Mathematics is the majestic structure conceived by man to grant him comprehension of the universe - LE CORBUSIER Gainor Roberts

Transcription

1 Mathematics is the majestic structure conceived by man to grant him comprehension of the universe - LE CORBUSIER 2014 Gainor Roberts

2 Which rectangle do you like the best? Left or right?

3 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

4 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

5 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?

6 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!

7 SO WHAT IS A RATIO? Remember this?

8 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

9 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

10 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

11 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

12 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.

13 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 times B and B is times C. Alternatively, C is of B and B is of A.

14 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

15 Phi the Golden Number!

16 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, etc.

17 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 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 and you get 8.09 Or take 34 and multiply by and you get or take 377 and multiply by and you get which is the same if you add 233 and 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!

18 How to make a Golden Rectangle without math

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20 WITH GEOMETRY (WELL SORT OF) ANOTHER WAY TO MAKE A GOLDEN RECTANGLE

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30 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!

31 PUTTING THEM ALL BACK TOGETHER

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40 HERE IS A GOLDEN SPIRAL

41 HERE IS THE INFINITY POINT

42 EACH LINE IS ALSO A GOLDEN RATIO

43 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 (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 , or roughly 26 inches. Now you have your Golden Ratio 16 to 26 Back to the art store to buy 26 inch stretchers

44 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 and you get 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!

45 Golden Rectangle Calculator

46 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

47 GOLDEN MEAN GAUGES Golden Mean Gauge. Introductory level Price: Extra Large Gauge 18 inches. Price Standard Golden Mean Gauge Price: Golden Mean Gauge (Steel) Price $ taken from the website

48 All sorts of interesting things follow this sequence; in nature, certain sea shells follow the Fibonacci sequence, as do breeding pairs of rabbits, certain plants grow leaves in a perfect Fibonacci sequence and the sunflower seeds follow the Fibonacci spiral sequence and astronomers see the Fibonacci spiral in black holes and galaxies! There are hundreds, or perhaps thousands of examples. Here are a few

49 photo by Gainor Roberts The Golden Spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral.

50 Gary Meisner has a very convincing discussion of this at

51 According to Gary Meisner on the website the Chambered Nautilus shell does not fit into a standard Golden Rectangle. The usual spiral is created by turning the rectangle 90 each time a rectangle is formed. He makes a very good, but complicated point, and offers visual proof that it only works when the rectangle is turned 180. Various people say it does follow the Fibonacci sequence and others say it doesn t. I think the diagrams to the left show, on top the 90 version, and on the bottom the 180 version speak for themselves.

52 PETAL ARRANGEMENTS By dividing a circle into Golden proportions, where the ratio of the arc length are equal to the Golden Ratio, we find the angle of the arcs to be degrees. In fact, this is the angle at which adjacent leaves are positioned around the stem. This phenomenon is observed in many types of plants.

53 PINECONES In the pinecone pictured, eight spirals can be seen to be ascending up the cone in a clockwise direction while thirteen spirals ascend more steeply in a counterclockwise direction.

54 PINEAPPLES One set of 5 parallel spirals ascends at a shallow angle to the right a second set of 8 parallel spirals ascends more steeply to the left and the third set of 13 parallel spirals ascends very steeply to the right

55 DAISY 1 2. we can see the phenomenon in almost two-dimensional form. 3. The eye sees twenty-one counterclockwise 4. and thirty-four logarithmic or equiangular spirals. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms of the Fibonacci sequence.

56 SUNFLOWER

57 THE FIBONACCI RABBITS This famous problem was first presented to the world in 1202 where we find on pages of the manuscript of Fibonacci s Liber Abacci Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, so as to find out how many pairs of rabbits will be born there in the course of one year, it being assumed that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their own birth. As the first pair produces issue in the first month, in this month there will be 2 pairs. Of these, one pair, namely the first one, gives birth in the following month, so that in the second month there will be 3 pairs. Of these, 2 pairs will produce issue in the following month, so that in the third month 2 more pairs of rabbits will be born, and the number of pairs of rabbits in that month will reach 5; of which 3 pairs will produce issue in the fourth month, so that the number of pairs of rabbits will then reach 8. Of these, 5 pairs will produce a further 5 pairs, which, added to the 8 pairs, will give 13 pairs in the fifth month. Of these, 5 pairs do not produce issue in that month but the other 8 do, so that in the sixth month 21 pairs result. Adding the 13 pairs that will be born in the seventh month, 34 pairs are obtained: added to the 21 pairs born in the eight month it becomes 55 pairs in that month: this, added to the 34 pairs born in the ninth month, becomes 89 pairs: and increased again by 55 pairs which are born in the tenth month, makes 144 pairs in that month. Adding the 89 further pairs which are born in the eleventh month, we get 233 pairs, to which we add, lastly, the 144 pairs born in the final month. We thus obtain 377 pairs: this is the number of pairs procreated from the first pair by the end of one year. ********** If you have a desire to know all the math involved in the computation of this Rabbit Problem go to this website: FIBONACCI - HIS RABBITS AND HIS NUMBERS and KEPLER BY Keith Tognetti, School of Mathematics and Applied Statistics University of Wollongong NSW 2522 Australia

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60 THE HUMAN BODY It is also worthwhile to mention that we have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and1 thumb on each hand. The ratio between the forearm and the hand is the Golden Ratio!

61 LEONARDO - THE VITRUVIAN MAN The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Vitruvius determined that the ideal body should be eight heads high. Leonardo s drawing is traditionally named in honor of the architect. This image demonstrates the blend of art and science during the Renaissance and provides the perfect example of Leonardo s deep understanding of proportion.

62 SOLAR SYSTEM AND THE UNIVERSE New findings reveal that the universe itself is in the shape of a dodecahedron, a twelve-sided geometric solid with pentagon faces, all based on phi. Saturn s magnificent rings show a division at a golden section of the width of the rings Curiously, even the relative distances of the ten planets and the largest asteroid average to phi

63 HURRICANE SANDY

64 SPIRAL GALAXY

65 ARCHITECTURE

66 NOTRE DAME CATHEDRAL

67 THE TAJ MAHAL

68 From the Joy of Thinking: The Beauty and Power of Mathematical Ideas LE CORBUSIER

69 The Great Pyramid of Egypt closely embodies Golden Ratio proportions. EGYPT There is debate as to the geometry used in the design of the Great Pyramid of Giza in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with certainty. There is evidence, however, that the design of the pyramid embodies these foundations of mathematics and geometry: Phi, the Golden Ratio that appears throughout nature. Pi, the circumference of a circle in relation to its diameter. The Pythagorean Theorem Credited by tradition to mathematician Pythagoras (about BC), which can be expressed as a² + b² = c².

70 The CN Tower in Toronto, the tallest tower and freestanding structure in the world, contains the golden ratio in its design. The ratio of observation deck at 342 meters to the total height of is or phi, the reciprocal of Phi! John Hamilton Andrews, architect Here is Plato s divided line!

71 DESIGN Upper case Phi is Φ and lower case phi is φ

72 FASHION CAR DESIGN HANDWRITING MEDICAL and

73 MUSIC Musical scales are based on Fibonacci numbers Note how the piano keyboard of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2 There are 13 notes in the span of any note through its octave. A scale is comprised of 8 notes, of which the 5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, that is the 1st note of the scale. Here is a Fibonacci reminder 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.

74 MOZART AND DEBUSSY BOTH USED THE GOLDEN RATIO IN THEIR COMPOSITIONS, AS DID MANY OTHER COMPOSERS AND POETS Musical instruments are often based on phi

75 ART

76 Leonardo The Last Supper. Golden Sections all over the place!

77 Note on Da Vinci s The Annunciation that the brick wall of the courtyard is at exact golden ratio proportions in relation to the top and bottom of the painting see the diagram on the next slide

78 I don t know if Leonardo planned on all the Golden Ratios in this painting, but it is fun to see how many associations we can fine. The Painting seems to be double Golden Rectangles, and that spiral joins the hand of the angel with the hand of Mary perfectly!

79 This self-portrait by Rembrandt ( ) is an example of a triangular composition. A perpendicular line from the apex of the triangle to the base cut the base in golden section. Did he mean to do this?

80 PENTAGRAMS Michelangelo's Holy Family... and Raphael Crucifixion are other examples, wherein the principle figure outlines the Golden triangle which can be used to locate one of its underlying pentagrams. The arms of the star form divided lines in Golden Ratios

81 SEURAT According to one art expert, Seurat "attacked every canvas by the golden section"

82 A Painting by Hans Hoffman, called Golden Rectangles. Obviously he is exploring the mystery of Phi in this painting.

83 Piet Mondrian Composition in Red, Yellow and Blue. Obviously Mondrian was fascinated by the mystery of phi in his artwork

84 M.C. ESCHER Escher was one of the most famous of the mathematically inspired artists. His extraordinary tessellations, Platonic Solids, polyhedrons, and other geometrically arranged shapes are fascinating to observe and in my opinion he is not as well known as he should be. Many of Escher's works contain impossible constructions, made using geometrical objects that cannot exist but are pleasant to the human sight. Some of Escher's tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter concerning hyperbolic geometry. Relationships between the works of mathematician Kurt Gödel, artist Escher, and composer Johann Sebastian Bach are explored in Gödel, Escher, Bach, a Pulitzer Prize-winning book.

85 Mark Rothko Violet, Green, Red 1951

86 Henri Cartier-Bresson

87 Sacrament of the Last Supper by Salvador Dali

88 This is not accidental. Dali was fascinated by mathematics and geometry and many of his monumental works are based on geometry and the Golden Ratio

89 Renaissance Art Composition and the Ukrainian Parliament Fight August 7, 2014 by Gary Meisner This photo and article was all over the internet August,

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91 Matisse L Escargot Escargot is the French word for snail. Many mollusk shells follow some form of the Fibonacci sequence or are in some way allied with the Golden Ratio. Here Is a golden spiral superimposed on top of Matisse s artwork

92 HOW TO USE THESE PRINCIPLES IN YOUR ARTWORK RULE OF THIRDS

93 RULE OF THIRDS

94 TO REALLY SIMPLIFY THINGS! Use ⅓ to ⅔ or 1:3 in your compositions Put the center of interest on one of the intersections or on one of the vertical or horizontal lines

95 Put the Center of Interest of your painting or photograph at the exact spot where the lines intersect and your painting will have magical and mystical properties!

96 Most papers and canvases are not Golden Rectangles so the dimensions do not compute mathematically to form phi. But here both Legal and Tabloid sizes compute to if you round them off a bit.

97 RABATMENT is another device used by artists through the centuries to compose and design their paintings. The concept is relatively simple and it can produce visually stunning paintings and photographs. It works with any size paper or canvas except squares. For this illustration I am using a Golden Rectangle. The idea is to find the square within the rectangle. There will be two of them horizontally and two if the canvas or paper is vertical.

98 RABATMENT RULE OF THIRDS On the left is Monet s Landscape at Vétheuil. Measure the short side of the canvas and flop it down to the long side and you will have a square. That imaginary line is the Rabatment and it is used over and over by artists through the centuries to design their canvases. Obviously when a canvas is all ready square finding the Rabatment is not possible. So using the Rule of Thirds on Monet s Houses of Parliament we find that he actually knew what he was doing! Monet was well trained in classical techniques and these compositions are no accident.

99 TO END WITH A SHORT STORY

100 In 1994 when I started the Feeling Series paintings I didn t know much about the Golden Ratio or much about the rules of composition, except maybe about the rule of thirds. When I assigned sizes to the canvases for these paintings the 48 x 30 size was chosen by picking random numbers out of the air. The Question: What is the psychological appeal of the Golden Rectangle and do artists use intuition or applied design in their works? How do we know if the Golden Ratio is present in an artwork, and do we know if it is by accident or by design? It wasn t until I had started to research the Golden Ratio for an article on Design* that I was writing for my students, that I went into my studio and measured the canvas on my easel and did the math. I was blown away by the fact that I had adopted the Golden Proportion to these three canvases purely by chance or intuition. *here is a link to that article.

101 RESOURCES The Internet has hundreds, maybe more, websites and blogs with diagrams and photographs about the Golden Ratio and Golden Rectangles. I have used many of them and I did not remember to give every image a credit. There is so much online about the Golden Ratio and its allied fields of mathematics it is overwhelming. I am not going to list all of them as a Google search can bring all this wealth to your own computer. But here are a few I found useful. One of the best resources for information on all this is here: by Gary Meisner. Some of you might be interested in his software for finding phi in your own compositions and photographs. Go to this website You can download a trial version and it is only $24.95 with coupon. and many other articles are here Great information here as well as many links to art related websites some interesting diagrams and information on phi in nature More Resources

102 MORE RESOURCES The Painter s Secret Geometry A Study of Composition in Art by Charles Bouleau published by Dover Publishing Co. The Great Courses ( The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas 24 Lectures on DVDs by Professor Michael Starbird and Professor Edward B. Burger The Golden Section Nature s Greatest Secret by Scott Olsen, Published by Walker & Co Scott Olsen is Professor of Philosophy at Central Florida Community College, Ocala, FL

103 THE END Thank you for watching