61. Golden Ratio -- Facts, Info, And Encyclopedia Article The golden ratio is also referred to as the golden mean, golden section, (Click link for more info and facts about Sacred geometry) Sacred geometry http://www.absoluteastronomy.com/encyclopedia/g/go/golden_ratio.htm

Golden ratio [Categories: Mathematical constants, Irrational numbers, Algebraic numbers] This article is about the mathematical ratio. For the (A follower of Aristotle or an adherent of Aristotelianism) Aristotelian concept of "golden mean" see (Click link for more info and facts about Nicomachean Ethics) Nicomachean Ethics The golden ratio is an (A real number that cannot be expressed as a rational number) irrational number , approximately 1.618, that possesses many interesting properties. Shapes defined by the golden ratio have long been considered (Click link for more info and facts about aesthetically) aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry and the ancient Pythagorean belief that reality is a numerical reality, except that numbers were not units as we define them today, but were expressions of ratios. The golden ratio is still used frequently in art and design. The golden ratio is also referred to as the golden mean golden section golden number or divine proportion The golden ratio was first studied by ancient mathematicians due to its frequent appearance in (The pure mathematics of points and lines and curves and surfaces) geometry . The golden ratio seems to have been understood and used by the Egyptians. The discovery of irrational numbers, numbers that cannot be represented as an exact ratio of two

62. Art History Glossary - G - Golden Ratio This ratio is a common theme in sacred geometry worldwide. Also Known Asgolden section, golden proportion, golden mean, phi ratio, sacred cut, http://arthistory.about.com/cs/glossaries/g/g_golden_ratio.htm

var zLb=8; zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Art History Homework Help ... Help w(' ');zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/7.htm','');w(xb+xb); FREE Newsletter Sign Up Now for the Art History newsletter! See Online Courses Search Art History Stay up to date! Email to a friend Print this page Elsewhere on the Web Golden Section in Art and Architecture Sacred Geometry Most Popular The Last Supper John vs. Mary Magdalene Art History Job, Fellowship and Internship Postings What are the "elements" of art ... Mentor Huebner: One Artist Show - Painting #5: Sexi Beach Ba... What's Hot Glossary: "stippling" Wassily Kandinsky Glossary: "Golden Ratio" Have you read the book? ... Early Renaissance Art Related Topics Painting Drawing / Sketching Ancient / Classical History Medieval History ... Archaeology Glossary From Shelley Esaak Your Guide to Art History FREE Newsletter. Sign Up Now! Definition: noun Golden Ratio is a term (with an astounding number of aliases) used to describe aesthetically pleasing proportioning within a piece. (There is, in fact, a mathematical formula with which one can calculate Golden Ratio. Please see the links at right for information on such, from those who are less right-brained than your Guide.) All math aside, some unknown genius figured out (millenia ago) that, in a work of art or architecture, if one maintained a ratio of small elements to larger elements that was the same as the ratio of larger elements to the whole, the end result was extraordinarily pleasing to the eye. This ratio is a common theme in sacred geometry world-wide.

63. Bibliography: L natural proportion, patterns, ratio, golden section,polygons, geometry in The golden mean Mathematics and the Fine Arts Doubleday, 1974 http://www.luckymojo.com/bibliol.html

Bibliography: L Lawlor, Anthony The Temple in the House: Finding the Sacred in Everyday Architecture G.P. Putnam's Sons, 1994 ISBN 0-87477-777-1 Keywords: natural proportion, patterns, ratio, golden section, spirals, polygons, polyhedra, Pythagoras, Fibonacci, geometry in art-music-architecture, megaliths, mounds, petroglyphs, houses of worship, funerary-cemetery-burial sites, vernacular architecture, archaeoastronomical observatories, astro-calendrical devices, labyrinths, symbolic landscaping, ley lines, feng shui, religion, myth, folklore, cosmology, religious iconography, freemasonic symbolism, number symbolism, prehistoric, ancient, medieval, renaissance, modern, Africa, Asia, Australia, Crete, Egypt, Europe, Greece, Meso-America, Middle East, North America, Pacific Islands, Rome, South America, world survey Comment: Perhaps the best recent book on the subject of vernacular sacred architecture; written by an architect. Highly recommended. CY Lawlor, Robert Sacred Geometry: Philosophy and Practice Thames and Hudson Ltd., 1982, 1990

64. Sacred Geometry -- What Is It? Bragdon seems so normal as he writes about the golden mean and creates a newdesign style based on projective geometry, that it comes as quite a shock http://www.luckymojo.com/sacreddefined.html

SACRED GEOMETRY WHAT IS IT? The term "sacred geometry" is used by archaeologists, anthropologists, and geometricians to encompass the religious, philosohical, and spiritual beliefs that have sprung up around geometry in various cultures during the course of human history. It is a catch-all term covering Pythagorean geometry and neo-Platonic gometry, as well as the perceived relationships between organic curves and logarithmic curves. Here are a few examples of how the "sacred" has entered into geometry in different eras and cultures: 1) The ancient Greeks assigned various attributes to the Platonic solids and to certain geometrically-derived ratios, investing them with "meaning." For example, the cube symbolized kingship and earthly foundations, while the Golden Section was seen as a dynamic principle embodying philosphy and wisdom. Thus a building dedicated to a god-king might bear traces of cubic geometry, while one dedicated to a heavenly god might have been constructed using Golden Section proportions. 2) When Hindus (ancient and modern) plan to erect any ediface for religious purposes, from a small wayside shrine to an elaborate temple, they first perform a simple geometric construction on the ground, establishing due East and West and constructing a square therefrom. (It's a simple, elegant piece of work, at about the level of high school geometry). Upon this dioagram they lay out the entire building. The making of this geometric construction is accomanied by prayers and other religious observances.

65. On Geometry And Advanced Energy Conversion The number Phi is usually represented as a golden mean Spiral , consisting of However, this spiral does look a lot like the one in the geometry of http://www.padrak.com/ine/ONGEOMETRY.html

Return to the INE Main Page On Geometry and Advanced Energy Conversion Patrick Bailey, INE President March 1996 There are quite a few areas in which the idea of the use of a particular geometry has overlapped with the advanced energy areas. Here is a summary of what I can remember off of the top of my head: 1. The Great Pyramid of Giza, Egypt, was built about the same time as the Sphinx, and I think it was definitely built way, way before the other structures there. It has a full base to height ratio of 2 to SQRT(Phi), where Phi is the Golden Ratio and is equal to (1+SQRT(5))/2. That gives an internal triangle with sides of 1-to-SQRT(Phi)-to-Phi. Several experimenters have noticed that four-sided pyramids build with this ratio exhibit strange properties when a base is aligned perpendicular to local magnetic north. I have not seen many papers on exact EM experimental results of the effects of this geometry, and perhaps we should! This would be a great idea for elementary and high school science projects! I also note that when a pyramid of this exact shape is cut along the base and along the edges and unfolded, one gets the Templar Cross. Coincidence?

66. The Sacred Geometry Of Sound So the golden mean proportions of the cosmos and our body temples are closelyaligned with And it is we ourselves who embody the geometry of the cosmos! http://www.spiritofmaat.com/archive/jan4/prns/williams.htm

Spirit of Ma'at: The Sacred Geometry of Sound by Healer/Musician Ani Williams Sacred Geometry and the Structure of Music Legend recounts how Orpheus was given a lyre by Apollo. By playing his lyre, Orpheus produced harmonies that joined all of Nature together in peace and joy. Inspired by this Orphic tradition of music and science, Pythagoras of Samos conducted perhaps the world's first physics experiment. By plucking strings of different lengths, Pythagoras discovered that sound vibrations naturally occurred in a sequence of whole tones or notes that repeat in a pattern of seven. This correlation of sound, matter, and consciousness is important. For as Stanford physicist William Tiller has proved, human consciousness imprints the space and matter of the universe. It is our intent that gives the direction and quality to Creation. The Music of Atomic Shapes The Platonic solids, basic shapes of Sacred Geometry, are five three-dimensional geometric forms of which all faces are alike. And each platonic solid represent one of the five elements of creation, as follows: These five Platonic solids comprise the alchemical dance of the elements and of Creation itself. My introduction to the spiritual power of sound began with an experience of this truth.

67. Human Form From Sacred Geometry Sacred geometry studies such primal systems which reveal the unity of the The golden mean is expressed in the 52 degree angle pyramid structure and Pi http://www.people.vcu.edu/~chenry/

SACRED GEOMETRY New Discoveries Linking The Great Pyramid to the Human Form Professor, Department of Sculpture Virginia Commonwealth University Richmond, Virginia This site is best viewed on Microsoft Internet Explorer 4.0 or higher with screen set to 1024 X 768 pixels, 24 bit ...16 million colors. Set ... View/Text Size ... to Meduim Click on thumbnails to view larger images. For more than twenty years, I have been studying the image generating properties of reflective spheres stacked in 52 degree angle pyramids. The 52 (51.827) degree angle slope of the sides of The Great Pyramid in Cairo, Egypt embodies the Golden Mean which is the ratio that is used in Nature to generate growth patterns in space. Sacred Geometry studies such primal systems which reveal the unity of the cosmos by representing the relationships between numbers geometrically. The Vesica Piscis is one of the most fundamental geometrical forms of this ancient discipline and it reveals the relationship between the The Great Pyramid and the 2 dimensional expansion of a circle of one unit radius R as shown in Figure 1. This relationship is more completely described in The New View Over Atlantis by John Michell published by Thames and Hudson. Figure 1 Vesica Piscis in 2 Dimensions In the early 1970s, I became very interested in the three dimensional representation of this geometry and I visualized this as a three dimensional pyramid inside two intersecting spheres shown in Figure 2.

68. Introduction To Phi And The Golden Mean The golden Section is a unique Ratio (or relationship between parts) (Histheory is that the PHI geometry of DNA implies a waveform preference for http://www.miqel.com/pure-math-patterns/visual-math-phi-golden.html

Home math patterns frontpage Introduction to the fascinating patterns of Visual Math Platonic / Vectoral Geometry Simple Iterative Fractals Natural fractals Minimal Surfaces ... Multi Dimensional Forms PHI: The Golden Ratio or Golden Section In Nature, Art, Science and Religion) The Golden Section is a unique Ratio (or relationship between parts) that seems to be preferred by Nature as the best geometry for growth, energy conservation, elegance and has some fundamental relationships to the platonic solids and the Mandelbrot set. It was formally discovered by the Greeks and incorporated into their art and architecture, but it has been shown to occur even in prehistoric art, possibly as a function of Man's natural affinity for it's beauty. The rectangle at left has a vertical edge length of 1 the horizontal or width of the rectangle is 1.618

69. The 'Golden Mean' In Number Theory The choice of the golden mean is dictated by number theory the golden mean is the of the same universal golden scale, based on a geometry of intervals http://www.maths.ex.ac.uk/~mwatkins/zeta/goldenmean.htm

The ' Golden Mean ' in number theory Planat, et. al. have subjected certain arithmetic functions to a type of analysis appropriate to signal processing and discovered a type of power law in the frequency spectrum possibly involving the Golden mean C. Castro, "Fractal strings as the basis of Cantorian-Fractal spacetime and the fine structure constant" [abstract:] "Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable p inary ( p inary, p prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p -adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature." Castro's observation possibly linking the 'Golden String' to a function central to the behaviour of certain eigenvalues in random matrix theory (which in turn appears to be deeply linked to the behaviour of the nontrivial zeros of the Riemann zeta function).

70. Geometry The golden mean (or golden Section), represented by the Greek letter phi, A really good book on this subject is Sacred geometry Philosophy and http://www.dial-a-teacher.com/geometry/page11.html

Phi The Golden Mean The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things. The decimal representation of phi is 1.6180339887499... . Pure Math Proportion is the relationship of the size of two things. Arithmetic proportion exists when a quantity is changed by adding some amount. Geometric proportion exists when a quantity is changed by multiplying by some amount. Phi possesses both qualities. If you study the Fibonacci series and the Golden Rectangle carefully, you will eventually realize that phi + 1 = phi * phi. The resulting proportion is both arithmetic and geometric. It is thus perfect proportion; you could think of it as the place on some imaginary graph where the curved line of multiplication crosses the straight line of addition. Nature In pure mathematics, an increase in size can be any imaginable number, even one like

71. Geometry Of Human Anatomy - MedHunters geometry of Human Anatomy. By Cynthia M Piccolo properties of the divineproportion, also known as the golden ratio, golden mean, and golden section. http://www.medhunters.com/articles/geometryOfHumanAnatomy.html

Home Magazine For Employers About Us ... EMPLOYERS Geometry of Human Anatomy By Cynthia M Piccolo In the days before plastic surgery gave us "perfection," one could argue that artists, mathematicians, engineers, and architects were the experts in the "perfect" human form. As far back as ancient Egypt, individuals had discovered the mystical properties of the divine proportion, also known as the golden ratio, golden mean, and golden section. Why "divine"? It is considered to be the proportions that constitute anything that is aesthetically pleasing. An example of a building designed using this ratio is the Parthenon atop the Acropolis in Athens. In art, the divine proportion can frequently be found, perhaps most famously in the proportions and placement of the participants in da Vinci's The Last Supper. But what about the body? The index finger is said to show the ratio: each section of the index finger, moving from the tip to the base of the hand, is larger by approximately 1.618. (1.618 is not a whole number, so the proportions don't have to be 100% exact, but the closer, the more "perfect.") Similarly, the ratio of the hand to the forearm is also 1.618. Who cares about perfection in hands, you say? Well, what about the perfect smile? According to phi-philes, in an aesthetically pleasing mouth, the front two teeth form a golden rectangle (which is said to be one of the most visually satisfying of forms, as it is formed with sides of 1 and 1.618). There is also a phi ratio in the height to width of the center two teeth. And the ratio of the width of the two center teeth to those next to them is also phi. And, the ratio of the width of the smile to the third tooth from the center is also phi.

72. Sacred Geometry Like the golden mean Spiral, this geometric progression has no beginning and noend. It is from the Harmonics of Music that all the laws of physics can be http://www.spiraloflight.com/sacred_geometry.htm

lightSource A Sacred Geometry Experience Sacred Geometry is the blueprint of Creation and the genesis of all form. It is an ancient science that explores and explains the energy patterns that create and unify all things and reveals the precise way that the energy of Creation organizes itself. On every scale, every natural pattern of growth or movement conforms inevitably to one or more geometric shapes. As you enter the world of Sacred Geometry you begin to see as never before the wonderfully patterned beauty of Creation. The molecules of our DNA, the cornea of our eye, snow flakes, pine cones, flower petals, diamond crystals, the branching of trees, a nautilus shell, the star we spin around, the galaxy we spiral within, the air we breathe, and all life forms as we know them emerge out of timeless geometric codes. Viewing and contemplating these codes allow us to gaze directly at the lines on the face of deep wisdom and offers up a glimpse into the inner workings of the Universal Mind and the Universe itself. The ancients believed that the experience of Sacred Geometry was essential to the education of the soul. They knew that these patterns and codes were symbolic of our own inner realm and the subtle structure of awareness. To them the “sacred” had particular significance involving consciousness and the profound mystery of awareness  the ultimate sacred wonder. Sacred Geometry takes on another whole level of significance when grounded in the experience of

73. Geoman Text The pentagram expresses the golden mean in all its proportions. Thus, byarranging the four most fundamental shapes in plane geometry using the most http://www.geoman.com/geomantext.html

G E O M A N A T O M Y James L. Fournier The new Vitruvian Man I discovered, which I have called Geoman, adds two of the three fundamental polygonal shapes in plane geometry, the equilateral triangle, and the pentagram, both in the golden mean relationship to the same circle represented with the square in Leonardo daVinci's Vitruvian Man. In Leonardo's drawing there are two of the four simplest shapes in plane geometry. These are the circle representing unity, or one, and the square representing four. I had noticed that the relationship between the circle and the square in Leonardo's drawing is the golden mean ratio. Thus the length of one side of the square is 1.618 times the radius of the circle. I found this by measuring Leonardo's drawing, but D'Arcy Thompson had also demonstrated by statistical analysis that on average the distance from the ground to a person's navel is the golden mean of their overall height. I also knew that the navel, our center, the center that we grow from, is also our center of mass, the point which we would spin about weightless in space, which is also the center that we find when we meditate and find our "center" (yellow in the diagram). Each of the three shapes shares the same edge length as the square, and thus each is in the golden mean relationship to the radius of the circle. The key to the success of the diagram is that the tip of the equilateral triangle protrudes through the base of the square, counter to the medieval teaching that the arm span is equal to the diagonal to the feet. Instead, the two points where the triangle meets the ground represent the centers of the feet, and the tip of the triangle represents the pointed toes as if in a pirouette.

74. Golden Triangle And Spiral The golden mean Spiral, the seven toroid cascade from the heart s and theK mer s of Cambodia, one universal geometry and number system unites them, http://www.akasha.de/~aton/GoldenTriSpiral.html

Golden Mean Triangle's The Golden Mean Spiral, the seven toroid cascade from the heart's cardio-rythm at coherent moments, known as Compassionate Love, which establishes the recursive fractal or mirror-into-mirror cascade, that we call the Pyramid Vortex of the Phoenix and Dove, can be made by the simplest Universal Keys of Trinity, that comprises this Universe's university for co-creatorship. The angles are exceptionally significant, for 72 years is 1 degree in the zodiac for our planets recursion through the 12 houses of HOR (Horus). Multiples and divisions of this harmonic number, are used by ALL the major indegenous cultures, from the Myth's of the Egyptians, Vedic Philosophers, to the Edda Runic Bards of Scandinavia, to the Mayan's, the Greeks, and the K'mer's of Cambodia, one universal geometry and number system unites them, as we will show in articles to be. But this sequence also enables us to make the Unified Field Cartography system, for navigation in Earths magnetic field Icosahedron, as you will find at the Geodyseum of Earths Biosphere (GEB) site, and to turn time into a Dodecahedron. This establihses an extraterrestrial Unified Field Universal Navigation system.

75. Geometry - A To Z Home's Cool Homeschooling Math geometry for kids through real life lessons build a bookcase, play games, The golden mean is a ratio that is present in the growth patterns of many http://homeschooling.gomilpitas.com/explore/geometry.htm

YOU ARE HERE: HOME EXPLORATIONS 4 KIDS MATH Click on the banner for recommended books and supplies for homeschooling A to Z Home's Cool Homeschooling Explorations 4 Kids I am Ann Zeise , your guide to the best and most interesting and useful sites and articles about home education on the web. Search All of A to Z Articles Calendar Curriculum Explorations 4 Kids Field Trips Jokes Laws Links Methods Older Kids Regional Religion/Cultural The Web Home New Kids Links One Exploration A Day Contact Ann Zeise ... Curriculum Shopping Site Index: A B C D ... Z Advanced Math Math 4 Teens Calculators History of Math High School Math Algebra Geometry Trigonometry Calculus Math 4 Kids Art Math Connection Calendars Clocks Logic Games ... Word Problems Math Curriculum Math Books Math Software Math Stuff Explorations 4 Kids Computer Literacy Drivers Ed Fine Arts General Interest ... Social Studies A to Z Home's Cool Home Articles Beginning to Homeschool Community Networking ... Methods, Styles

76. SAPAAN Vol. 1 Spring 2003 - Hidden Geometry In Music Of Bach And Pitch Bb3 at time point 24 of section A is at the golden mean of local apexes Pitch C2 at the nadir of section B is at the geometric mean at time point http://www.sapaan.com/vol1/saggau.htm

77. Guillaume Morel :: The Golden Number The golden mean, or golden Ratio, or golden Number is one of these In bothArithmetics geometry, has many properties that we cannot develop here. http://www.guillaumemorel.com/en-gold.htm

Agent Author Consultant Manager ... Webmaster Golden Mean Keys: golden mean, 1.618, divine, proportion, ratio, mathematics, paintings, aesthetics, harmony Presentation: The Golden Mean, or Golden Ratio, or Golden Number is one of these mysterious irrational numbers, like e or (Pi). It is often called or (Phi). My research: The value of The positive result of the equation "X²=X+1" gives value: NB: you can find the value of the Golden Number with 2000 places here is also the limit of the ratio of Fibonacci series numbers: Lim [n ] ( U[n+1]/U[n]) = where U is defined as follow: Fibonacci serie U[n+2]=U[n+1]+U[n] U[0] = 0, U[n]=1 U[n] values are: 0, 1 , 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. Mathematical properties of has many properties that we cannot develop here. The basic knowledge to keep in mind with the Golden Number is: and ' are very close as far as their decimal part is identical and they have the following properties: '=-1 and Geometrically, is defined as follow: Euclide (~300BC) called this geometrical drawing "to divide a line in mean & extreme ratio"

78. Zausner Golden Mean - 1/18/97 Excerpt on properties of numbers and dynamical geometry of aesthetics from The groups of eight and five stairs also exhibit the golden mean in the http://www.blueberry-brain.org/goldmean/zausner.htm

The citation for the dissertation is: Zausner, T. (1993). Transformation of Personality and Emergence of Self in Titian’s “Presentation of the Virgin to the Temple. Excerpt on properties of numbers and dynamical geometry of aesthetics from CHAPTER XVI: ASCENSION The Stairs Eliade remarks that ascension is often seen as a climbing of stairs, and cites the ceremonial staircase in the Orphic and in the Mithraic initiations. He says that stair symbolism is culturally universal and part of the archaic content of the human mind. According to Jung, stairs symbolize the process of psychic transformation in which the contents of the unconscious are brought into conscious awareness. My research suggests that in the humanist symbolism of the painting, Mary's ascension is emphasized by the upper bank of five stairs and by the number of stairs in the staircase. (Fig. 11) Schneider discovered that the angle of the upper five stairs is different from the angle of the lower eight stairs. He believes Titian did this deliberately because the lower stairs are formed of shorter stone rectangles, while the upper stairs are formed of slightly longer ones. (Fig. 11, 34)

79. Sacred Geometry It is this principle of oneness underlying all geometry that permeates the The golden ratio (aka phi ratio aka sacred cut aka golden mean aka divine http://www.meta-religion.com/Mathematics/Articles/sacred_geometry.htm

to promote a multidisciplinary view of the religious, spiritual and esoteric phenomena. About Us Links Search Contact ... Science home Religion sections World Religions New Religious Groups Ancient Religions Spirituality ... Extremism Science sections Archaeology Astronomy Linguistics Mathematics ... Contact The Web Metareligion More sections: Subscribe to the Metareligion Newsletter Photography Submit your site Suggest an article ... Polls Please, help us sustain this free site online. Make a donation using Paypal: Sacred Geometry by Bruce Rawles Courtesy of http://www.intent.com/sg/ In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical principle of the inseparable relationship of the part to the whole. It is this principle of oneness underlying all geometry that permeates the architecture of all form in its myriad diversity. This principle of interconnectedness, inseparability and union provides us with a continuous reminder of our relationship to the whole, a blueprint for the mind to the sacred foundation of all things created.

80. Golden Ratio The golden ratio is also referred to as the golden mean, golden section, The number φ turns up frequently in geometry, in particular in figures http://www.algebra.com/algebra/about/history/Golden-ratio.wikipedia

Golden ratio Regular View Dictionary View (all words explained) Algebra Help my dictionary with pronunciation , wikipedia etc Golden ratio It has been suggested that this article or section be merged with Golden Mean Discuss This article is about the mathematical ratio. For the Aristotelian concept of "golden mean" see Nicomachean Ethics The golden ratio is an irrational number , approximately 1.61803..., that possesses many interesting properties. Shapes defined by the golden ratio have long been considered aesthetically pleasing in Western cultures, reflecting nature's balance between symmetry and asymmetry and the ancient Pythagorean belief that reality is a numerical reality, except that numbers were not units as we define them today, but were expressions of ratios. The golden ratio is still used frequently in art and design. The golden ratio is also referred to as the golden mean golden section golden number divine proportion or sectio divina Contents History A startlingly quick proof of irrationality Alternate forms Mathematical uses ... History History The golden ratio was first studied by ancient mathematicians due to its frequent appearance in geometry . The golden ratio may have been understood and used by the Egyptians. The discovery of irrational numbers, numbers that cannot be represented as an exact ratio of two

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