For the past month or so I have been reading and have almost finished Math and the Mona Lisa, which I bought back around 2004 when it was first published by a long-ago friend, Bülent Atalay, who is a theoretical physicist and who teaches at U.Va., Mary Washington University and Princeton University. Atalay is also an accomplished, published artist.
I have had an abiding interest in the “Fibonacci Series” and the related mathematical “Golden Ratio” for many years. I started out as a physics major in college, but I decided to switch to the dreaded “social sciences” because I hit the “brick wall” of calculus, in which I was sadly lacking. I did pass the intro physics course, but barely. Nevertheless, I have enjoyed continuing to dabble in that stuff as a layperson.
The main premise of Atalay’s book is to explore the wonderful conjuncture of math, science and art in the person of Leonardo da Vinci, who painted The Last Supper and the Mona Lisa and also explored and enunciated many scientific truths. One of da Vinci’s main thrusts was to study the occurrence of the so-called “Golden Ratio” or “Golden Section,” expressed as the Greek letter ϕ (phi, pronounced “fee,” not “fye”), which is mostly a rectangular proportion that is frequently manifest in both art and nature and is inherently pleasing to most viewers’ eyes. The mathematical ratio for ϕ is 1.618 : 1, being the mathematical ratio of the long side of the rectangle to the short side and which also happens to be manifest in a string of numerals called the “Fibonacci Series,” first expressed by Leonardo of Fibonacci, a contemporary of da Vinci, discussed below.
The Golden Rectangle is reflected in the façade of the Parthenon, the proportions of playing cards, 5 x 8 index cards, etc. It is also manifest in certain ratios of human figures as in the human face and the height of the navel to the overall height of the figure!
The essence of the Golden Ratio is that as any such structure or body gets larger and larger (like a plant or tree adding buds or branches or a growing chambered nautilus), it does so logarithmically by adding an exact “SQUARE” to such a previous “rectangle,” sized equivalent to the long side of that previous rectangle, such that the resultant and larger combined rectangular structure is 1.618 larger than the square being so tacked on! And, the previous rectangle happens to be 0.618 of that same square, the mathematical inverse of ϕ (1/ϕ)! Then the resultant combination of the prior “Golden Rectangle” added to the new square becomes it’s own “Golden Rectangle”! Consider: if the value of the added “square” is arbitrarily stated as “1,” then the combination is 1.0 + 0.618 = 1.618!
The “Fibonacci Series” is defined as the series of numbers in which each is the total of the two preceding numerals. Leonardo de Fibonacci expressed it as a pair of rabbits maturing then breeding in a closed space, then the resultant pair of offspring mature and also breed two offspring, and the first pair breed again, all of which then mature and breed further, etc. The Series is: 0,1,1,2,3,5,8,13,21,34,55,89,144, etc. Above 5 and 8, dividing the next higher number by the immediately lower number closely yields the Golden Ratio. (The ratio expressed by 5 and 8 is 1.6, pretty close!) Atalay shows a photo of a stained-glass window at the UN created by the famous artist, Marc Chagall, which has 5 square panes x 8 square panes, being itself such a manifestation of the Golden Ratio. Consider adding “squares” beginning with 8 x 8, then 13 x 13, then 21 x 21, etc. in a spiral fashion, and the growth” pattern mentioned above is readily apparent. As a growing body may be restricted by physical space (like the nautilus shell), it may well be forced to spiral out to the side as it grows. So, the new growth produces a “SQUARE” of sorts because new tissue must spread more or less equally in all directions, and that is basically what a “square” is—it is spread out from side to side and end to end equally, and it is going to be defined in scope by the size of the existing tissue from which it emanates.
Years ago I read H. E. Huntley’s book, Divine Proportion, about these topics. I also want to read Mario Livio’s book about the same thing. Years ago another friend also gave me Jay Hambidge’s book, Dynamic Symmetry, etc. which explores how the Golden Ratio is manifest in the proportions of ancient Greek pottery. Several years ago I attended a lecture by Roger Penrose, the theoretical physicist. I bought his book, Emperor’s New Mind, about the math of interlocking tiles and geometric figures. I tried to read the book at the time but found too difficult to continue. I am fascinated by mathematical “coincidences” found in nature. I also am reading a book, Physics for Gearheads, which has a lot of interesting material related to auto mechanics. I am a certified auto mechanic as well as a retired lawyer and auctioneer.
Some of this stuff may be over the reader’s head. Atalay’s book has been a tough slog for me, but I really want to know about it all. Leonardo da Vinci was an amazing individual whose contributions to both art and science cannot be overstated.
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