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Mathematics Is the Majestic Structure Conceived by Man to Grant Him Comprehension of the Universe”- LE CORBUSIER

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Mathematics Is the Majestic Structure Conceived by Man to Grant Him Comprehension of the Universe”- LE CORBUSIER “Mathematics is the majestic structure conceived by man to grant him comprehension of the universe”- LE CORBUSIER ©2014 Gainor Roberts Which rectangle do you like the best? Left or right? This is a Golden Rectangle If given a choice between the two many people will choose this one. There are many theories about why this is so. It has been used in art, and architecture and industrial design for centuries…perhaps since we began making pictures and it gets really interesting in studying nature, cosmology and mathematics VARIOUS NAMES BY WHICH PHI IS KNOWN GOLDEN MEAN GOLDEN RATIO GOLDEN PROPORTION DIVINE PROPORTION GOLDEN SECTION EXTREME RATIO MEDIAL SECTION DIVINE SECTION GOLDEN CUT GOLDEN NUMBER So what does this have to do with anything? Why the fuss about the Golden Ratio anyway? How does this have anything to do with my life and my art? PHI Pronouncing Phi May 13, 2012 by Gary Meisner Phee, Phi, Pho, Phum™ … or how do you say Φ? The generally accepted pronunciation of phi is fi, like fly. Most people know phi as “fi,” to rhyme with fly, as its pronounced in “Phi Beta Kappa.” In Dan Brown’s best selling book “The Da Vinci Code,” however, phi is said to be pronounced fe, like fee. Gary Meisner has a website that I mention frequently in this PowerPoint. His website is a wonderful “go to” source of great information about the Golden Ratio and all things associated with it. First and foremost it seems reasonable to figure out how to say PHI properly. The link below will give you a lengthy article with variations and proper ways to look, and sound like you know what you are talking about! https://www.goldennumber.net/pronouncing-phi/ SO WHAT IS A RATIO? Remember this? PROPORTION I teach drawing and many of my students seem to have a great deal of difficulty understanding the concept of proportion, but in art it is essential, if you are attempting to be realistic and not skew reality. Children get their proportions all wrong (perhaps I should say unreal, rather than wrong). In any case their house is the same size as the father and the tree is smaller than Dad. We think this is charming, but when it comes to classical drawing it is not charming at all. The illustrations on the left show how we can use a sighting stick to find out the relative proportions of the little box to the big box. We “measure” with a stick (or pencil or brush) and sight the little box’s width, and without moving our thumb from the stick, we move it to the big box and see that the big box is approximately one and a half times the size of the small box. Photos by Gainor Roberts Using a sighting stick to find proportions is a good way to "measure" your subject. Knowing this will help you draw the proportions correctly. So many problems in drawing are optical illusions, and figments of what we think we "know" about something, but do not be fooled! Our eye and brain are not accurate measurers, for most of us, and we need to have as much accurate information about our subject to create believable and accurate drawings and paintings. And so this is a ratio. We can say that the big box is one and a half times as big as the little box. Looking at the two of them with out measuring doesn’t tell us that at all. We usually need visual aids and perhaps math, if you are so inclined to get it right Proportions of the human head, if seen straight on and not tipped, will show the eyes are usually midway between the top of the head and the chin. But when we look at a person we tend to see the eyes higher in the skull and make the forehead smaller. This is a very typical mistake that many artists make. Proportions and ratios are a very important part of our anatomy and our art! From Understanding Drawing by Gainor Roberts However more accurate measurements of the head and face are based on thirds and this scheme follows the dimensions of the Golden Ratio as applied to human anatomy From Understanding Drawing by Gainor Roberts OK, I get it about ratio and proportion. But what’s the “golden” part? A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. What? That statement makes no sense to my right brain at all! That is a quote from Plato living between 428 and 348 BC in Greece. Euclid had a great influence on philosophy, mathematics and geometry, who lived in Egypt. Both of these men were so brilliant and they set the groundwork for millennia of thinkers and discoverers, as well as later geniuses in the Renaissance and into our modern times. Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi (Φ ) is simply the ratio of the line segments that result when a line is divided in one very special and unique way. Divide a line so that: the ratio of the length of the entire line (A) to the length of larger line segment (B) is the same as the ratio of the length of the larger line segment (B) to the length of the smaller line segment (C). This happens only at the point where: A is 1.618 … times B and B is 1.618 … times C. Alternatively, C is 0.618… of B and B is 0.618… of A. So, to continue, here is some algebra for the math heads in our midst. This is the solution to Plato and Euclid’s line divided into unequal lengths Phi 1.6180339887…..the Golden Number! FIBONACCI Fibonacci was born in 1170 in Pisa, and was known as Leonardo Bonacci or Leonard of Pisa, which was shorted to Fibonacci, or son of Bonacci. He was a mathematical genius who wrote, in the early 13th century the Liber Abaci, or Book of Calculation. Having traveled extensively he concluded that trying to do arithmetic with Roman Numerals was much more difficult than using the Hindu-Arabic number system. He spread the word through his teachings and book. Centuries later this numerical sequence was named after him. We are lucky to have been saved by Fibonacci from having to add and subtract, or horrors, divide using Roman Numerals! 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377....etc. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377....etc. The sequence is derived by adding 0 +1 which equals 1. Then add 1 +1 which equals 2. Then add 2 +1 which equals 3. Then add 2 +3 which equals 5 and so on. Each number is the sum of the previous two and each number approximates the previous number multiplied by the Golden Section 1.618! The numbers become more accurate as they grow larger. Take 5 and multiply by 1.618 and you get 8.09 Or take 34 and multiply by 1.618 and you get 55.012 or take 377 and multiply by 1.618 and you get 609.986 which is the same if you add 233 and 377…..610. Is this magic? Many people thought so, and it was kept a secret by various groups and societies through to modern times. It was so mysterious it was given Divine associations by many throughout history each of these pairs are the dimensions of a GOLDEN RECTANGLE give or take a few decimal places! How to make a Golden Rectangle without math WITH GEOMETRY (WELL SORT OF) ANOTHER WAY TO MAKE A GOLDEN RECTANGLE We can do this exercise an infinite number of times but I don’t have a microscope! The divisions go on and on forever, but for our purposes we are going to stop now….it is sort of tiresome making these golden rectangles! PUTTING THEM ALL BACK TOGETHER HERE IS A GOLDEN SPIRAL HERE IS THE INFINITY POINT EACH LINE IS ALSO A GOLDEN RATIO But there is an easier way to make a golden rectangle! Suppose you have some stretcher bars for a 16 x 20 canvas, but you want to make Golden Rectangles instead. How do I find out how long I need to make the new stretcher bars? Multiply by 1.618 (yeah, I have to use a calculator to do this!) So I decide to have one side be 16 inches. So the other side has to be 25.8888…, or roughly 26 inches. Now you have your Golden Ratio 16 to 26 Back to the art store to buy 26 inch stretchers Suppose I want to go the other way and figure out what a golden rectangle would be based on 28 going smaller not larger. Divide 26 by 1.618 and you get….16.0692 and so you will go to the art store and buy 16 inch stretcher bars, unless you are very picky and order from specialty suppliers the exact measurements! Golden Rectangle Calculator http://www.miniwebtool.com/golden-rectangle-calculator/?a=30 And for the fanatic Golden Ratio addict you can buy an instrument to measure your distances. Evidently Dentists and Surgeons use these tools to measure body parts and teeth. taken from the website http://www.goldennumber.net/ GOLDEN MEAN GAUGES Golden Mean Gauge.
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