]

The Jesuits and Globalization edited by Thomas Banchoff and José Casanova

The Society of Jesus, commonly known as the Jesuits, is the most successful and enduring global missionary enterprise in history. Founded by Ignatius Loyola in 1540, the Jesuit order has preached the Gospel, managed a vast educational network, and shaped the Catholic Church, society, and politics in all corners of the earth. Rather than offering a a global history of the Jesuits or a linear narrative of globalization, Thomas Banchoff and José Casanova have assembled a multidisciplinary group of leading experts to explore what we can learn from the historical and contemporary experience of the Society of Jesus—what do the Jesuits tell us about globalization and what can globalization tell us about the Jesuits?