eureka moments: 3+3=5

In the grand scheme of things nature is much bigger in comparison to the individual. Nonetheless, man has found it irresistible to try to prove his superiority. A lost cause. Our ego is always searching for divine patterns; from the bible thumpers like Harold Camping to artists like DaVinci and even Jackson Pollock, or a Carl Jung; there seems to be an innate need to establish that we are the superior part of nature. Which means that nature is divisible and has a hierarchy.Intentionally or not, artists constantly borrow from the patterns that make up the natural world.In each Each effort to create art, we recreate nature ourselves , albeit on a miniature scale. It is as if by attempting to reproduce it we are somehow able, futilely to master it.The question then is of chicken an egg proportions: Is art a product of the individual or of nature? Is nature the art or the artist?

---Prevalent in the major works of Leonardo Da Vinci and underlying many of his design compositions, is the Golden Ratio ("Golden Mean"), a ratio of approximately 1:1.618, found in nature and creation, and inherent in the Fibonacci sequence. The Golden Rectangle, the Golden Triangle, and the Golden Pyramid, all based on the Golden Ratio are all appear prominent in the work of Leonardo Da Vinci.---Read More:

There has always been a long standing affinity between the mathematical and the aesthetic going back to ancient times. These two modes of reasoning and communicating have contributed to the growth of both fields; the aesthetic is always present in mathematical thinking and contributes to the growth of mathematical knowledge. The golden ratio. The irrational number 1.618. Who would have foreseen that such an innocuous number discovered by Euclid would end up having implications for numerous natural phenomena ranging from the leaf and seed arrangements of plants to the structure of the crystals of some metal alloys, and from the arts to the financial analysis of stocks and bonds.

James Gillray. The Pursuit of Knowledge. read more:

The Golden Ratio got its name due to its constant involvement in beauty and perfection, hence it is also called the Divine Ratio. Interestingly, most objects that humans deem aesthetically pleasing, adhere to the Golden Ratio.Even in everyday life, humans compare what they see to an ingrained yet subconscious Golden Ratio. This is where we get the notion of “mathematical beauty”. In a mathematically beautiful face, every proportion from the eyes, ears, mouth, nose, and in-between, matches up perfectly to the Golden Ratio.

...have created a 'beautification' program which takes a photo of a face, then adjusts distances in the features to make it more attractive, using a complicated mathematical algorithm. Unfortunately, the software isn't perfect and produces odd results. When a photo of Brigitte Bardot was used, the after photo seems to be less striking. Some critics are incensed at the idea that math could determine what makes beauty: "Yet, like the many other attempts to use objective principles or even mathematical formulas to define beauty, this software program raises what psychologists, philosophers and feminists say are complex, even disturbing, questions about the perception of beauty and a beauty ideal. To what extent is beauty quantifiable? Does a supposedly scientific definition merely reflect the ideal of the moment, built from the images of pop culture and the news media?... Read More:

Many books claim that if you draw a rectangle around the face of Leonardo da Vinci’s Mona Lisa, the ratio of the height to width of that rectangle is equal to the Golden Ratio. No factual basis exists to indicate that Leonardo consciously used the Golden Ratio in the Mona Lisa’s composition, nor to where precisely the rectangle should be drawn. Nevertheless, one has to acknowledge the fact that Leonardo was a close personal friend of Luca Pacioli, who published a three-volume treatise on the Golden Ratio in 1509 entitled Divina Proportione, and it is possible Botticelli was also familiar with this work.

---Second, the mathematical mind is a melancholy one, for which the most striking evidence is the famous etching Melencolia I by Albrecht Dürer. Dürer was melancholy. True, he was an artist, but (as the authors take pains to point out) he was an artist who liked mathematics....----Read More:

—Does the unconscious play a part in maths, too? “I’m sure it does,” Carla says, “though it’s hard to express how exactly; I’m not a psychologist. It’s the place from where we all bring our ideas. I think there is a connection between dreams and maths. Dreams have their own logic, that doesn’t immediately make sense. In mathematics, too, we use many different types of logic, sometimes counterintuitive ones….”This reminds me of another picture in the exhibition, Caravaggio by John Trobaugh…. ( Image removed by request)

…It’s about taking body pieces and attaching them to parts of the body where they don’t belong. This is something that could happen in a dream, but it’s also very systematic: taking things apart, permuting them and reassembling them in a different order, is something we do in maths a lot. Both maths and dreams allow us to break the boundaries of reality. Maybe it’s all about expressing the infinite world we have inside of us, and which in our real life can only come out in finitely many ways.” Read More:

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---A self portrait of the artist reflecting on the "eureka" moment and realization that mathematics was an integral component of understanding and explaining the the universe. This piece also pays homage to the artists past career as a potter and his past experience as a prisoner of arithmetic.--- Read More:


Is there a mathematics behind our known mathematics? :David Pimm’s essay (“Drawing on the image in mathematics and art”) caught me by surprise and kept me up late reading it. In it, he describes how one particular aesthetic (the purity of iconoclasm) permeated and became ritual in both the Bourbaki movement and in the reformed English Church. He contends that notions of “purity” (in the religious and also in the “pure math” sense) led to the same sense of iconoclasm: rejection of figures (geometric or human) and an attempt to get at Truth directly, without any mediating interference.Read More:


---There is evidence that some artists have used the golden ratio intentionally, such as Salvador Dali in his work Sacrament of the Last Supper. One instance of this ratio can be found in the pentagons behind the figure of Jesus, which, as already stated, contain golden ratios. It is not known why Dali chose to include golden ratios in his work, however.--- Read More: image:

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Among the most important irrational numbers of the past, the square root of two probably figures as the most important one. I think I’d call it the Silver Section, just to contrast it to the Golden Section. There’s this Danish guy, Brunus, who wrote two volumes on sacred geometry. He’s sort of coming from a freemason background or viewpoint. Sacred geometry was made by builders who were high initiates of the secret societies and so on. So he makes a very interesting case for this Silver Section, which is basically just the square root of two. Of course that was the number, at least in historical times, that first confounded the Pythagorean number mystics. They related to the number mysticism of the musical scale; all of the musical intervals are rational numbers – although sometimes the important ones have huge numerators and denominators. To do that kind of arithmetic took phenomenal skill – multiplication and division of very large numbers. And they of course hoped you could approach any number this way. And approximately you can, but exactly you can’t. And that was in versions of the history of the mathematics of ancient Greece. It was the Pythagoreans who first proved that the square root of two is not a rational number.

Well it was long, long before then that the square root of two was found in sacred architecture. The pyramid at Giza has the Silver Section in it, as well as the Golden Mean. So Brunus makes this extensive argument on the priority of the importance of the square root of two, and this magic rectangle the Silver Section. …Read More:

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