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findings.
GOLDEN RATIO DO IT YOURSELF PROJECTS Golden Section Gauge. Make your own golden section gauge using the template below. It will work best if you construct it from heavy cardboard stock or plastic. Drill holes and place a brad at each of the indicated points. When the gauge is adjusted the middle arm will always show the golden section or phi point between the two outerarms.
GOLDEN RATIO POINT OF THE EARTH Site visitors did alternate measures based on the miles shown by Google Earth from the North Pole to Mecca to the South Pole, submitting that the golden ratio point falls within the boundaries of Islam’s holy city. They found the North-South distance to be 19980 kilometers, and divided by 1.618034 to arrive at a distance of 7,631.68, which they said was within Mecca’s city limits.PHI MANDALAS
Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and the MUSIC AND THE FIBONACCI SEQUENCE AND PHI GOLDEN RATIO MYTH, MATH AND MISUNDERSTANDING (FOR DEBUNKERS) The 2009 study claimed to have debunked the relationship of the golden ratio to beauty and claimed to have discovered two “new golden ratio” measurements that defined attractiveness. The faces identified as the most attractive, however, had over a dozen goldenratios in
DIVINE PROPORTION/GOLDEN RATIO IN THE ART OF DA VINCIDA VINCI’S ILLUSTRATIONS APPEAR IN PACIOLI’S BOOK “THE DIVINE PROPORTION”DA VINCI CREATED THE ILLUSTRATIONS FOR THE BOOK “DE DIVINA PROPORTIONE” (THE DIVINE PROPORTION) BY LUCA PACIOLI. IT WAS WRITTEN IN ABOUT 1497 AND F...GOLDEN RATIOS / DIVINE PROPORTIONS APPEAR IN RENAISSANCE ART IN PAINTINGS OF RELIGIOUS IMPORTANCEDA VINCI’S USE OF THE DIVINE PROPORTION IS EVIDENT IN SOME OF HIS OWN WORKS, EVEN BEFORE HIS COLLABORATION WITH PACIOLI. THE PAINTINGS I REVIEWED S...DA VINCI’S “THE LAST SUPPER” HAS MANY CLEAR EXAMPLES OF DIVINE PROPORTIONSPERHAPS ONE OF THE BEST ILLUSTRATIONS OF ITS USE IS IN “THE LAST SUPPER,” PAINTED BETWEEN 1494 AND 1498. VARIOUS DESIGN AND ARCHITECTURAL FEATURES...DA VINCI’S MONA LISA AND THE GOLDEN RATIOONE OF DA VINCI’S MOST FAMOUS PAINTINGS IS “LA JACONDE,” THE MONA LISA. THIS PAINTING WAS BEGUN IN ABOUT 1503 AND WORK ON IT CONTINUED FOR YEARS. I...ANALYSIS OF DA VINCI’S VITRUVIAN MANANOTHER OF DA VINCI’S MOST FAMOUS WORKS IS THAT OF THE VITRUVIAN MAN, CREATED AROUND 1490. THE OFFICIAL TITLE OF THE DRAWING IS “LE PROPORZIONI DEL...GOLDEN RATIOS ARE EASY TO IDENTIFY AND APPLY WITH SIMPLE TOOLSDIVINE PROPORTIONS ARE QUITE EASY FOR AN ARTIST TO APPLY. ALL IT TAKES IS A SIMPLE TWO PRONG GAUGE THAT PIVOTS AT ITS GOLDEN RATIO POINT. YOU TAKE...NOT EVERY DA VINCI’S PAINTING SHOWS CLEAR EVIDENCE OF GOLDEN RATIOSIF YOU REVIEW ALL OF DA VINCI’S PAINTINGS, YOU WILL LIKELY NOT FIND CLEAR EVIDENCE OF THE GOLDEN RATIO IN MANY OF THEM. HE MAY HAVE USED IT IN MANY... Divine proportions are quite easy for an artist to apply. All it takes is a simple two prong gauge that pivots at its golden ratio point. You take a measure on one side and then simply flip it around to get the golden ratio of that measure. Another popular design is the threeprong gauge.
FACIAL BEAUTY AND THE “NEW” GOLDEN RATIO (OR IS IT JUST 1UNIVERSITY STUDY DECLARES A “NEW” GOLDEN RATIO FOR FACIAL BEAUTY BUT VALIDATES PHI, THE GOLDEN RATIO, AS THE BASIS FOR PERCEPTIONS OF BEAUTY.A UNIVERSITY STUDY (PDF) BY PAMELA M. PALLETT, STEPHEN LINK AND KANG LEE AT THE UNIVERSITY OF TORONTO AND UNIVERSITY OF CALIFORNIA, SAN DIEGO ANNOU...BUT ALL THAT GLITTERS ISN’T GOLD …WITH THIS ARTICLE, I THINK YOU’LL AGREE THAT THE SEARCH IS OVER, THE PROOF EXISTS AND THE REAL GOLDEN RATIO IS STILL QUITE SPECIAL. IRONICALLY, THE...THE METHODOLOGY OF MEASURING AND CALCULATING THE “NEW” GOLDEN RATIOLET’S TAKE A DEEPER LOOK AT THE STUDY AND HOW THE “NEW” GOLDEN RATIOS ARE ACTUALLY BASED ON THE GOLDEN RATIO OF 1.618. THE RESEARCHERS CREATED A SE...THE CONCLUSIONS WERE FLAWED IN NOT UNDERSTANDING HOW THE GOLDEN RATIO IS FOUND IN THE HUMAN FACETHIS IS WHERE THE CONCLUSIONS OF THE STUDY WERE FLAWED. THE STUDY IN FACT ACTUALLY CONFIRMED THAT PERCEPTIONS OF BEAUTY ARE INDEED BASED ON THE GOL...THE FACES SELECTED AS MOST ATTRACTIVE SHOW CLEAR GOLDEN RATIO PROPORTIONSTHE FACE BELOW, JUDGED AS MOST ATTRACTIVE IN THE STUDY FOR VERTICAL PROPORTIONS, ILLUSTRATES THE FOLLOWING FACIAL FEATURES BASED ON GOLDEN RATIOS O...THE “NEW” GOLDEN RATIO OF 0.46 IS EASILY DERIVED FROM THE TRUE GOLDEN RATIO OF 1.618THE “NEW” GOLDEN RATIOS DECLARED BY THE STUDY CAN BE EASILY RECONCILED TO THE TRUE GOLDEN RATIO AS SHOWN BELOW. THE NUMBERS SHOWN ON THE PHOTO BELO...DOES THE GOLDEN RATIO DEFINE BEAUTY? THE SEARCH IS OVER, THE PROOF EXISTS AND IT’S STILL THE GOLD STANDARDAS REFERENCED ABOVE, SCIENCE DAILY QUOTED THE ONE OF THE RESEARCHERS AS SAYING “PEOPLE HAVE TRIED AND FAILED TO FIND THESE RATIOS SINCE ANTIQUITY”... The “new” vertical golden ratio of 0.36 is close to 0.38, which is simply the golden ratio reciprocal variation of 1/1.618 squared. Their vertical ratio of 0.36 varies from this, but was based on the hairline at the top of the forehead, which is a highly variable feature in thehuman face.
RAPHAEL AND THE GOLDEN RATIO IN RENAISSANCE ART Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance and lived from 1483 to 1520. He is recognized as one of the three great masters of that period, accompanied by Michelangelo and Leonardo da Vinci. His work is admired for its form, composition, and visual achievement of the ideal ofhuman
THE NAUTILUS SHELL SPIRAL AS A GOLDEN SPIRAL Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. THE GOLDEN RATIO: PHI, 1.618 Golden Ratio, Phi, 1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the Face. One source with over 100 articles and latestfindings.
GOLDEN RATIO DO IT YOURSELF PROJECTS Golden Section Gauge. Make your own golden section gauge using the template below. It will work best if you construct it from heavy cardboard stock or plastic. Drill holes and place a brad at each of the indicated points. When the gauge is adjusted the middle arm will always show the golden section or phi point between the two outerarms.
GOLDEN RATIO POINT OF THE EARTH Site visitors did alternate measures based on the miles shown by Google Earth from the North Pole to Mecca to the South Pole, submitting that the golden ratio point falls within the boundaries of Islam’s holy city. They found the North-South distance to be 19980 kilometers, and divided by 1.618034 to arrive at a distance of 7,631.68, which they said was within Mecca’s city limits.PHI MANDALAS
Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and the MUSIC AND THE FIBONACCI SEQUENCE AND PHI GOLDEN RATIO MYTH, MATH AND MISUNDERSTANDING (FOR DEBUNKERS) The 2009 study claimed to have debunked the relationship of the golden ratio to beauty and claimed to have discovered two “new golden ratio” measurements that defined attractiveness. The faces identified as the most attractive, however, had over a dozen goldenratios in
DIVINE PROPORTION/GOLDEN RATIO IN THE ART OF DA VINCIDA VINCI’S ILLUSTRATIONS APPEAR IN PACIOLI’S BOOK “THE DIVINE PROPORTION”DA VINCI CREATED THE ILLUSTRATIONS FOR THE BOOK “DE DIVINA PROPORTIONE” (THE DIVINE PROPORTION) BY LUCA PACIOLI. IT WAS WRITTEN IN ABOUT 1497 AND F...GOLDEN RATIOS / DIVINE PROPORTIONS APPEAR IN RENAISSANCE ART IN PAINTINGS OF RELIGIOUS IMPORTANCEDA VINCI’S USE OF THE DIVINE PROPORTION IS EVIDENT IN SOME OF HIS OWN WORKS, EVEN BEFORE HIS COLLABORATION WITH PACIOLI. THE PAINTINGS I REVIEWED S...DA VINCI’S “THE LAST SUPPER” HAS MANY CLEAR EXAMPLES OF DIVINE PROPORTIONSPERHAPS ONE OF THE BEST ILLUSTRATIONS OF ITS USE IS IN “THE LAST SUPPER,” PAINTED BETWEEN 1494 AND 1498. VARIOUS DESIGN AND ARCHITECTURAL FEATURES...DA VINCI’S MONA LISA AND THE GOLDEN RATIOONE OF DA VINCI’S MOST FAMOUS PAINTINGS IS “LA JACONDE,” THE MONA LISA. THIS PAINTING WAS BEGUN IN ABOUT 1503 AND WORK ON IT CONTINUED FOR YEARS. I...ANALYSIS OF DA VINCI’S VITRUVIAN MANANOTHER OF DA VINCI’S MOST FAMOUS WORKS IS THAT OF THE VITRUVIAN MAN, CREATED AROUND 1490. THE OFFICIAL TITLE OF THE DRAWING IS “LE PROPORZIONI DEL...GOLDEN RATIOS ARE EASY TO IDENTIFY AND APPLY WITH SIMPLE TOOLSDIVINE PROPORTIONS ARE QUITE EASY FOR AN ARTIST TO APPLY. ALL IT TAKES IS A SIMPLE TWO PRONG GAUGE THAT PIVOTS AT ITS GOLDEN RATIO POINT. YOU TAKE...NOT EVERY DA VINCI’S PAINTING SHOWS CLEAR EVIDENCE OF GOLDEN RATIOSIF YOU REVIEW ALL OF DA VINCI’S PAINTINGS, YOU WILL LIKELY NOT FIND CLEAR EVIDENCE OF THE GOLDEN RATIO IN MANY OF THEM. HE MAY HAVE USED IT IN MANY... Divine proportions are quite easy for an artist to apply. All it takes is a simple two prong gauge that pivots at its golden ratio point. You take a measure on one side and then simply flip it around to get the golden ratio of that measure. Another popular design is the threeprong gauge.
FACIAL BEAUTY AND THE “NEW” GOLDEN RATIO (OR IS IT JUST 1UNIVERSITY STUDY DECLARES A “NEW” GOLDEN RATIO FOR FACIAL BEAUTY BUT VALIDATES PHI, THE GOLDEN RATIO, AS THE BASIS FOR PERCEPTIONS OF BEAUTY.A UNIVERSITY STUDY (PDF) BY PAMELA M. PALLETT, STEPHEN LINK AND KANG LEE AT THE UNIVERSITY OF TORONTO AND UNIVERSITY OF CALIFORNIA, SAN DIEGO ANNOU...BUT ALL THAT GLITTERS ISN’T GOLD …WITH THIS ARTICLE, I THINK YOU’LL AGREE THAT THE SEARCH IS OVER, THE PROOF EXISTS AND THE REAL GOLDEN RATIO IS STILL QUITE SPECIAL. IRONICALLY, THE...THE METHODOLOGY OF MEASURING AND CALCULATING THE “NEW” GOLDEN RATIOLET’S TAKE A DEEPER LOOK AT THE STUDY AND HOW THE “NEW” GOLDEN RATIOS ARE ACTUALLY BASED ON THE GOLDEN RATIO OF 1.618. THE RESEARCHERS CREATED A SE...THE CONCLUSIONS WERE FLAWED IN NOT UNDERSTANDING HOW THE GOLDEN RATIO IS FOUND IN THE HUMAN FACETHIS IS WHERE THE CONCLUSIONS OF THE STUDY WERE FLAWED. THE STUDY IN FACT ACTUALLY CONFIRMED THAT PERCEPTIONS OF BEAUTY ARE INDEED BASED ON THE GOL...THE FACES SELECTED AS MOST ATTRACTIVE SHOW CLEAR GOLDEN RATIO PROPORTIONSTHE FACE BELOW, JUDGED AS MOST ATTRACTIVE IN THE STUDY FOR VERTICAL PROPORTIONS, ILLUSTRATES THE FOLLOWING FACIAL FEATURES BASED ON GOLDEN RATIOS O...THE “NEW” GOLDEN RATIO OF 0.46 IS EASILY DERIVED FROM THE TRUE GOLDEN RATIO OF 1.618THE “NEW” GOLDEN RATIOS DECLARED BY THE STUDY CAN BE EASILY RECONCILED TO THE TRUE GOLDEN RATIO AS SHOWN BELOW. THE NUMBERS SHOWN ON THE PHOTO BELO...DOES THE GOLDEN RATIO DEFINE BEAUTY? THE SEARCH IS OVER, THE PROOF EXISTS AND IT’S STILL THE GOLD STANDARDAS REFERENCED ABOVE, SCIENCE DAILY QUOTED THE ONE OF THE RESEARCHERS AS SAYING “PEOPLE HAVE TRIED AND FAILED TO FIND THESE RATIOS SINCE ANTIQUITY”... The “new” vertical golden ratio of 0.36 is close to 0.38, which is simply the golden ratio reciprocal variation of 1/1.618 squared. Their vertical ratio of 0.36 varies from this, but was based on the hairline at the top of the forehead, which is a highly variable feature in thehuman face.
RAPHAEL AND THE GOLDEN RATIO IN RENAISSANCE ART Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance and lived from 1483 to 1520. He is recognized as one of the three great masters of that period, accompanied by Michelangelo and Leonardo da Vinci. His work is admired for its form, composition, and visual achievement of the ideal ofhuman
THE NAUTILUS SHELL SPIRAL AS A GOLDEN SPIRAL Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. THE GOLDEN RATIO: PHI, 1.618 Golden Ratio, Phi, 1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the Face. One source with over 100 articles and latestfindings.
PHI, THE GOLDEN RATIO AND GEOMETRY The ratio of AG to AB is Phi. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating an angle of 51.83 degrees, the cosine of which is phi, or 0.618. A ruler and compass can be used to construct the “golden rectangle,” as shownby
THEOLOGY - THE GOLDEN RATIO: PHI, 1.618 - GOLDEN RATIO Is there meaning hidden in Phi, the symbol for the Golden Number? The use of the Greek letter Phi Phi to represent the golden number 1.618 is generally said to acknowledge Phidias, a 5th century B.C. sculptor and mathematician of ancient Greece, who studied phi and created sculptures for the Parthenon and Olympus. The APPEARANCES OF PHI, THE GOLDEN RATIO, IN THE SOLAR SYSTEM 149,600. 2.5833. 4.1521. Curiously enough we find: Ö Period of Venus * Phi = Distance of the Earth. Ö 2.5490 * 1.6180339 = 1.5966 * 1.6180339 = 2.5833. In addition, Venus orbits the Sun in 224.695 days while Earth orbits the Sun in 365.242 days, creating a ratio of 8/13 (both Fibonacci numbers) or 0.615 (roughly phi.) MUSIC AND THE FIBONACCI SEQUENCE AND PHI Musical scales are related to Fibonacci numbers. The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as: There are 13 notes in the span of any note through its octave. A scale is composed of 8 notes, of which the 5th andTHE PHI FORMULA
Phi – 1 = 1 / Phi. This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements: A = B + C. and. A/B = B/C. A is to B as B is to C, where. A is 161.8% of B and B is 161.8% of C, and. B is 61.8% of A and C is 61.8% of B. Let n be any integer other than 5 and you won’t BEAUTY IN THE HUMAN FACE AND THE GOLDEN RATIO 164 A.D. Rome. 1794 A.D. Dr. Marquardt has another section of his site that explores face variations by age, sex and ethnic group. Here he observes that “Each archetypal mask appears to be a species non-ethnic configuration. That is the more a face, any face, looks like the mask, the more attractive it THE DESIGN OF LIFE AND THE GOLDEN RATIO The design of life is based on a “Golden Ruler™, a ruler composed of golden ratios within golden ratios. It has long been known that the Golden Section, (also Golden Ratio or Divine Proportion), appears in certain proportions of living organisms. In 1997, I began to wonder how universal this was and how far it GOLDEN RATIO, BEAUTY AND DESIGN In October 2015, Dr. Keith Devlin was interviewed by science writer Julia Calderone in an article titled “The one formula that’s supposed to ‘prove beauty’ is fundamentally wrong.” Professor Devlin is a accomplished Ph.D. in mathematics at Stanford University and NPR’s Math Guy. His comments have reach and impact, but his views on the golden ratio are not always accurate, complete SEURAT AND THE GOLDEN RATIO IN ART COMPOSITION. Seurat and the Golden Ratio in Art Composition. Georges Seurat (1859–1891) was a French Post-Impressionist painter, who according to Wikipedia, “is noted for his innovative use of drawing media and for devising the technique of painting known as pointillism.”. It continues, “His large-scale work A Sunday Afternoon on the Island ofLa
PHI, THE GOLDEN RATIO AND GEOMETRY The ratio of AG to AB is Phi. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating an angle of 51.83 degrees, the cosine of which is phi, or 0.618. A ruler and compass can be used to construct the “golden rectangle,” as shownby
HUMAN GROWTH AND DEVELOPMENT Refinement of adult skills, parenting role. 55. Elder adult. Fulfillment of adult skills, serving, retirement begins with eligibility for Medicare, Social Security and AARP. 89. Completion. Insight and wisdom into life. The Fibonacci series found in the human age numbers below relate to both Phi (the golden ratio) and the Fibonacci and GoldenPHI MANDALAS
Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and theTHE PHI FORMULA
Phi – 1 = 1 / Phi. This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements: A = B + C. and. A/B = B/C. A is to B as B is to C, where. A is 161.8% of B and B is 161.8% of C, and. B is 61.8% of A and C is 61.8% of B. Let n be any integer other than 5 and you won’t QUANTUM GRAVITY, REALITY AND THE GOLDEN RATIO Quantum Gravity Research (QGR) is a research organization which states its mission as: “ QGR’s mission is to discover the geometric first-principles unification of space, time, matter, energy, information and consciousness. Our team of physicists, mathematicians and chemists are hard at work on what we call “emergence theory,”a new
CELEBRITY FACES AND THE GOLDEN RATIO: THIS TIME FOR REAL. This first gallery compares the fit of four of the celebrity faces to the horizontal golden ratio proportions often found in attractive faces. The most basic of these is found in the distance between the pupils and bottom of the chin in relation to the lip/teeth line and tip of the nose. Photos need to be taken from directly in front of the DIVINE PROPORTION/GOLDEN RATIO IN THE ART OF DA VINCI Divine proportions are quite easy for an artist to apply. All it takes is a simple two prong gauge that pivots at its golden ratio point. You take a measure on one side and then simply flip it around to get the golden ratio of that measure. Another popular design is the threeprong gauge.
MICHELANGELO AND THE ART OF THE GOLDEN RATIO IN DESIGN. The article claimed, incorrectly, that there was no previous association of Michelangelo’s works with the golden ratio. It identified the very same golden ratio touch point as the one reported here two years earlier. It also claimed, however, a new finding. It claimed that the fingers touched at the golden ratio point of thelength of all the
SEURAT AND THE GOLDEN RATIO IN ART COMPOSITION.SEE MORE ONGOLDENNUMBER.NET
THE NAUTILUS SHELL SPIRAL AS A GOLDEN SPIRAL Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. PHI, THE GOLDEN RATIO AND GEOMETRY The ratio of AG to AB is Phi. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating an angle of 51.83 degrees, the cosine of which is phi, or 0.618. A ruler and compass can be used to construct the “golden rectangle,” as shownby
HUMAN GROWTH AND DEVELOPMENT Refinement of adult skills, parenting role. 55. Elder adult. Fulfillment of adult skills, serving, retirement begins with eligibility for Medicare, Social Security and AARP. 89. Completion. Insight and wisdom into life. The Fibonacci series found in the human age numbers below relate to both Phi (the golden ratio) and the Fibonacci and GoldenPHI MANDALAS
Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and theTHE PHI FORMULA
Phi – 1 = 1 / Phi. This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements: A = B + C. and. A/B = B/C. A is to B as B is to C, where. A is 161.8% of B and B is 161.8% of C, and. B is 61.8% of A and C is 61.8% of B. Let n be any integer other than 5 and you won’t QUANTUM GRAVITY, REALITY AND THE GOLDEN RATIO Quantum Gravity Research (QGR) is a research organization which states its mission as: “ QGR’s mission is to discover the geometric first-principles unification of space, time, matter, energy, information and consciousness. Our team of physicists, mathematicians and chemists are hard at work on what we call “emergence theory,”a new
CELEBRITY FACES AND THE GOLDEN RATIO: THIS TIME FOR REAL. This first gallery compares the fit of four of the celebrity faces to the horizontal golden ratio proportions often found in attractive faces. The most basic of these is found in the distance between the pupils and bottom of the chin in relation to the lip/teeth line and tip of the nose. Photos need to be taken from directly in front of the DIVINE PROPORTION/GOLDEN RATIO IN THE ART OF DA VINCI Divine proportions are quite easy for an artist to apply. All it takes is a simple two prong gauge that pivots at its golden ratio point. You take a measure on one side and then simply flip it around to get the golden ratio of that measure. Another popular design is the threeprong gauge.
MICHELANGELO AND THE ART OF THE GOLDEN RATIO IN DESIGN. The article claimed, incorrectly, that there was no previous association of Michelangelo’s works with the golden ratio. It identified the very same golden ratio touch point as the one reported here two years earlier. It also claimed, however, a new finding. It claimed that the fingers touched at the golden ratio point of thelength of all the
SEURAT AND THE GOLDEN RATIO IN ART COMPOSITION.SEE MORE ONGOLDENNUMBER.NET
THE NAUTILUS SHELL SPIRAL AS A GOLDEN SPIRAL Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. PHI / GOLDEN RATIO PRODUCTS AND SERVICES Book: The Golden Ratio “The Golden Ratio – The Divine Beauty of Mathematics” was written by GoldenNumber.net site author, Gary B. Meisner. The book has an engaging balance between being an informative educational textbook and a visually inviting 9.2″ x 11.4″ coffee table book, with over 250 illustrations on 224 pages. Available in sixlanguages
LIGHT, THE HUMAN BODY, CHAKRAS AND THE GOLDEN RATIO Kalana Dido of France made a very insightful discovery in May 2021 that unites the visual spectrum of light, the proportions of the human body, the chakras (i.e, the centers of spiritual energy in the human body in Hindu tradition) and the Golden Ratio.THE PHI FORMULA
Phi – 1 = 1 / Phi. This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements: A = B + C. and. A/B = B/C. A is to B as B is to C, where. A is 161.8% of B and B is 161.8% of C, and. B is 61.8% of A and C is 61.8% of B. Let n be any integer other than 5 and you won’t GOLDEN RATIOS IN ENERGY RADIATION AND VIBRATIONS Unusual signals at Phi Hertz are being received on Earth. ELFRAD (Extremely Low Frequency Research And Development), a self-funded research organization which tracks radiation on Earth through an array of antennas, has found an extremely low frequency signal at phi, or 1.618033 Hertz, or cycles per second. Their site says the following: “This chart indicates theCHAKRAS ARCHIVES
Kalana Dido of France made a very insightful discovery in May 2021 that unites the visual spectrum of light, the proportions of the human body, the chakras (i.e, the centers of spiritual energy in the human body in Hindu tradition) and the Golden Ratio. THE DESIGN OF LIFE AND THE GOLDEN RATIO The design of life is based on a “Golden Ruler™, a ruler composed of golden ratios within golden ratios. It has long been known that the Golden Section, (also Golden Ratio or Divine Proportion), appears in certain proportions of living organisms. In 1997, I began to wonder how universal this was and how far it CELEBRITY FACES AND THE GOLDEN RATIO: THIS TIME FOR REAL. This first gallery compares the fit of four of the celebrity faces to the horizontal golden ratio proportions often found in attractive faces. The most basic of these is found in the distance between the pupils and bottom of the chin in relation to the lip/teeth line and tip of the nose. Photos need to be taken from directly in front of the BEST FACIAL BEAUTY ANALYSIS PHOTO TIPS TO REVEAL GOLDEN RATIOS 1. Orientation of the head, and eliminating the impacts of perspective. As simple as it sounds, one of the most important aspects of any image used for facial analysis is that the subject be looking directly into the camera. The front of the face must be parallel to the lens of the camera. There must be no tilt of the head either up,down, left
DR. LEVIN'S GOLDEN MEAN GAUGES Achieve excellence in whatever you do! The Golden Ratio, the phi relationship, has long been used by craftsmen and professionals of many fields to understand and achieve beauty, balance and harmony in their work. Artists, architects, designers, engineers, musicians, photographers, sculptors, surgeons and stock analysts alike use this relationship to achieve excellence in their results. MEISNER BEAUTY GUIDE FOR GOLDEN RATIO FACIAL ANALYSIS Meisner Beauty Guide for Golden Ratio Facial Analysis. March 21, 2016 by Gary Meisner 8 Comments. A beatiful female face. HW: A golden rectangle frames the head. Pupils to lip center line to chin. V2: Pupils to nostrils to chin. V3: Pupils to nose flare to PHI, THE GOLDEN RATIO AND GEOMETRYANTENNA PHI AND THETATHETA AND PHI building a small circular house with golden ratio (from argentina) hi dear people, im starting a project of building my circular little house here in argentina and would like to learn about the golden ratio for defining the height of the walls and the height and angles of the triangular roof that will sit on those walls for a diameter of a circle of 6.8 meters. im a bit lost, is it posible PHI AND FIBONACCI IN KEPLER AND GOLDEN TRIANGLES The Pythagorean 3-4-5 triangle is the only right-angle triangle whose sides are in an arithmetic progression. 3 + 1 = 4, and 4 plus 1 = 5. The Kepler triangle is the only right-angle triangle whose side are in a geometric progression: The square root of phi times Φ = 1 and 1times Φ = Φ.
MUSIC AND THE FIBONACCI SEQUENCE AND PHIPHI MANDALAS
Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and the GOLDEN RATIOS IN ENERGY RADIATION AND VIBRATIONS Unusual signals at Phi Hertz are being received on Earth. ELFRAD (Extremely Low Frequency Research And Development), a self-funded research organization which tracks radiation on Earth through an array of antennas, has found an extremely low frequency signal at phi, or 1.618033 Hertz, or cycles per second. Their site says the following: “This chart indicates thePHI IN THE BIBLE
Although perhaps not immediately obvious, phi and the golden section also appear in the Bible. Also see the Theology page. The Ark of the Covenant is uses Fibonacci numbers, approximating a Golden Rectangle In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten PI, PHI AND FIBONACCI Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways: The Pi-Phi Product and its derivation through limits The product of phi and pi, 1.618033988 X 3.141592654, or 5.083203692, is found in golden geometries: Golden Circle Golden Ellipse Circumference = p * Φ Area = p * Φ Ed Oberg and Jay A. Johnson PENROSE TILING AND PHI Tiling in 5-fold symmetry was thought impossible! Areas can be filled completely and symmetrically with tiles of 3, 4 and 6 sides, but it was long believed that it was impossible to fill an area with 5-fold symmetry, as shown below: 3 sides 4 sides 5 sides leaves gaps 6 sidesThe solution was
DIVINE PROPORTION/GOLDEN RATIO IN THE ART OF DA VINCI Da Vinci’s Mona Lisa and the Golden Ratio. One of Da Vinci’s most famous paintings is “La Jaconde,” the Mona Lisa. This painting was begun in about 1503 and work on it continued for years. THE NAUTILUS SHELL SPIRAL AS A GOLDEN SPIRAL Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. PHI, THE GOLDEN RATIO AND GEOMETRYANTENNA PHI AND THETATHETA AND PHI building a small circular house with golden ratio (from argentina) hi dear people, im starting a project of building my circular little house here in argentina and would like to learn about the golden ratio for defining the height of the walls and the height and angles of the triangular roof that will sit on those walls for a diameter of a circle of 6.8 meters. im a bit lost, is it posible PHI AND FIBONACCI IN KEPLER AND GOLDEN TRIANGLES The Pythagorean 3-4-5 triangle is the only right-angle triangle whose sides are in an arithmetic progression. 3 + 1 = 4, and 4 plus 1 = 5. The Kepler triangle is the only right-angle triangle whose side are in a geometric progression: The square root of phi times Φ = 1 and 1times Φ = Φ.
MUSIC AND THE FIBONACCI SEQUENCE AND PHIPHI MANDALAS
Phi-based geometric shapes produce interesting mandala designs. A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation. Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and the GOLDEN RATIOS IN ENERGY RADIATION AND VIBRATIONS Unusual signals at Phi Hertz are being received on Earth. ELFRAD (Extremely Low Frequency Research And Development), a self-funded research organization which tracks radiation on Earth through an array of antennas, has found an extremely low frequency signal at phi, or 1.618033 Hertz, or cycles per second. Their site says the following: “This chart indicates thePHI IN THE BIBLE
Although perhaps not immediately obvious, phi and the golden section also appear in the Bible. Also see the Theology page. The Ark of the Covenant is uses Fibonacci numbers, approximating a Golden Rectangle In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten PI, PHI AND FIBONACCI Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways: The Pi-Phi Product and its derivation through limits The product of phi and pi, 1.618033988 X 3.141592654, or 5.083203692, is found in golden geometries: Golden Circle Golden Ellipse Circumference = p * Φ Area = p * Φ Ed Oberg and Jay A. Johnson PENROSE TILING AND PHI Tiling in 5-fold symmetry was thought impossible! Areas can be filled completely and symmetrically with tiles of 3, 4 and 6 sides, but it was long believed that it was impossible to fill an area with 5-fold symmetry, as shown below: 3 sides 4 sides 5 sides leaves gaps 6 sidesThe solution was
DIVINE PROPORTION/GOLDEN RATIO IN THE ART OF DA VINCI Da Vinci’s Mona Lisa and the Golden Ratio. One of Da Vinci’s most famous paintings is “La Jaconde,” the Mona Lisa. This painting was begun in about 1503 and work on it continued for years. THE NAUTILUS SHELL SPIRAL AS A GOLDEN SPIRAL Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. PHI AND FIBONACCI IN KEPLER AND GOLDEN TRIANGLES The Pythagorean 3-4-5 triangle is the only right-angle triangle whose sides are in an arithmetic progression. 3 + 1 = 4, and 4 plus 1 = 5. The Kepler triangle is the only right-angle triangle whose side are in a geometric progression: The square root of phi times Φ = 1 and 1times Φ = Φ.
THE GOLDEN SECTION / GOLDEN RATIO GoldenNumber.Net explores the appearance of Phi, 1.618 (also known as the Golden Ratio, Golden Mean, Golden Section or Divine Proportion, in mathematics, geometry, life and the universe and shows you how to apply it, and its applications are limitless: Art Architecture Design of any kind – Graphics, logos, products, fashion, web sites and more Photo composition, photo cropping matting GOLDEN RATIO DO IT YOURSELF PROJECTS Golden Ruler Grid™ If you have a printer and transparencies, you can print the “Golden Ruler”© grid below and use it to see the golden section and phi in various objects. Simply hold the transparency in front of you and match the lines up to the object to see if its designis based on
LIGHT, THE HUMAN BODY, CHAKRAS AND THE GOLDEN RATIO Kalana Dido of France made a very insightful discovery in May 2021 that unites the visual spectrum of light, the proportions of the human body, the chakras (i.e, the centers of spiritual energy in the human body in Hindu tradition) and the Golden Ratio. GOLDEN RATIOS IN ENERGY RADIATION AND VIBRATIONS Unusual signals at Phi Hertz are being received on Earth. ELFRAD (Extremely Low Frequency Research And Development), a self-funded research organization which tracks radiation on Earth through an array of antennas, has found an extremely low frequency signal at phi, or 1.618033 Hertz, or cycles per second. Their site says the following: “This chart indicates the THEOLOGY - THE GOLDEN RATIO: PHI, 1.618 - GOLDEN RATIO Is there meaning hidden in Phi, the symbol for the Golden Number? The use of the Greek letter Phi Phi to represent the golden number 1.618 is generally said to acknowledge Phidias, a 5th century B.C. sculptor and mathematician of ancient Greece, who studied phi and created sculptures for the Parthenon and Olympus. ThePHI IN THE BIBLE
Although perhaps not immediately obvious, phi and the golden section also appear in the Bible. Also see the Theology page. The Ark of the Covenant is uses Fibonacci numbers, approximating a Golden Rectangle In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten GOLDEN RATIOS IN GREAT PYRAMID OF GIZA SITE TOPOGRAPHY Overview: Analysis of the site of the Great Pyramid of Giza reveals that the positions and relative sizes of the pyramids may be based on the golden ratio. Evidence of the Golden Ratio in the Great Pyramid complex. There are many pyramid theories and questions as PENROSE TILING AND PHI Tiling in 5-fold symmetry was thought impossible! Areas can be filled completely and symmetrically with tiles of 3, 4 and 6 sides, but it was long believed that it was impossible to fill an area with 5-fold symmetry, as shown below: 3 sides 4 sides 5 sides leaves gaps 6 sidesThe solution was
PI, PHI AND FIBONACCI Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways: The Pi-Phi Product and its derivation through limits The product of phi and pi, 1.618033988 X 3.141592654, or 5.083203692, is found in golden geometries: Golden Circle Golden Ellipse Circumference = p * Φ Area = p * Φ Ed Oberg and Jay A. Johnson* Φ
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THE GOLDEN RATIO: PHI, 1.618 (SEE INDEX)
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FEATURED ARTICLES: MATH OR MYTH? AN OVERVIEW AND THE EVIDENCE GOLDEN RATIO OVERVIEW What makes a single number so interesting that ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it? It's a number that goes by many names. This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section. It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in "De Divina Proportione" in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know Phi was brought even more into the mainstream of pop culture. The allure of “The Da Vinci Code” was that it creatively integrated fiction with both fact and myth from art, history, theology and mathematics, leaving the reader never really knowing what was truth and … More on Math, Myth and Truth GOLDEN RATIO MYTH, FACT AND MISUNDERSTANDING: THE MISSING EVIDENCE There are many misconceptions and misrepresentations about the golden ratio. Some look too fervently for patterns and say it exists where it really doesn't. Some whose goal is to debunk golden ratio myth say it doesn't exist where it really does, missing the obvious and often not stating what proportions appear instead. People on both sides often just repeat what they've heard rather than personally performing the analysis required to support their conclusions. Intelligence and education are not always factors in getting to the truth, as even Ph.D.'s in mathematics sometimes get it wrong. As the author of this site since 1997, I've changed my views and the information on this site as well. Let's look at some of the common points of confusion and debate, covering beauty, the Parthenon, the UN Secretariat Building, the Great Pyramid, Nautilus shell, use by famous artists (Da Vinci, Botticelli, Seurat, etc.) and other topics. I'll provide objective answers, with additional evidence in the … More on Math, Myth andTruth
LATEST BLOG ARTICLES GARY MEISNER’S LATEST TWEETS ON THE GOLDEN RATIO Here are the latest Tweets on the golden ratio and related topics by Gary Meisner, founder and author of GoldenNumber.net and developer of PhiMatrix golden ratio analysis and design software: Tweets by GaryBMeisner … More on this article DR. GEORGE MARKOWSKY’S “MISCONCEPTIONS ABOUT THE GOLDEN RATIO”REVIEWED
Debunking the Debunkers of Golden Ratio Myths. Overview: This article reviews Dr. George Markowsky's article on eight misconceptions about the golden ratio. He starts his analysis by establishing a very generous ±2% range for considering evidences of the golden ratio as acceptable. In four cases though, evidences for the original claims are missed or ignored. In two cases, the measurements are taken incorrectly. In two cases, the measurements taken are too limited to be conclusive. In one case, no measurements are taken at all. The arguments are accompanied by weak or incomplete evidence, which is used to dismiss the measurements that vary from the golden ratio by less than 1%. A flawed assumption is made that there are "infinitely numbers near Phi" that could have been used instead. This is of course true in theoretical mathematics, but false with respect to physical measurements in the real world, which are limited by significant figures of accuracy. In addition, the number of … More on thisarticle
DISCOVERY OF THE GOLDEN RATIO IN MICHELANGELO’S “THE CREATION OFADAM.”
2015 study announces the discovery of the Golden Ratio in the Sistene Chapel. Clinical Anatomy published an article on July 17, 2015 by a team from Brazil titled "More than a neuroanatomical representation in The Creation of Adam by Michelangelo Buonarroti, a representation of the Golden Ratio." (PDF here.) The excitement of the authors in making this discovery for themselves is understandable, and their creativity is commendable. The discovery though was not entirely new, and it does not appear to be entirely accurate, nor complete. The Brazilian study though motivated me to take a deeper look at Michelangelo's composition of his paintings in the Sistine Chapel. This led to much richer evidence yet of Michelangelo's use of the golden ratio, also known as the Divine Proportion. These new findings are presented in a separate article, Michelangelo and the Art of the Golden Ratio in Design. Analysis of the findings of the 2015 Brazilian study. The authors of the Brazilian study … More on this article ANOTHER FASTONE BY FASTCODESIGN ON THE GOLDEN RATIO? Was it routine site maintenance or censorship at FastCoDesign? Why delete reader comments? On April 13, 2015, FastCoDesign.com took a stand on design with an article by John Brownlee titled “The Golden Ratio: Design’s Biggest Myth – The Golden Ratio is Total Nonsense in Design. Here’s Why.” (Find article here.) By the end of June 2015, the article had hundreds of comments from readers, almost entirely negative, as readers expressed their anger, disappointment and frustration at the ignorance, inaccuracies and bias in the poorly researched and lamely written article. And then all the comments just disappeared! Descriptions of the article used by readers included "appalling, sensationalist, dangerous, stupid, under-informed nonsense, an exercise in ignorance and conceit, seething with misinformation, misleading, utter nonsense, profound ignorance, lame, entirely incorrect, click bait, what a troll, simplistic, naive, puerile, opinionated, unsophisticated, boring, fallacious, … More onthis article
APPLICATIONS FOR BETTER ART, DESIGN AND COMPOSITION GOLDEN RATIO, PHI AND FIBONACCI COMMEMORATIVE POSTAGE STAMPS Governments around the world commemorate the Golden Ratio, Phi and Fibonacci sequence. Stamp collecting is one of the world's most popular educational hobbies, and is enjoyed by people of all ages and nationalities. The most popular stamp ever is US 1993 Elvis stamp, with over 124 million collected. So, the Golden Ratio and Fibonacci sequence may not compete with Elvis in popularity anytime soon, but they too have been recognized with commemorative stamps issued by governments around the world. In addition to adding some "phi" to "philately," they offer some interesting expressions of golden ratio concepts, as well as its application in design. Liechtenstein 2013 Commemorative Fibonacci Sequence and Phi Stamp set: The Principality of Liechtenstein, a landlocked micro-state bordered by Switzerland and Austria, issued a set of three stamps in 2013 that illustrate the Fibonacci sequence and its relationship to the golden ratio. Note that interesting presentation of concepts: … More on Art and Design THE GOLDEN RATIO IN CHARACTER DESIGN, CARTOONS AND CARICATURES The golden ratio is Stephen Silver's secret weapon of character design. Stephen Silver is a character designer and art teacher who has designed characters for Disney Television Animation, Sony Feature Animation and Nickelodeon Animation. In this role he has designed the style of the shows such as "Kim Possible", Kevin Smith's "Clerks" the animated series, "Danny Phantom" and many others. He is also the author and artist of seven books on the art of character design, caricature, sketching and life drawing. In addition to his freelance work and lectures, he owns an art school called Silver Drawing Academy in Simi Valley, CA. He has two online character design courses online at www.schoolism.com. Those are some great credentials. And what does he say is his "secret weapon" for character design? Yes, it's the golden ratio. The golden ratio is flawless because it gives the "small, medium and large" that design is built upon. He describes the golden ratio is just perfect, flawless, … More on Art and Design GOLDEN RATIOS IN GREAT PYRAMID OF GIZA SITE TOPOGRAPHY Overview: Analysis of the site of the Great Pyramid of Giza reveals that the positions and relative sizes of the pyramids may be based on the golden ratio. Evidence of the Golden Ratio in the Great Pyramid complex. There are many pyramid theories and questions as to who built the pyramids in ancient Egypt. It's commonly known though in Egyptology that the proportions of the Great Pyramid of Egypt are within inches of a golden ratio-based pyramid. This is discussed in detail in my article Phi, Pi and the Great Pyramid. Some say, however, that this single piece of evidence is just a simple coincidence. The primary rationale given to deny this claim is that there is no written historical evidence that the ancient Egyptians had any knowledge of the golden ratio. That's a reasonable objection, but what if the evidence showed that the application of the golden ratio at Giza was not limited to the Great Pyramid? That evidence is presented in this article. The golden ratio connection … More on Art and Design MICHELANGELO AND THE ART OF THE GOLDEN RATIO IN DESIGN AND COMPOSITION New study unveils Michelangelo's extensive use of the golden ratio in the Sistine Chapel. In 2013, I reported that Michelangelo used the golden ratio in his painting "The Creation of Adam" in the Sistine Chapel. This Divine Proportion appears at the point at which Adam's finger is touched by the finger of God, as God breathes life into Adam. In July 2015, Clinical Anatomy published an article by a team from Brazil titled "More than a neuroanatomical representation in The Creation of Adam by Michelangelo Buonarroti, a representation of the Golden Ratio." (PDF here.) The article claimed, incorrectly, that there was no previous association of Michelangelo's works with the golden ratio. It identified the very same golden ratio touch point as the one reported here two years earlier. It also claimed, however, a new finding. It claimed that the fingers touched at the golden ratio point of the length of all the paintings on the ceiling of the Sistine Chapel. My analysis of this claim … More on Art and Design GOOGLE LOGO AND THE GOLDEN RATIO IN DESIGN New Google logo design finds visual harmony using the Golden Ratio. Google's design follows in the footsteps of Leonardo da Vinci and other masters When Luca Pacioli published "The Divine Proportion" in 1509 (with illustrations by Leonardo da Vinci), he described his work on this "golden ratio" of 1.618 as a "very delicate, subtle and admirable teaching" that would "delight in diverse questions touching on a very secret science." Johannes Kepler later called it "a precious jewel" of geometry. The designers at Google have apparently found its value too, as we see when we study and appreciate the underlying design of Google's new logo, iconic G, the microphone icon and even the layout of the Google search page. This is the kind of thoughtful design work that follows in the footsteps of Leonardo, Michelangelo, Raphael, Botticelli, Seurat, Le Corbusier and other masters of design, and that would make Pacioli proud. Here's a version without the arrows for a clearer … More on Art and Design RAPHAEL AND THE GOLDEN RATIO IN RENAISSANCE ART Raphael was one of three Master artists of the Renaissance Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance and lived from 1483 to 1520. He is recognized as one of the three great masters of that period, accompanied by Michelangelo and Leonardo da Vinci. His work is admired for its form, composition, and visual achievement of the ideal of human grandeur. One of his most famous works is The School of Athens, a fresco in the Apostolic Palace in the Vatican. It captures the spirit of the Renaissance, and is revered as his masterpiece. It was painted between 1509 and 1511. The School of Athens: Inspired by a union of art and mathematics It was also in 1509 that Luca Pacioli published the book De Divina Proportione (The Divine Proportion), with illustrations by Leonardo da Vinci. MonaLisa.org reports that The School of Athens "incorporates many of the mathematical theories of Luca and Leonardo." "Civilisation" author Kenneth … More on Art andDesign
LEONARDO DA VINCI, SALVATOR MUNDI AND THE DIVINE PROPORTION See the YouTube video "Leonardo da Vinci's "Salvator Mundi" and the Divine Proportion." 2017 Update - Salvator Mundi, "the last Leonardo" sells for a record-breaking $450,312,500, decimating the prior world record for the most expensive work of art sold at auction. See Christie's article and learn below some of what makes this piece so incredible and valuable. In 2011, the discovery of a lost painting by Leonardo da Vinci was announced to the world. This painting, Salvator Mundi, had been in the art collection of King Charles I of England in 1649, was auctioned in 1763 and then lost for many years. Its recovery was led by Robert Simon, an art historian and private art dealer, and it was restored by Dianne Dwyer Modestini. Many unique qualities of this painting led experts to confirm that it is indeed an original work of Leonardo da Vinci, one of only fifteen now in existence. Da Vinci and the Golden Ratio in Art Composition. A lesson for all artists and designers. This article … More on Art and Design DA VINCI AND THE DIVINE PROPORTION IN ART COMPOSITION Leonardo Da Vinci has long been associated with the golden ratio. This association was reinforced in popular culture in 2003 by Dan Brown's best selling book "The Da Vinci Code." The plot has pivotal clues involving the golden ratio and Fibonacci series. In 2006, the public awareness of the association grew when the book was turned into a movie starring veteran actor Tom Hanks. Da Vinci's association with the golden ratio, known in his time as the Divine proportion, runs much longer and deeper. Da Vinci's illustrations appear in Pacioli's book "The Divine Proportion" Da Vinci created the illustrations for the book "De Divina Proportione" (The Divine Proportion) by Luca Pacioli. It was written in about 1497 and first published in 1509. Pacioli was a contemporary of Da Vinci's, and the book contains dozens of beautiful illustrations of three-dimensional geometric solids and templates for script letters in calligraphy. The original manuscript can be viewed online … More on Art and Design THE UN SECRETARIAT BUILDING, LE CORBUSIER AND THE GOLDEN RATIO Some claim that the design of the United Nations headquarters building in New York City exemplifies the application of the golden ratio in architecture. Debunkers of the golden ratio say no, that this is just another groundless myth to be dispelled. Let's look at the history of its design, the sources of the claims and mathematics of the dimensions. Perhaps we can come to a solid conclusion that can be agreed upon by all. A lead architect of the UN Building, Le Corbusier, created a system of design based on the golden ratio The building, known as the UN Secretariat Building, was started in 1947 and completed in 1952. The architects for the building were Oscar Niemeyer of Brazil and and the Swiss/French architect Le Corbusier. On Corbusier, Wikipedia states: Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. Explaining the Modulor, Wikipedia states: Le Corbusier developed the Modulor in the long tradition of Vitruvius, … More on Art and Design LOGO DESIGN USING THE GOLDEN RATIO Phi is often applied in product logos. From Renaissance artists of the 1500's to graphic artists of today, phi is recognized for its ability to give a sense of aesthetic appeal in balance and harmony of design. Product logos represent an image that must make a positive and memorable impact on the conscious and subconscious minds of consumers, so it is no surprise to find phi proportions in many logo design of major companies. The Phi grid proportions are provided by PhiMatrix software. See other examples of logo design at the PhiMatrix site. Note: The logos presented on this page are for illustration purposes only of principles of graphic design and are do not imply in any way an endorsement of, or affiliation with, this web site or its affiliate web sites by the companies shown. Note how every dimensions of each letter of this logo is apparently based on proportions of phi (first golden ratio) or phi squared (second golden … More on Art and Design PRODUCT DESIGN AND MARKETING APPLICATIONS OF THE GOLDEN MEAN Phi is used in the design of many consumer products. Phi has been used to bring beauty, balance and harmony to some of the world's greatest art and architecture. It is also used to add style and appeal in the marketing and design of everyday consumer products. The applications are endless, but are illustrated by a few of the products below. Click on the image for a larger version. The gauge shown here was developed by Dr. Eddy Levin and is offered for sale on this site. Photos with the gauges are also courtesy of Dr. Levin. Photos with golden ratio grid lines were created with PhiMatrix software. For another example, note the dimensions of the classic Hewlett Packard HP12C Financial calculator. The official dimensions on the HP site are 5” x 3.1” which has a ratio of 1.6129: 5” x 3” would have been the simplest, most obvious English measure dimensions. That would have resulted in a width to height ratio of 1.6667. Dimensions of 5” x 3.2” would have resulted in a … More on Artand Design
ASTON MARTIN, JAMES BOND AND THE GOLDEN RATIO What do James Bond, Aston Martin and the Golden Ratio have in common? James Bond, also known as 007, drove an Aston Martin DB5 in the movies GoldFinger and GoldenEye, and Aston Martin is now boasting its application of the Golden Ratio in the design of its latest DB9 and Rapide S automobiles. The Aston Martin Rapide S is described as: "Breathtaking Proportions - The ‘Golden Ratio’ sits at the heart of every Aston Martin. Balanced from any angle, each exterior line of Rapide S works in concert and every proportion is precisely measured to create a lithe, pure form. Our engineering follows the same principle. A near perfect weight distribution ensures Rapide S is balanced in form and balanced in function." The Aston Martin DB9 is described as: "Perfectly Proportioned - Every inch of DB9's form is designed for elegance and balance. The simple beauty of nature guides the design of DB9, with the 'golden ratio' setting all proportions. The result is a profile where every line, dimension … More on Artand Design
STOCK MARKET AND FOREX TRADING ANALYSIS PHI AND FOOTBALL STRATEGY’S GOLDEN RATIO RULE Using the Golden Ratio to win at football. The golden ratio is found in many places, but here's one you might not have expected: It can be found in football gaming strategy when making the "point after touchdown" decision as to whether to kick for 1 extra point or go for a 2 point conversion. This applies to the situation when a team is trailing by 14 points late in a football game, and then scores a touchdown to be trailing the other team by 8 points. So what should you do: Kick for 1 point after touchdown (PAT) or go for a 2 point conversion? Most football teams kick the PAT in this situation, but in fact it is (almost) always better to go for 2. The reason is related to … More on Markets/Gaming STOCK MARKET ANALYSIS, PHI AND THE FIBONACCI SEQUENCE Human expectations occur in a ratio that approaches Phi. Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative. Phi and Fibonacci numbers are used to predict stocks Phi (1.618), the Golden Mean and the numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8, ...) have been used with great success to analyze and predict stock market moves, known as retracements. Forbes ASAP featured a story on the work of scientist Stephen Wolfram in cellular … More on Markets/Gaming WHAT DETERMINES OUR PERCEPTIONS OF BEAUTY? MEISNER BEAUTY GUIDE FOR GOLDEN RATIO FACIAL ANALYSIS Facial golden ratios are simple, clear and abundant. Since my first investigation of human facial proportions in 1997, it seemed clear that the most important features that define a human face are the simplest and most obvious: An oval head and the positions and dimensions of the eyes and the mouth. As evidence, consider the "smiley face." It expresses a variety of facial expressions with nothing more than a circular head, two eyes and a mouth line: We immediately recognize this icon as a face, but it clearly does not represent the proportions of a human face. Let's instead express these same facial markers with normal human proportions, and with the simplest possible representation. This requires a vertical oval, two lines for the eyes and a line for the mouth. The resulting face will look something more like this: This simple facial configuration is easily mapped to very natural human proportions with the use of four simple golden ratio … More on Beauty HOW TO TAKE PHOTOS AND MEASUREMENTS FOR FACIAL BEAUTY ANALYSIS AND IDENTIFICATION OF GOLDEN RATIO PROPORTIONS Best practice techniques for facial and beauty analysis. Numerous studies have been done to investigate the existence and/or relationship of golden ratio proportions in the human face. Surprising few get it right, so the results of many of these studies are inaccurate and misleading. There are four important keys to selecting photographs that are suitable for facial measurements. Next, it is critical to understand and measure the facial markers that commonly reflect golden ratio proportions. In this article I'll cover these key points, the common mistakes made and the best practices to yield results that are both accurate and meaningful. 1. Orientation of the head, and eliminating the impacts of perspective. As simple as it sounds, one of the most important aspects of any image used for facial analysis is that the subject be looking directly into the camera. The front of the face must be parallel to the lens of the camera. There must be no tilt of the head either up, down, left or … More onBeauty
THE GOLDEN RATIO, BEAUTY AND DESIGN: IT’S TIME TO ‘FACE’ THEFACTS.
Another attempt to debunk the Golden Ratio is fundamentally wrong. In October 2015, Dr. Keith Devlin was interviewed by science writer Julia Calderone in an article titled "The one formula that's supposed to 'prove beauty' is fundamentally wrong." Professor Devlin is a accomplished Ph.D. in mathematics at Stanford University and NPR's Math Guy. His comments have reach and impact, but his views on the golden ratio are not always accurate, complete, consistent, objective or supported by evidence. Earlier in 2015, he was featured by a leading online design magazine in an article titled "The Golden Ratio: Design’s Biggest Myth – The Golden Ratio is Total Nonsense in Design. Here’s Why." The article drew hundreds of negative responses (which were later censored and deleted by the magazine) because of its inaccuracies and misrepresentations. A highly regarded NYC designer, Darrin Crescenzi, described the article as "dangerous" and "under-informed nonsense," motivating him to write a … More onBeauty
CELEBRITY FACES AND THE GOLDEN RATIO: THE REAL STORY. Do these celebrities owe their good looks to the Golden Ratio? Click on images for full resolution. When a joke becomes science, then science becomes a joke. A designer in Russia morphed several celebrity faces as a joke, but it somehow morphed into "scientific evidence" in yet another attempt to debunk the golden ratio. This time it was on Wired.com, a major online magazine site. (Find article here.) Igor Kochmala created some fun images of celebrity faces that he morphed to fit nonsensical patterns of golden ratio spirals. His blog page, called "Yet-another-Fibonacci-homework," humorously featured his "Dr. Fibonazi's Plastic Surgery Clinic." There he transformed five celebrity faces and the Mona Lisa to goofy looking images to supposedly "bring natural harmony" to their faces, as shown below. Click on any image for full resolution. The blog post was picked up by dozens of online magazine sites. Smash.com described the images as "hilarious distortions." … More on Beauty THE WORLD’S MOST PERFECT FACE: JOAN SMALLS? ELLE SAYS YES! GOLDENRATIO SAYS …
In 2014, Elle Magazine announced “Meet the very real woman behind the planet’s most perfect face: It’s Joan Smalls. Supermodels, over? As if. Joan Smalls wields the kind of flawless, unequivocal beauty that is both a throwback to the past and a sign that fashion is finally facing the future.” The Elle article describes Joan's background and rise to supermodel status, saying, "The hair is long and straight, the bone structure aquiline, the smile dazzling. No wonder Smalls is kicking up the kind of fashion-industry excitement that comes along only a handful of times each decade. Over the past year, she has snagged the number-one spot on Models.com, having pulled in a rumored $3.5 million—making her the eighth-highest-paid model in the world, according to Forbes." (Follow her on Facebook, Instagram and Twitter.) Is "flawless, unequivocal beauty" marketing hype or mathematical fact? LA Times reporter Rene Lynch was intrigued, but wanted to do some investigative reporting on this … More on Beauty FACIAL ANALYSIS AND THE MARQUARDT BEAUTY MASK "Beauty is in the phi of the beholder." It has long been said that beauty is in the eye of the beholder and thought that beauty varies by race, culture or era. The evidence, however, shows that our perception of physical beauty is hard wired into our being and based on how closely the features of one's face reflect phi in their proportions. The Golden Ratio appears extensively in the human face, as demonstrated in a 2009 university study on attractiveness and as illustrated by the video below of Florence Colgate, Britain's "Most Perfect Face" of 2012, : http://www.youtube.com/watch?v=kKWV-uU_SoI The image analysis shown in the video was done with PhiMatrix Golden Ratio Design and Analysis Software But let's take a deeper look yet at beauty based on analysis of the evidence. A template for human beauty is found in phi and the pentagon Dr. Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery. Dr. Marquardt performed … More on Beauty BEAUTY, THE PERFECT FACE AND THE GOLDEN RATIO, FEATURING FLORENCECOLGATE
What determines the beauty of a perfect face? We may never answer the question "how many angels can dance on the point of a needle," but how about this one: "How many Divine proportions ratios are there in the face of an angel?" The YouTube video and information in this article may give some insight into the answer, with details on the contest, a recap of explanations of beauty that fall short and new insights and a video illustrating the impact of the golden ratio in beauty. The British contest in 2012 to find "Britain's Perfect Face" draws over 8,000 entries, and finds Florence Colgate as the winner British company Lorraine Cosmetics sponsored a contest in 2012 to find “Britain’s Perfect Face.” Billed as the "Naked Competition," and seeking the most perfect makeup-free face, contestants were required to have completely natural faces, without make-up, botox or cosmetic surgery. The competition drew entries from 8,045 contestants, who were initially screened by a panel that … More on Beauty FACIAL BEAUTY AND THE "NEW" GOLDEN RATIO (OR IS IT JUST 1.618 INDISGUISE?)
University study declares a "new" golden ratio for facial beauty but validates Phi, the Golden Ratio, as the basis for perceptions of beauty. A university study (PDF) by Pamela M. Pallett, Stephen Link and Kang Lee at the University of Toronto and University of California, San Diego announced that a "new" golden ratio had been found in perceptions of beauty in the human face. The study also said there was "little support for the Golden Ratio" that "dates back to antiquity, when the ancient Greeks believed beauty was represented by a Golden Ratio of 1:1.618." Science Daily, in reporting on this study, quoted the one of the researchers as saying "People have tried and failed to find these ratios since antiquity" and "there was never any proof that the golden ratio was special. As it turns out, it isn't." But all that glitters isn't gold ... With this article, I think you'll agree that the search is over, the proof exists and the real Golden Ratio is still quite special. … More on BeautyWHAT IS PHI?
WHAT IS PHI? (THE BASICS OF THE GOLDEN RATIO) Phi for "Neo-Phi-tes:" Phi ( Φ = 1.618033988749895... ), most often pronounced fi like "fly," is simply an irrational number like pi ( p = 3.14159265358979... ), but one with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation. Phi is the basis for the Golden Ratio, Section or Mean The ratio, or proportion, determined by Phi (1.618 ...) was known to the Greeks as the "dividing a line in the extreme and mean ratio" and to Renaissance artists as the "Divine Proportion" It is also called the Golden Section, Golden Ratio and the Golden Mean. Phi, like Pi, is a ratio defined by a geometric construction Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi () 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 …More on Phi
WHAT IS THE FIBONACCI SEQUENCE (AKA FIBONACCI SERIES)? Leonardo Fibonacci discovered the sequence which converges on phi. In the 1202 AD, Leonardo Fibonacci wrote in his book "Liber Abaci" of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi. This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning "son of (the) Bonacci". Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . . This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze. Click to enlarge. The relationship of the Fibonacci sequence to the golden ratio is this: The ratio of each successive pair of … More on Phi THE GOLDEN SECTION / GOLDEN RATIO GoldenNumber.Net explores the appearance of Phi, 1.618 (also known as the Golden Ratio, Golden Mean, Golden Section or Divine Proportion, in mathematics, geometry, life and the universe and shows you how to apply it, and its applications are limitless: Art Architecture Design of any kind – Graphics, logos, products, fashion, web sites and more Photo composition, photo cropping matting and framing Personal beauty and facial or dental cosmetic procedures to enhance beauty Stock market and FOREX analysis The Golden Section is a ratio based on a the number Phi, 1.618... The Golden Section or Ratio is is a ratio or proportion defined by the number Phi (= 1.618033988749895... ) It can be derived with a number of geometric constructions, each of which divides a line segment at the unique point where: the ratio of the whole line (A) to the large segment (B) is the same as the ratio of the large segment (B) to the small segment (C). In other words, A is to B as B is to … More on Phi APPEARANCES IN LIFE AND NATURE IS THE NAUTILUS SHELL SPIRAL A GOLDEN SPIRAL? Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. Some say yes, but offer no proof at all. Some show examples of spirals, but incorrectly assume that every equi-angular spiral in nature is a golden spiral. Several university math professors say no, but they only compared the nautilus spiral to the spiral created from a golden rectangle. Another university professor says no, but only measured height and width of the entire shell. Let's look at this objectively and solve this mystery and debate. The Golden Spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral. A traditional Golden Spiral is formed by the nesting of Golden Rectangles with a Golden Rectangle. This resulting Golden Spiral is often associated with … More onNature
APPEARANCES IN THE COSMOS STARS THAT PULSATE TO THE GOLDEN RATIO A research article called "Strange Nonchaotic Stars" published at the American Physical Society reported the discovery of a class of white-blue variable stars that pulsate in a fractal pattern at frequencies close to the golden ratio. Download the paper in PDFformat
here: strange-nonchaotic-stars-pulsate-golden-ratio-1501.01747v2 Pulsations described as "strange nonchaotic behavior." The discovery was reported by John F. Lindner, Vivek Kohar, Behnam Kia, Michael Hippke, John G. Learned, and … More on the Cosmos PHI AND THE SOLAR SYSTEM The dimensions of the Earth and Moon are in Phi relationship, forming a Triangle based on 1.618. The illustration shows the relative sizes of the Earth and the Moon to scale. Draw a radius of the Earth (1). Draw a line from the center point of the Earth to the center point of the Moon (square root of Phi). Draw a line to connect the two lines to form a Golden Triangle (Phi). Using dimensions from Wikipedia and geometry's classic Pythagorean Theorem, this is expressed … More onthe Cosmos
UNIQUE PROPERTIES IN GEOMETRYPHI AND GEOMETRY
Phi (Φ) was described by Johannes Kepler as one of the "two great treasures of geometry." (The other is the Theorem of Pythagoras.) Phi appears in many basic geometric constructions. 3 lines: Take 3 equal lines. Lay the 2nd line against the midpoint of the 1st. Lay the 3rd line against the midpoint of the 2nd. The ratio of AG to AB is Phi, the Golden Ratio. (Contributed by Jo Niemeyer) 3 sides: Triangle Insert an equilateral triangle inside a circle, add a line at the midpoint of the two sides and extend that line to the circle. The ratio of AG to AB is Phi. 4 sides: Square Insert a square inside a semi-circle. The ratio of AG to AB is Phi. 5 sides: Pentagon Insert a pentagon inside a circle. Connect three of the five points to cut one line into three sections. The ratio of AG to AB is Phi. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating … More on Geometry UNIQUE PROPERTIES IN MATHEMATICS FIBONACCI 60 REPEATING PATTERN The last digit of the numbers in the Fibonacci Sequence form a pattern that repeats after every 60th number: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1 This pattern can be seen in the following list of the first 72 Fibonacci … More on Mathematics MATHEMATICS OF PHI, 1.618, THE GOLDEN NUMBER Phi, Φ, 1.618…, has two properties that make it unique among all numbers. If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or Φ² = Φ + 1. If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or 1 / Φ = Φ - 1. These relationships are derived from the dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C). This relationship is expressed mathematically as: A = B + C, and A / B = B / C. Solving for A, which on both sides give us this: B + C = B²/C Let's say that C is 1 so we can determine the relative dimensions of the line segments. Now we simply have this: B + 1 = B² This can be rearranged as: B² - B - 1 = 0 Note the various ways that this equation can be rearranged to express the relationship of the line segments, and also Phi's unique … More on MathematicsPOWERS OF PHI
Phi has a unique additive relationship. The powers of phi have unusual properties in that they are related not only exponentially, but are additive as well. We know that: Phi 2 = Phi + 1 Which is the same as: Phi 2 = Phi 1 + Phi 0 And this leads to the fact that for any n: Phi n+2 = Phi n+1 + Phi n Thus each two successive powers of phi add to the next one! n Phin 0 1.000000 1 1.618034 2 2.618034 3 4.236068 4 6.854102 5 11.090170 6 17.944272 Powers of Phi and its reciprocal: Another little curiosity involves taking phi to a power and then adding or subtracting its reciprocal: For any even integer n: Phi n + 1 / Phi n = a whole number For any odd integer n: Phi n - 1 / Phi n = a whole number Examples are shown in the tables below: for n = even integers n Phin 1/Phi n Phi n + … Moreon Mathematics
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