Math and Mona Lisa

This book captured my attention partly to learn the scientific basis of attraction in people. Why are we attracted to handsome and beautiful persons – apart from aural, karmic and other esoteric connections, what is true scientific basis that nature imposes for such benign or fatal attraction? Apart from being fair which need not be an ingredient for being pretty – what else is in play to capture our visual cue relentlessly? Have you ever watched certain movie or photograph unconsciously spellbound by the beauty, charm, charisma and spontaneity and all combined just hooks us on that heroine or hero appearing on screen?! Such is the captivating sense of beauty – they very innate characteristic makes us swoon and invokes our artistic expression partly as we can never possess them physically but alas artistically to chisel them in our imagination and give a structure, beauty and everything possible to make it a masterpiece and perhaps impose fate and enjoy the journey being a master!

Understanding beauty requires introspection into ‘nature of science’, and ‘nature of art’, then followed by ‘art of nature’ and ‘science of art’ to understand this whole edifice – first understanding what science and art stands for and how they are expressed and then how art is exposed in nature and deduce the science behind to further understand art itself through this scientific perspective. A grand theme dealt by Bulent Atalay with an art and science backdrop of famous Italian renaissance artists and scientists. This book goes by two headings in one : Math and the Mona Lisa – The Art and Science of Leonardo da Vinci.

Understanding Numbers – numbers in the game and a bit of history
Since antiquity apart from Indian and Chinese mathematical contributions, mathematics grew significantly under Islamic mathematicians and scientists. Al-khawrazmi succeeded in formulating a self-consistent al-jabr or algebra which was instrumental to 17th century analytic geometry and calculus. Al-Haytham’s study of optics and vision compares favorably with the work performed by Leonardo da Vinci five centuries later. Persian scholar Omar Khayyam published algebra up through 3rd degree and showed how geometric solutions to the equations could be obtained. As Islamic contributions declined, possibly as new intellectual movement then favored dogma and faith over reason and direct evidence signaling the ascendance of pre-medieval radicalism and then slowly science and mathematics took root in Europe to gain traction by itself trampled by Christian dogma for certain period until Galileo and Kepler came in to picture. Leonardo Fibonacci, during this period of tumultuous ascendance of science in Christian minds, published his ‘Liber Abaci’ (the book of Abacus) wherein he proposed a numbering system heavily borrowed from Arabs with some difference in notation (hence we have the symbols 0 to 9) and he also introduced the place-value concept where each position represents a value multiplied by 10 to the power of that place. Fibonacci’s rabbit problem ultimately gave rise to famous Fibonacci Series – a school problem we all solved which is also fundamental to all beauty in nature by which we are inspired but never knew it has Fibonacci to vouch for it. Here second column is nothing but the Fibonacci series – you arrive at it by adding the preceding row value! Rabbit problem has it own restrictions and bylaws leading to rule we deduced in the previous sentence.

Months Number of Pairs Ratios (later/former)
first 1 1
second 1 2
third 2 1.5
fourth 3 1.666
fifth 5 1.60
sixth 8 1.625
seventh 13 1.615
eighth 21 1.619

Here as you notice the ‘Number of Pairs’ column, adding 1st and 2nd term yields 3rd, 2nd and 3rd yields 4th and so forth. The ratios of pairs oscillate up and down around approximately 1.62. By 17th and 18th terms, it has a value 2584/1597 ~= 1.618 rounded to 3 places and is denoted by phi and variously known as the “golden mean”, the “golden ratio”, the “divine proportion”. Geometric constructions associated with Golden Triangle, Golden Rectangle, Golden Point, Golden Pyramid and the logarithmic spiral.

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Nature of Science

In prehistoric times when ceramics and metallurgy were in vogue, it was kind of technology sans science. in contrast to technology, science is a system of knowledge – the orderly and systematic comprehension. description and explanation of natural phenomena, constrained by logic and mathematics. Science unlike technology , has progressed fits and starts, its course sometimes entirely retrograde in direction. From fall of Rome to renaissance, it regressed and at times evolved independently. In renaissance science we reinvented and industrial revolution took it to the next level. The real beginning of scientific revolution could be attributed the historic publication of monumental book ‘De revolutionibus’ by Copernicus in mid-sixteenth century and culminated with Newton to connect mathematics and natural law that was preceded by Galileo and Kepler. The present understanding of science is through a journey of 2 millennia and still ongoing and Leonardo’s significance in this scheme of science is he developed some scientific principles that were done by ancients and subsequently forgotten and some fields of science and technology that would never be reinvented for centuries. With this  backdrop of science’s progression, let’s look at nature of art.

Nature of Art
throughout history and antiquity, certain numbers and ratios have been used consistently in creations and architectures. This divine proportion is approximated in prosaic every objects including credit cards, playin cards, index cards and of course monumental pieces that include ancient and renaissance masterpieces. In  most cases these numbers & ratios were used subconsciously and inadvertently whereas in Leonardo’s case they were thoroughly experimented and applied with full premeditation. Pyramids and the Parthenon, creations two millennia apart, employ divine proportions in them. ‘Pyramid Power’ attributing special properties to the pyramid shape for preserving food, retarding ageing and maintaining the sharp edge of the razor – has gained legitimacy as part of popular culture but there’s neither scientific evidence nor physical principle that can prove these powers. Nonetheless, a number of remarkable mathematical relations exist in pyramids that can be supported by science. Egyptian pyramids exhibit golden triangle and golden ratio.  For the one in Khufu, height to base length of one edge is 1.57 nearly 1.62, the ratio of altitude of a face to one half its base length is 1.62, ratio of combined area of faces to base area is 1.62. A similar calculations on Khafre ends up same and they 2 are golden pyramids. Whereas east and west façade of Parthenon show golden rectangle, their length-to-width ration is 1.62. Even Greek and roman sculpture exhibit phi wherein the ratio of height o height-to-navel is close to 1.62.

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Art of Nature
It is the nature of art that these mathematical shapes and figures in 2 and 3 dimensions find their way inexorably in works of art, and conversely it is the art of nature that these shapes and figures are displayed by nature’s own creations.the key to patterns and regularities in nature lies in space filling, as well as in biological and physical dynamics. Ultimately, it is the physical forces that give the creations of nature – both animate and inanimate, and both at microscopic and macroscopic level – their symmetries and shapes. Logarithmic spirals are commonplace and can be found in shavings from wood plane, claws of cat, the cochlea of inner ear and the human lip curved gently outward. Human anatomy and face also exhibit phi.

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Science of Art
The theory of linear perspective is of central importance as a tool for the painter to create an illusion of depth – the appearance of three dimensions on a two dimensional canvas. Euclid formalized geometry, but the use of perspective in ancient art was based on artist’s intuition rather than on any mathematical authority. But modern day image and drafting software can seamlessly do perspectives. Perhaps an understanding of 1-point, 2-point and 3-point perspectives is crucial to science of art. In 1-point, typically a painting has a rectangle or square façade and slopes to center (vanishing point) to create an illusion of 3d. In 2-point, the onlooker is looking sideways at the painting- hence there are 2 vanishing points on the horizon line of  viewer’s eyesight. In 3-point, there’s vertical elevation being displayed to onlooker.

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Mona Lisa
Maths behind Mona Lisa is clear in the figure below

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I hope you have enjoyed the Math behind Mona Lisa – summarized and excerpted from the book and appreciate nature’s beauty from a scientific point of view and perhaps apply these principles in your own design – be it UI, decorations or even home building.

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