The Golden Ratio was first expressed by the ancient Greek philosopher Euclid over two thousand years ago. This ratio is referred to as PHI (pronounced: fee) and is expressed mathematically as 1:1.618 (continuing on into infinity and never repeating). This ratio is found throughout nature and has been recognized as a fundamental component of all things that man has found aesthetically pleasing.
Recognizing the inherently pleasing nature of things “in the ratio”, man has employed the Golden Ratio throughout time in some of the most remarkable and inspiring achievements.
These include the Parthenon in Greece,
the ancient Pyramids in Egypt, DaVinci’s “Mona Lisa” and Stradivarius’ Violins.
Even greater than these achievements are the naturally occurring examples of things “in the ratio” such as the graceful curves of the Nautilus shell,
the patterns within a sunflower, the position of the stars in the heavens and the shape of our own DNA.
Want to know more about PHI or the Golden Ration ? Where would one find it? Here's what Terrell gave us as examples.
50+ short Golden Proportion facts and bite sized thoughts
1. The Golden Ratio not only looks better, it sounds better. That’s why some manufacturers use it in the construction of audio components.
http://www.cardas.com/
2. The Golden Ratio is aesthetically appealing to people. That’s why it appears so often in art used either consciously or unconsciously used by the artist.
http://milan.milanovic.org/math/english/gold/index.html
3. According to his sister, Maria Anna (Nannerl), Mozart was fascinated by numbers and mathematics. Many of his sonatas divide into two parts at the golden section point.
4. The Golden Ratio, 1.618…, is the only number that when squared gives the same answer as if added to the number 1.
5. If the Golden Ratio makes the front of a building look fantastic, imagine what it can do for the backside of a woman.
6. Modern architects like Corbusier and Jose Luis Sert used the Golden Ratio in the design of their buildings.
7. Because the pentagram is composed of five lines each divided into Golden Sections it was used as the symbol of the Pythagoreans, an ancient Greek society of numerologists.
8. The oldest surviving written mention of the ratio is in Euclid’s Elements a book of geometry that dates back almost 2500 years.
9. The UN building is composed of three Golden Section rectangles.
10. The Golden Ratio is a non-repeating decimal. Blue Ratio is a non-repeating jean based on the Golden Ratio. No two are ever exactly alike. They are as unique as the people who wear them.
11. The ratio has many names including: Golden Ratio, Golden Number, Golden Section, Golden Mean, Golden Proportion, Divine Proportion and Continuous Proportion.
12. Leonardo Da Vinci may have used the ratio in his study of human proportions.
13. You can see the ratio everywhere in nature – in the proportions of some flowers and seashells even in clusters of stars.
14. In the middle ages, an Italian mathematician named Fibonacci discovered a sequence of numbers that follow the ratio. You create the sequence by starting with 1 and adding the preceding number to get the next number which produces this sequence: 0 – 1 – 1 – 2 – 3 – 5 – 8 –13 – 21 – 34 – 55 – 89… The higher the number, the closer it gets to the ratio.
15. Since prehistoric times, artists, architects and designers have used the ratio to give harmony and unity to art. It was even used in the design of the Great Pyramids and the Parthenon.
16. Johannes Kepler, the famous 16th century mathematician and astronomer, said "Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio (the Golden Ratio). The first we may compare to a measure of gold; the second we may name a precious jewel."
17. The Athenian architect and sculptor, Phidias (490 to 432 BC) used the ratio in his design for the Parthenon, which was the temple for Athena, Greek goddess of Wisdom, science, and art. Modern mathematicians use the Greek letter phi, for Phidias, to represent the ratio.
http://www.recoveredscience.com/const305goldenprehistory.htm
18. A line is divided into the Golden Section when the short segment is in the same proportion to the long segment as the long segment is to the whole line.
19. Expressed as numeric ratio the Golden Ratio is 1:1.6180339887498949… and on and on forever without repeating.
20. By using the Golden Section, Oxford University Professor Roger Penrose discovered tiles with five-fold symmetry that can cover an infinite plane without repeating. When they appear naturally in crystals, they are called quasicrystals, and, until fairly recently, were thought to be impossible.
http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey=Penrose+tiling&gwp=8&curtab=2222_1
21. Composer Bela Bartok used the Golden Section to determine where to put the point of climax. No off color jokes please.
22. Some orthodontists use a “Golden Mean (Ratio) Gauge” to ensure the perfect placement of their patient’s teeth.
http://www.goldenmeangauge.co.uk/
23. A software company called Atrise sells a computer template and grid software to help you design on your computer according to the Golden Ratio.
http://www.atrise.com/golden-section/
24. Cosmologists in France and the US say that space could be finite and shaped like a dodecahedron -- a twelve-sided polygon closely related to the Golden Ratio. They say this shape helps explain measurements of the radiation left over from the big bang.
http://physicsweb.org/articles/news/7/10/5
25. The golden ratio is the only number whose square can be produced simply by adding 1 and whose reciprocal by subtracting 1.
26. Superimpose a square on a rectangle whose length-to-breadth is in the golden ratio and what remains is another, smaller golden rectangle.
27. In 1996, the Golden Ratio was expressed as a decimal number carried out 10 million places without repeating.
28. The 15th-century Italian mathematician and friar, Luca Pacioli, equated the Golden Ratio with the incomprehensibility of God.
29. Inspired by Adolf Zeising’s book, Der goldene Schnitt, German psychologist, Gustav Fechner, conducted surveys to see if the golden rectangle had psychological aesthetic impact. Fechner made thousands of measurements of commonly seen rectangles, such as writing pads, books, playing cards, windows, and found that most were close to Phi. He also tested people’s preferences and found most people prefer the shape of the golden rectangle. His findings were published in 1876, and his experiments were repeated by Witmar (1894), Lalo (1908) and Thorndike (1917).
http://en.wikipedia.org/wiki/Golden_Mean
30. The Triumphal Arch of Constantine and the Coliseum, both in Rome, are examples ancient use of golden relationships in architecture.
31. Fibonacci numbers are part of phylotaxis appearing in three different spiral arrangements: 1. vertically in leaves spiraling up a stem, 2. horizontally as on the flat head of the sunflower, and, 3. tapered or rounded as on pinecones and pineapples.
http://www.goldenmeangauge.co.uk/fibonacci.htm
32. In 1998, a book on phylotaxis called "Symmetry in Plants" was published as a multidisciplinary study by 44 scientists, all leaders in their fields, including chapters by botanists, mathematicians crystallographers and molecular geneticists.
http://www.goldenmeangauge.co.uk/fibonacci.htm
33. Quasi-crystals have five-fold symmetry, which means they make a pattern that looks the same when rotated by multiples of one-fifth of 360. In the 1990s, physicists in Switzerland and the US imaged the microscopic terrain of the surface of such crystals. They found flat "terraces" punctuated by abrupt vertical steps that come in two predominant sizes in Golden Ratio proportion to each other. Once believed to be impossible, quasi-crystals were first observed in 1984 in an aluminiun-manganese alloy (Al6Mn). Since then, quasicrystals have been found in other substances.
http://brain.web-us.com/sleeperdeltaPhi.htm
34. A quick proof of the irrationality of the Golden Ratio is if a/b is a fraction in lowest terms, then b/(a - b) is in even lower terms — which, since they are equal, is a contradiction.
http://en.wikipedia.org/wiki/Golden_ratio#A_startlingly_quick_proof_of_irrationality
35. French composer Erik Satie used the Golden Section in several of his pieces, including Sonneries de la Rose+Croix. http://en.wikipedia.org/wiki/Golden_ratio#A_startlingly_quick_proof_of_irrationality
36. The Golden Ratio is an irrational number.
37. You can use the Golden Ratio to create Golden Rectangles, Golden Triangles, Golden Spirals – even three dimensional shapes.
38. Sacred Geometries are shapes and patterns that are revered as holy because believers see revealed truth in the beauty of their formation. The Golden Ratio is part of the sacred geometry of many cultures.
http://www.charlesgilchrist.com/SGEO/YSGCB.html
www.intent.com/sg
http://www.floweroflife.org/
www.sangraal.com
39. Bonsai artists have acknowledged using the Golden Ratio in the design of their miniature trees.
http://www.bonsai4me.com/AdvTech/ATGolden%20Section.html
40. Credit cards come very close to Golden Rectangle proportions.
41. There are special proportioning tools that help dentists, artists and architects quickly find Golden Ratio proportions.
42. Triads, which are the western musical harmonies, are based on the Fibonacci numbers one, three, and five.
43. DNA measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. 34 and 21 are Fibonacci numbers related to the Golden Ratio.
44. The Golden Ratio can be used to create fractals.
http://www.fortunecity.com/emachines/e11/86/friction.html
45. The Golden Ratio is found in human proportions, nature, art, music and architecture. It is one of the world’s most ubiquitous numbers.
http://www.miqel.com/pure-math-patterns/visual-math-phi-golden.html
46. There have been many books relating directly and indirectly to the Golden Ratio including recently: Gnomon, from Pharaohs to Fractals by Midhat J. Gazale, The Golden Section by Hans Walser and The Golden Ratio by Mario Livio.
47. The Golden Ratio relates to the patterns of lightning bolts.
48. The Golden Ratio is related to population growth. In fact, it was a puzzle based on the expanding population of rabbits that lead Fibonacci to discover the set of numbers that bears his name.
49. Facial proportions vary. However, many artists and plastic surgeons use the Golden Ratio as a guide to creating the perfect proportioned face.
http://www.plasreconsurg.com/pt/re/prs/abstract.00006534-200409150-00041.htm;jsessionid=DZrMJ50QQ1uqPWGbrkuG5h9rGyn8k2mmMsCGlTV98Fo1TYlLpX3E!1389088241!-949856145!9001!-1
50. In Western musical scales the ratios of a justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of Fibonacci sequence, making them the closest low integer ratios to the golden ratio.
51. The Golden Ratio can be found in all types of natural and cultured crystals, the hexagonal geometry of snowflakes, creatures with spiral exoskeletons like some snails and shellfish, things that branch in fractals like lightning and rivers, the geometric and atomic patterns of metals.
http://www.infinitetechnologies.co.za/articles/geometry1.html
52. The first known name for the Golden Ratio was coined nearly 2500 years ago by Euclid who called it "extreme and mean ratio". The term "Divine Proportion" was first used by the 15th century mathematician, Luca Pacioli in his three volume work by the same name. The name Golden Ratio first appeared in 1835, in a book written by the mathematician Martin Ohm.
53. A pentagram inscribed within a pentagon creates a variety of Golden figures including Golden Sectioned lines, and both varieties of Golden Triangles.
54. The Golden Ratio is present in the pentagram, the five Platonic solids, in fractal geometry and certain crystal structures, and in Penrose tiles.
55. The number of petals of most flowers is a Fibonacci number. An iris has 3 petals, a primrose 5, a delphinium 8, ragwort 13, an aster 21, daisies 13, 21, or 34 – all Fibonacci numbers related to the Golden Ratio.
56. The seeds in the center of a sunflower follow a pattern of 21 to 34 spirals running clockwise, and 34 to 55 running counterclockwise. Pine cones have 5 clockwise and 8 counterclockwise spirals, pineapples has 8 clockwise and 13 counterclockwise spirals – all Fibonacci numbers related to the Golden Ratio.
57. Leaves wind around the stems of trees and plants according to the Golden Ratio. The number of complete turns of the spiral before another leaf grows directly above the first is generally a Fibonacci number related to the Golden Ratio.
58. The path followed by a diving Pergrine falcon is a spiral that turns on a constant angle related to the Golden Section.
Now let's get to the important stuff-- how it applies to denim!!
The Boyfriend Fit
Narrow BootCut .
Skinny Jeans
