A&O – ART & Mathematics












“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)



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.







  • Unsupported image type.International Society for Mathematical and Computational Aesthetics: http://www.rci.rutgers.edu/~mleyton/ISMA.htm
  • David Bergamini 1963. Mathematics. Time Inc. (this book has a section on “The mathematics of beauty in nature and art” but does not address why a particular formulation is “beautiful” from a biological/sensory/perceptual perspective). Doczi, Gyorgy 1981.
  • The Power of Limits. Shambhala, Boulder. (“proportional harmonies in nature, art, and architecture”) Mitchison, G.J. 1977. Phyllotaxis and the Fibonacci series.  Science 196:270-275.   Issues of space, proportion, scale and perspective are included in Nathan Cabot Hale’s  1972 book, Abstraction in Art and Nature: A Program of Study for Artists, Teachers, and Students (Watson-Guptill Publications, N.Y.)

 The logarithmic spiral: its mathematical regularities appear to involve intrinsic attractiveness .  computational paleontology has devised programs to model the log spiral in seashells and a mathematical analysis of the golden mean as a problem in aesthetics from the MathSoft Corporation

from Cracking The Da Vinci Code By Keith Devlin / Photography by Dan Winters / DISCOVER Vol. 25 No. 06 | June 2004 



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 http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html



  • Stevens, P.S. 1974. Patterns in Nature. Atlantic–Little, Brown & Co., Boston)  The Golden Mean, Spirals, Symmetry, Equilibrium and Dynamic.  
  • Mathematics, Geometry, and Beauty.  (R. Arnheim 1974. Art and Visual Perception; Bloomer, Carolyn M. 1976. Principles of Visual Perception. Van Nostrand Reinhold, NY; M.E. Huntly, The Divine Proportion: A Study in Mathematical Beauty; Christopher Williams 1981. Origins of Form, Architectural Book Publ Co.; Phillip C. Rittersbush 1968. The Art of Organic Forms, Smithsonian);

Unsupported image type.  “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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  • Unsupported image type. Are Fractals a window on the resonance between art and nature?  [more]
  • Unsupported image type. Jackson Pollock’s fractal patterns:  “Can science be used to further our understanding of art? This question triggers reservations from both scientists and artists. However, for the abstract paintings produced by Jackson Pollock in the late 1940s, the answer is a resounding “yes”. [article from physics world magazine] [more]







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.”
























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.  

































what do you think of this expression of sacred geometry in popular culture:

The Fibonacci in Lateralus













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