(Early morning, a Fall day in 1969, junior year in high school…) Five minutes. In five minutes we get the news as to which side we are going to argue today. 10 minutes after that we go on…
It’s the day’s first debate match; my partner K. is buried in her 3×5 note cards, one set in a brown box (Position A), the other in a black (Position B). She’s in the zone… I’m also looking at my note cards; a single question going through my head: Why are the note cards 3×5?
Why are the note cards 3×5?… A second thought is creeping into my head: maybe there’s an attention deficit problem in play here. I look over at K. again. Still locked in, getting geared up (she’s actually a brilliant debater, I’m smart enough to realize she’s our closer). I’m still wondering why a 3×5 size? Why not 3×4? 4×6?…maybe 5×7?
Well… If we start taking ratios of the long side to the short, we come up with the following: 4/3 = 1.33, 6/4 = 1.5, 7/5 = 1.4. The ratios are all different, and none particularly pleasing as a shape….
5×8? Yes, much more like it. Also appearing much like the 3×5… When we calculate the ratio of the long side to the short for each we start weaving our way around the number 1.6, the almost magical number that the mathematician, Franciscan, and FOL (Leonardo…yes, that Leonardo…) Luca Pacioli called The Divine Proportion. It’s The Golden Ratio. In its earliest form (as far as we know, studied by the Greeks in the 5th century BC), the investigation utilized two line segments (A and B; A the longer, B the shorter). Add the two line segments together (A+B). The ratio: A+B is to A as A is to B. The number: 1.618033988749…. Like its more famous cousin Pi, an irrational number.
Both the 3×5 and 5×8 rectangles are examples of The Golden Rectangle (there are more). However, there is a lot more than just more rectangles… With respect to plane geometry shapes, there is also the Golden Angle of a circle, the Golden Triangle, Golden Rhombus, and Pentagon. It gets even more involved. In the realm of three dimensional objects we have the Regular Dodecahedron and the Icosahedron, both 3-D cousins of the pentagon. All built upon the Golden Ratio.
And then there is the Golden Spiral…
Leonardo of Pisa (1170 – 1250?) is not as well known as the other Leonardo, but his contributions have shaped our world. He is, for example, responsible for bringing Arabic numerals to the western world. In certain circles he is most famous for his presentation of a numerical sequence, best known to us as The Fibonacci Sequence. In concept it is simple enough: usually starting with 0 and 1, each number in the sequence is the sum of the preceding two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… With increasing n the ratio between two numbers in the sequence converges on…Golden. It’s hard to visualize when reported as a string of numbers, but when displayed graphically the Golden Spiral is revealed:
The rectangle reflects the Golden Ratio, the swirl pattern the Golden Spiral. Note the composition. If we were to place the center of the flower in the center of the rectangle, it simply would not work…
So…photographers are familiar, looking for the best (most beautiful?…) composition…
Salvador Dalí is widely regarded as the greatest surrealist painter the world has known. I suspect that there are few unfamiliar with his The Persistence of Memory, with the melting watches (his nod to Einstein and the implications of the Theory of Relativity). Dalí was many things, but first and foremost, a brilliant technician as a painter. Presenting a work in two dimensions but seeing in three…
Below is one of his religious pieces (there are many): The Sacrament of the Last Supper:
He created the piece as a golden rectangle. The golden spiral allowed him to balance the elements in the painting, including the outstretched arms floating above. And the unusual architecture and windows?… The scene is in perspective from an interior horizontal edge of a dodecahedron…a three dimensional expression of the Golden Ratio.
1.618033988749…. Defining Beauty.
