ART & ORGANISM
ATTRIBUTES of ART: notes on MATHEMATICS
UNDER CONSTRUCTION – MANY LINKS INCOMPLETE
an observation unexpected by most art lovers:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty –a beauty cold and austere, like that of sculpture.”
(Bertrand Russell Philosophical Essays (1910) no. 4)
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DO the AESTHETIC SENSIBILITIES IN ART AND MATHEMATICS COMPARE?
How deeply can we look? Read Newman (2014)
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Can we reconcile the cool logic of mathematics with the warm emotionality of creative art?
Is this an issue of logic and emotion, sapience and sentience?
Can mathematics be “naturalized”? (made it continuous with nature?)
Did Fibonacci (for example) show that nature and mathematics are continuous?
…or is this our idealization of imperfect perceptions?
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Is mathematics intrinsically beautiful? (or is it another dimension of how we interpret the world that can be perceived in very pleasing ways? The work of some artists have been found to expresses mathematical principles or regularities–have they converged on them intuitively?
Is the necessary “truth” of mathematical configurations necessarily beautiful (in the sense of the poet John Keats)(and look at Dennis Overbye’s comments, below)
Are the aesthetic representations of mathematical statements more-or-less useful … might they be complementary?
What is the interplay of FIXED and FLEXIBLE elements of our nature? How is our nature (including our tastes, and what we are more or less attracted to) determined? — this is the issue of determinism.
The view that BEAUTY could be viewed OBJECTIVELY is troubling to some people, who see it as a bastion of a unique dimension of subjectivity.
Some mathematical regularities seem deeply connected to what holds our attention, what we find attractive — these are a priori in that they precede the forming of taste by our culture, our parents and peers, and our experiences. Could this be related to the way in which our brains operate rather than to some quality in the phenomenon perceived? Mathematical regularities may not define our taste or aesthetic choices, but they are at least powerful influences on them.
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Mathematical Beauty. Visual Mathematics, Jackson Pollack, chaos, fractals
and The Mandelbrot set: watch a You Tube of an accessible description of its discovery and development
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MATHEMATICS MATHEMATICS AND MUSIC:
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. CAN BEAUTY BE MATHEMATICALLY REPRESENTED? is MATHEMATICS BEAUTIFUL? is MATHEMATICS an INTRINSIC ATTRIBUTE of BEAUTY?
The logarithmic spiral: its mathematical regularities appear to involve intrinsic attractiveness .
. Emerging scholarship, enlarging interdisciplinary connections. For example, take a look at:
Abstract: It is not unanimous among scientists if there is beauty in science. Some deny it. Mental clarity of conclusions when captured in simple looking equations is mathematical beauty. This we also find in the Euclidian geometry when performing the Golden Section and by deriving the Golden or Devine Number in golden rectangles, spirals and the Golden Angle. The Golden Section is considered as most beautiful and used in architecture and art. It is found everywhere in nature, e.g. in the pentagram of flowers, in the spirals of the shells of snails and Nautilus and even in galaxies of space. The Golden Angle in plants is realized in the phyllotaxis of spirals of leaf rosettes, in fruit stands and in the cones of conifers and cycads. It optimizes packing of modules such as seeds and fruits as well as the capture of light by leaves for photosynthesis and the fitness of productivity. Although we can mathematically deduce it and scientifically explain its role in organization and formation of patterns of structure and function, we cannot explain why we find it beautiful. In a methodological dualism esthetics and beauty are transcendental categories besides science. Or are the pleasant sensations of the Golden Section elicited by different stimuli to which our brain is adapted? Perhaps the Golden Section found everywhere in the entire universe is a link between natural science and the transcendental dimension, while a flower of a rose remains both a complex scientific system and an object of overwhelming beauty.” Keywords: Golden angle; Golden number; Fibonacci-series; Light capture; Phyllotaxis; Transcendence
Abstract. HUMANS and certain other species find symmetrical patterns more attractive than asymmetrical ones. These preferences may appear in response to biological signals1–3, or in situations where there is no obvious signalling context, such as exploratory behaviour4,5 and human aesthetic response to pattern6–8. It has been proposed9,10 that preferences for symmetry have evolved in animals because the degree of symmetry in signals indicates the signaller’s quality. By contrast, we show here that symmetry preferences may arise as a by-product of the need to recognize objects irrespective of their position and orientation in the visual field. The existence of sensory biases for symmetry may have been exploited independently by natural selection acting on biological signals and by human artistic innovation. This may account for the observed convergence on symmetrical forms in nature and decorative art11. (endnotes to references in original paper on-line)
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from Cracking The Da Vinci Code By Keith Devlin / Photography by Dan Winters / DISCOVER Vol. 25 No. 06 | June 2004 |
. GOLDEN MEAN SOME aspects of the Fibonacci series in ancient aesthetics might be more myth than history: ” . . . the entire story about the Greeks and the golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value. But he seemed more interested in the mathematics than the architecture, for he gave the golden ratio a decidedly unromantic label: extreme and mean ratio. The term divine proportion first appeared with the publication of the three-volume work of that name by the 15th-century mathematician Luca Pacioli. Calling f “golden” is even more recent: In 1835 it appeared in a book written by the mathematician Martin Ohm” (from Cracking The Da Vinci Code By Keith Devlin / Photography by Dan Winters / DISCOVER Vol. 25 No. 06 | June 2004 . . . [more] . BEAUTY and MATHEMATICS — explore Fibonacci numbers and all their ramifications at https://fibonacci.com/nature-golden-ratio/
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PATTERNS
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“All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section.” (Christopher D. Green (1994) Perception, 24, 937-968.) ” . . . there seems to be, in fact, real psychological effects associated with the golden section, but that they are relatively sensitive to careless methodological practices.” [complete article] . | |
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FRACTALS
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The Most Seductive Equation in Science: Beauty Equals Truth By DENNIS OVERBYE (NYTimes on the Web March 26, 2002) In the fall of 1915, Albert Einstein, living amid bachelor clutter on coffee, tobacco and loneliness in Berlin, was close to scrawling the final touches to a new theory of gravity that he had pursued through mathematical and logical labyrinths for nearly a decade. But first he had to see what his theory had to say about the planet Mercury, whose puzzling orbit around the Sun defied the Newtonian correctness that had long ruled the cosmos and science. The result was a kind of cosmic “boing” that changed his life. Einstein’s general theory of relativity, as it was known, described gravity as warped space-time. It had no fudge factors —— no dials to twiddle. When the calculation nailed Mercury’s orbit Einstein had heart palpitations. Something inside him snapped, he later reported, and whatever doubt he had harbored about his theory was transformed into what a friend called “savage certainty.” He later told a student that it would have been “too bad for God,” if the theory had been subsequently disproved. The experience went a long way toward convincing Einstein that mathematics could be a telegraph line to God, and he spent most of the rest of his life in an increasingly abstract and ultimately fruitless pursuit of a unified theory of physics. Rare indeed is the scientist who has not at one point or other been seduced by the beauty of his own equations and dumbfounded by what the physicist Dr. Eugene Wigner of Princeton once called the “unreasonable effectiveness of mathematics” in describing the world. The endless fall of the moon, the fairy glow of a rainbow, the crush of a nuclear shock wave are all explicable by scratches on a piece of paper, that is to say, equations. Every time an airplane safely touches down on time, a computer boots up, or a cake comes out right, the miracle is recreated. “The most incomprehensible thing about the universe is that it is comprehensible,” Einstein said. Math is the language of physics, but is it the language of God? Mathematicians often say that they feel as if their theorems and laws have an objective reality, like Plato’s perfect realm of ideas, which they do not create or construct as much as simply discover. But the equating of math with reality, others say, consigns vast arenas of experience to the darkness. There are no mathematical explanations yet for life, love or consciousness. “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality,” said Einstein. He maintained that it should be possible to explain scientific principles in words to a child, but his followers often argue that words alone cannot convey the glories of physics, that there is a beauty apparent only to the mathematically adept. That inhuman beauty has long been a lodestone for physicists, says Dr. Graham Farmelo, a physicist at the Science Museum in London and an editor of “It Must Be Beautiful: Great Equations of Modern Science.” “You can write it on the palm of your hand and it shapes the universe,” Dr. Farmelo said of Einstein’s gravitational equation, the one that produced heart palpitations. He compared the feeling of understanding such an equation to the emotions you experience “when you take possession of a great painting or a poem.” In the hopes of getting the rest of us to take possession some of our intellectual heritage, Dr. Farmelo recruited scientists, historians and science writers to write about the life and times of 11 of the most powerful or notorious equations of 20th century science. The book is partly a meditation on mathematical beauty, possibly a difficult concept for many Americans right now as they confront their tax forms. But as Dr. Farmelo noted in an interview, even the most recalcitrant of us have had glimpses of mathematical grace when, say, our checkbooks balanced. Imagine that your withholdings always turned out to be exactly equal to the tax you wind up owing. Or that your car’s odometer turned over to all zeros every year on your birthday no matter how far you thought you had driven. Such occurrences would be evidence of patterns in your financial affairs or driving habits that might be helpful in preparing tax returns or scheduling car maintenance. The pattern most highly prized in recent modern physics has been symmetry. Just as faces and snowflakes are prettier for their symmetrical patterns, so physical laws are considered more beautiful if they keep the same form when we change things by, for example, moving to the other side of the universe, making the clocks run backward, or spinning the lab around on a carousel. A good equation, Dr. Farmelo said, should be an economical compression of truth without a symbol out of place. He looks for attributes like universality, simplicity, inevitability, an elemental power and “granitic logic” of the relationships portrayed by those symbols. There is, for example, Einstein’s E=mc², which Dr. Peter Galison, a Harvard historian and physicist, describes in the book as “a metonymic of technical knowledge writ large,” adding, “Our ambitions for science, our dreams of understanding and our nightmares of destruction find themselves packed into a few scribbles of the pen.” When it comes to the quest for beauty in physics, even Einstein was a piker compared with the British theorist Paul Dirac, who once said “it is more important to have beauty in one’s equations than to have them fit experiment.” An essay by Dr. Frank Wilczek, a physics professor at the Massachusetts Institute of Technology, recounts how the 25-year-old Dirac published an equation in 1928 purporting to describe the behavior of the electron, the most basic and lightest known elementary particle at the time. Dirac had arrived at his formula by “playing around” in search of “pretty mathematics,” as he once put it. Dirac’s equation successfully combined the precepts of Einstein’s relativity with those of quantum mechanics, the radical rules that prevail on very small scales, and it has been a cornerstone of physics ever since. But there was a problem. The equation had two solutions, one representing the electron, another representing its opposite, a particle with negative energy and positive charge, that had never been seen or suspected before. Dirac eventually concluded that the electron (and it would turn out every other elementary particle) had a twin, an antiparticle. In Dirac’s original interpretation, if the electron was a hill, a blob, in space, its antiparticle, the positron, was a hole —— together they added to zero, and they could be created or destroyed in matching pairs. Such acts of creation and annihilation are now the main business of particle accelerators and high-energy physics. His equation had given the world its first glimpse of antimatter, which makes up, at least in principle, half the universe. The first antimatter particle to be observed, the antiproton, was found in 1932, and Dirac won the Nobel Prize the next year. His feat is always dragged forth as Exhibit A in the argument to show that mathematics really does seem to have something to do with reality. “In modern physics, and perhaps in the whole of intellectual history, no episode better illustrates the profoundly creative nature of mathematical reasoning than the history of the Dirac equation,” Dr. Wilczek wrote. In hindsight, Dr. Wilczek writes, what Dirac was trying to do was mathematically impossible. But, like the bumblebee who doesn’t know he can’t fly, through a series of inconsistent assumptions, Dirac tapped into a secret of the universe. Dirac had started out thinking of electrons and their opposites, the “holes,” as fundamental entities to be explained, but the fact that they could be created and destroyed meant that they were really evanescent particles that could be switched on and off like a flashlight, explains Dr. Wilczek. What remains as the true subject of Dirac’s equation and as the main reality of particle physics, he says, are fields, in this case the electron field, which permeate space. Electrons and their opposites are only fleeting manifestations of this field, like snowflakes in a storm. As it happens, however, this quantum field theory, as it is known, must jump through the same mathematical hoops as Dirac’s electron, and so his equation survives, one of the cathedrals of science. “When an equation is as successful as Dirac’s, it is never simply a mistake,” Dr. Steven Weinberg, a 1979 Nobel laureate in physics from the University of Texas, writes in an afterword to Dr. Farmelo’s book. Indeed, as Dr. Weinberg has pointed out in an earlier book, the mistake is often in not placing enough faith in our equations. In the late 1940’s, a group of theorists at George Washington University led by Dr. George Gamow calculated that the birth of the universe in a Big Bang would have left space full of fiery radiation, but they failed to take the result seriously enough to mount a search for the radiation. Another group later discovered it accidentally in 1965 and won a Nobel Prize. Analyzing this lapse his 1977 book, “The First Three Minutes,” Dr. Weinberg wrote: “This is often the way it is in physics. Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world.” . |
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RESONANCES BETWEEN ART and NATURE: might real or perceived resonances with fundamental qualities of the universe as revealed by science (looking with a competence or precision or resolution beyond that of our ordinary senses) evoke or reinforce a sense of order, of harmony, of personal “belonging” in the world. Is this what conveys “beauty” and a coherence that enhances whatever sense of “truth” is immanent, emergent. |
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what do you think of this expression of sacred geometry in popular culture: | |
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02/05/2011 … 2020
Read: commentary by Stefano Balietti (2020) on “The human quest for discovering mathematical beauty in the arts.” PNAS November 3, 2020 117 (44) 27073-27075;
EXCERPTS from Balietti (2020):
“In the words of the twentieth-century British mathematician G. H. Hardy, “the human function is to ‘discover or observe’ mathematics” (1). For centuries, starting from the ancient Greeks, mankind has hunted for beauty and order in arts and in nature. This quest for mathematical beauty has led to the discovery of recurrent mathematical structures, such as the golden ratio, Fibonacci, and Lucas numbers*, whose ubiquitous presences have been tantalizing the minds of artists and scientists alike. The captivation for this quest comes with high stakes.
“…art is the definitive expression of human creativity, and its mathematical understanding would deliver us the keys for decoding human culture and its evolution…”
“… fairly recently that the scope and the scale of the human quest for mathematical beauty was radically expanded by the simultaneous confluence of three separate innovations:
(1) The mass digitization of large art archives,
(2) the surge in computational power, and
(3) the development of robust statistical methods to capture hidden patterns in vast amounts of data…”
“Starting from its inception, marked by the foundational work by Birkhoff (3), progress in the broad field of computational aesthetics has reached a scale that would have been unimaginable just a decade ago. The recent expansion is not limited to the visual arts (2) but includes music (4), stories (5), language phonology (6), humor in jokes (7), and even equations (8)…” for a comprehensive review, see reference below.
“In PNAS, Lee et al. (10) extend this quest by looking for statistical signatures of compositional proportions in a quasi-canonical dataset of 14,912 landscape paintings spanning the period from Western renaissance to contemporary art (from 1500 CE to 2000 CE).”
* https://en.wikipedia.org/wiki/Lucas_number https://en.wikipedia.org/wiki/Lucas_sequence
Reference for review of computational aesthetics: M. Perc (2020) , Beauty in artistic expressions through the eyes of networks and physics. J. R. Soc. Interface 17, 20190686 (2020). Google Scholar