41. Golden Ratio -- From MathWorld The golden ratio, also known as the divine proportion, golden mean, or golden Ogilvy, CS Excursions in geometry. New York Dover, pp. 122134, 1990. http://mathworld.wolfram.com/GoldenRatio.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Constants Golden Ratio ... Lambrou Golden Ratio The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagram decagon and dodecagon . It is denoted , or sometimes (which is an abbreviation of the Greek "tome," meaning "to cut"). The term "golden section" ( goldene Schnitt ) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley's 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002, pp. 5-6). has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers . It is also a so-called Pisot Number Given a rectangle having sides in the ratio is defined such that partitioning the original

42. Sacred Geometry/ Golden Mean Spiral And Flower Of Life Geometry Sacred geometry/ golden mean Spiral And flower of life geometry, sacred geometry site. http://www.gr1innovations.com/sacred_geometry.html

Top Ten Programs Add your program Astrology Auctions ... Games Get: paid to read email paid to surf paid for referrals paid to search ... Web Hosting Promote with these tools! Free Advertising Paid Advertising Search Engine Ranking Tutorial new mars pictures ... popular google keywords Sacred Geometry, Golden Mean Spiral And flower of life geometry Make $150 per day at home, click here What is Sacred Geometry? Well I am glad you asked because I love to talk about sacred geometry. Sacred Geometry is a set of geometric shapes which in short are used to create the universe. The sacred shapes which compose everything in the universe are derived from the Flower of life, which is a set of geometric shapes, composed entirely of circles, inside the set of circles you will find all the building tools to create a universe, such as triangles, the star of David, golden spirals, the golden section, hexagons, pentagons, merkaba stars, you name it the flower of life has it. Have you ever looked at flowers and admired their beauty? Do you know why you find them beautiful? Well the reason is that their sacred geometric shape is one that is often seen in nature, nature is composed of many geometry shapes, when the ones that are most familiar and beautiful are represented in nature, the geometry in us, (as we ourselves are geometrically composed) relates to the flower, and our sub concious understands the shape.

43. Chapter 15, Golden Mean 8 Yet the immediate use of the golden mean in geometry, during Plato s time.was even more remarkable. The Section was actually the key to the geometric http://www.anselm.edu/homepage/dbanach/pyth4.htm

Selections from Julia E. Diggins, String, Straightedge, and Shadow Viking Press, New York , 1965. (Illustrations by Corydon Bell) 15. THE GOLDEN AGE AND THE GOLDEN MEAN The second half of the 5th century B.C. was the Golden Age of Greece. This was the period of her most beautiful art and architecture, and some of her wisest thinkers besides. Both owed much to the popular new study of geometry. By the start of the next century, geometry itself was entering its own classic age with a series of great developments, including the Golden Mean. The times were glorious in many ways. The Persian invaders had been driven out of Hellas forever, and Pericles was rebuilding Athens into the most beautiful city in the world. At his invitation, Greek mathematicians from elsewhere flocked into the new capital. From Ionia came Anaxa- goras, nicknamed "the mind." From southern Italy and Sicily came learned Pythagoreans and the noted Zeno of Elea. And their influence was felt over all Athens. High on the hill of the Acropolis rose new marble temples and bronze and painted statues. Crowds thronged the vast new open-air theater nearby, to hear immortal tragedies and comedies by the greatest Greek playwrights. These splendid public works were completed under the direction of the sculptor Phidias and several architects, all of whom knew and used the principles of geometry and optics. "Success in art," they insisted, "is achieved by meticulous accuracy in a multitude of mathematical proportions." And their buildings had a dazzling perfection never seen before-the beauty of calculated geometric harmony.

44. PHILOSOPHY OF SCIENCE Jill Britton’s excellent slide show introduction to the golden mean. Flat geometry and the golden Section. By Ron Knott. http://www.anselm.edu/homepage/dbanach/ph31a.htm

Greek Science and The Golden Section Ancient Science and Mathematics Selections from Julia E. Diggins String, Straightedge, and Shadow Viking Press, New York 1965. (Illustrations by Corydon Bell) Now back in print! Purchase at http://wholespiritpress.com/string.htm Chapters 8, 9: Thales Chapters 11, 12: Pythagoras and his Theorem ... Homage to Pythagoras One of the most comprehensive sets of resources on Pythagoras on the net Selection from Burnet's Early Greek Philosophy on Pythagoras from the site above. A collection of ancient quotes and sources on Pythagoras from the same site, originally from Arthur Fairbanks, ed. and trans. The First Philosophers of Greece, 1898. Harmony and Proportion Plato and Pythagoras on Music; Alberti and Palladio on architectural space. Pythagoras on Proportion in Music From the above site, MathGYM Pythagoreans activity pages. The Pythagorean Theorem essay Activity Prove the theorem graphically. Another applet that proves the theorm graphically from U. of British Columbia 43 proofs of the Theorem from the Cut the Kot site by Alex Bogomolny Here is an applet that demonstrates the method used in the Diggins book from this site.

45. *Sacred Geometry* - Tribe.net golden mean Fusion Anomaly golden Mena Spiral and the Merkaba golden SectionRatio - Phi Sacred geometry and the Archdome=golden mean http://sacredge0metry.tribe.net/?r=10535

46. The Golden Section In Greek Art And geometry itself made tremendous strides in the golden Age and the darker time How he did thiswith his work on the golden mean and his new theory of http://milan.milanovic.org/math/english/golden/golden4.html

Fibonacci Numbers and the Pascal Triangle Search only in Mathematics Golden Section The Divine Proportion The Golden Section in Greek art Pythagoras (560-480 BC), the Greek geometer, was especially interested in the Golden Section, and proved that it was the basis for the proportions of the human figure. He showed that the human body is built with each part in a definite Golden Proportion to all the other parts. Pythagoras' discoveries of the proportions of the human figure had a tremendous effect on Greek art. Every part of their major buildings, down to the smallest detail of decoration, was constructed upon this proportion. The Parthenon was perhaps the best example of a mathematical approach to art. Throughout history, the ratio for length width of rectangles of 1.6180339887... has been considered the most pleasing to the eye.The Greek sculptor Phidias sculptured many things including the bands of sculpture that run above the columns of the Parthenon. There are golden rectangles throughout this structure which is found in Athens, Greece. Phidia widely used the Golden Ratio in his works of sculpture. Selections from Julia E.

47. DaVinci Code, Star Tetrahedron, Golden Mean, Sacred Geometry, Dodecahedron, Ayah In geometry this ratio is found in a golden mean rectangle, the uniqueness ofwhich is found in its ability to selfreplicate. By starting with a simple http://www.heartoftheinitiate.com/articles_davincicreator01.htm

Article by Ralph Miller There are many mysteries hidden inside of Leonardo DaVinci's drawing called "Proportions of the Human Figure." It seems likely that DaVinci had an understanding of certain hidden secrets that had been passed on from ancient sources. It is interesting that DaVinci very cleverly hid most of these mysteries. For some reason he just did not put them in plain view. DaVinci wanted them to be discovered. Example of a fractal image In the Ayahuasca temples, many people who participate in ceremonies emerge in a world where all of reality can be represented in very complex mathematical formulas or relationships. The incredible design and beauty of creation, has an underlying organization that is unbelievably complicated and elegant. People see fractal patterns, which are part of these infinitely complicated geometric patterns. DaVinci was also aware of the mathematical anomaly called the 'Golden Mean' that, as it turns out, is found throughout nature. The Golden Mean is a relationship between three different dimensions that form a ratio. The ratio itself is actually a unique number called Phi symbolized as the Greek letter Phi or

48. DaVinci Code, Star Tetrahedron, Golden Mean, Sacred Geometry, Dodecahedron, Ayah In geometry this ratio is found in a golden mean rectangle, Diagram 11 Theactual geometric replication of the golden mean starts from an infinite http://www.heartoftheinitiate.com/articles_davincicreator11.htm

Article by Ralph Miller There are many mysteries hidden inside of Leonardo DaVinci's drawing called "Proportions of the Human Figure." It seems likely that DaVinci had an understanding of certain hidden secrets that had been passed on from ancient sources. It is interesting that DaVinci very cleverly hid most of these mysteries. For some reason he just did not put them in plain view. DaVinci wanted them to be discovered. Example of a fractal image In the Ayahuasca temples, many people who participate in ceremonies emerge in a world where all of reality can be represented in very complex mathematical formulas or relationships. The incredible design and beauty of creation, has an underlying organization that is unbelievably complicated and elegant. People see fractal patterns, which are part of these infinitely complicated geometric patterns. DaVinci was also aware of the mathematical anomaly called the 'Golden Mean' that, as it turns out, is found throughout nature. The Golden Mean is a relationship between three different dimensions that form a ratio. The ratio itself is actually a unique number called Phi symbolized as the Greek letter Phi or

49. Sacred Geometry, Phi Ratio, Golden Mean sacred geometry, phi ratio, golden mean. sacred geometry, phi ratio, goldenmean. The Conference Our purpose is to offer a joyful learning space where http://www.sacredgeometryconference.com/SacGeoarchive/

50. Sacred Geometry, Phi Ratio, Golden Mean An author and scientist, Bruce will define the basics of Sacred geometry in terms Hermetic Insights, golden Ratio Proportions, Archetypes, Polygons, http://www.sacredgeometryconference.com/rawles.html

51. Shop Mosaic Online Catalog: MIND: Golden Mean Sacred Geometry Book The golden mean, the heart of sacred geometry, is a special relationship betweena part and a whole that acts as a formula for harmony and beauty. http://www.mosaicdesignstudio.com/catalog/mind/6180.html

The Golden Mean, the heart of sacred geometry, is a special relationship between a part and a whole that acts as a formula for harmony and beauty. This 64 page book describes the profound significance of the Golden Mean and sacred geometry as well as the many ways it appears in nature, art, and mathematics, combining enjoyable reading with in-depth insight in this marvel. This book, by Now and Zen, is great for artists or anyone who is interested in the behaviors of the spirits. Golden Mean Sacred Geometry book: This page and the contents of this page may not be copied in any way, shape, or form without the written consent of MOSAIC, Ltd. SHB 8/99

52. Geometry In Art & Architecture Unit 2 The golden ratio is also called extreme and mean ratio. According to Euclid, Geometric Construction of the golden Ratio http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html

Description and Requirements The Book Bibliography Syllabus ... Introduction The Great Pyramid Music of the Spheres Number Symbolism Polygons and Tilings The Platonic Solids ... Early Twentieth Century Art The Geometric Art of M.C. Escher Later Twentieth Century Geometry Art Art and the Computer Squaring the Circle in the Great Pyramid "Twenty years were spent in erecting the pyramid itself: of this, which is square, each face is eight plethra, and the height is the same; it is composed of polished stones, and jointed with the greatest exactness; none of the stones are less than thirty feet." -Heroditus, Chap. II, para. 124. Slide 2-1: The Giza Pyramids and Sphinx as depicted in 1610, showing European travelers Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 22 Outline: The Great Pyramid The Golden Ratio Egyptian Triangle Squaring the Circle ... Reading The Great Pyramid Slide 2-2: The Great Pyramid of Cheops Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 205 We start our task of showing the connections between geometry, art, and architecture with what appears to be an obvious example; the pyramids, works of architecture that are also basic geometric figures.

53. Sacred Geometry Home Page Sacred geometry is an ancient art and science which reveals the nature of our The golden ratio (aka phi ratio aka sacred cut aka golden mean aka divine http://www.intent.com/sg/

Sacred Geometry Home Page by Bruce Rawles 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. The Sphere (charcoal sketch of a sphere by Nancy Rawles) Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm. The Circle The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine.

54. The Golden Section And The Golden Rectangle geometry has two great treasures one is the Theorem of Pythagoras; the other,the division of a line A New Solution for the Parthenon s golden mean http://www.mlahanas.de/Greeks/GoldenSection.htm

The Golden Section and the Golden Rectangle Michael Lahanas Der Goldene Schnitt und das Goldene Rechteck Geometry has two great treasures: one is the Theorem of Pythagoras ; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. Johannes Kepler Euclid in Book 6, Proposition 30 shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line". Euclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (ie is the same as the ratio AG/AB). AB/AG = AG/GB. If AB = 1 and AG = x then GB = 1-x and 1/x = x/(1-x) and it follows x = 1-x, i.e. x - x + 1 = The golden section was found by the Pythagoreans who used the Pentagram formed, by the diagonals of a regular Pentagon , as a symbol of their school. It was probably known much earlier in Egypt as some ratio of lengths and heights of Pyramids suggest. Pythagoras may have obtained this knowledge when he visited Egypt. Theano of Thurii Theorem of the golden mean.

55. Sacred Places: Sacred Geometry It is this principle of oneness underlying all geometry that permeates the The spiral is important when discussing aspects of the golden mean because http://www.arthistory.sbc.edu/sacredplaces/sacredgeo.html

Christopher L. C. E. Witcombe Sweet Briar College Syllabus INTRODUCTION Sacredness Caves Stones Mountains Trees Water Forms in the Landscape BIBLIOGRAPHY Sacred Places An exploration of how and why places become invested with SACREDNESS and how the SACRED is embodied or made manifest through ART and ARCHITECTURE Sacred Geometry Tiffiney Whitmire "All things throughout our universe seem to follow the same fundamental blueprint or geometric patterns. These geometrical archetypes, which reveal to us the nature of each form and its virational resonances. They are also symbolic of the underlying metaphysical principle of the inseperable relationship of the part to the whole. It is this principle of oneness underlying all geometry that permeates the architecture of all inseparability and union provides us with a continous reminder of our relationship to the wholea blueprint for the mind to the sacred foundation of all things created." We call this blueprint " Sacred Geometry Flower of Life Sacred Geometry is a term that is used by archaeologists, anthropologists, and geometricians. It includes the religious, philosohical, and spiritual beliefs that have surrounded geometry in many various cultures throughout history. It covers Pythagorean geometry as well as the relationships between organic curves and logarithmic curves. The "sacred" aspect of geometry has evolved as a result of different cultures. I have provided five different views of how different cultures have made geometry become sacred. Later, I will explain the meaning of certain numbers, as they play an important role in sacred geometry. A few examples of sacred sites will also be provided. These examples will provide you with a real sense of how sacred geometry is used in sacred sites.

56. Golden Mean - Definition Of Golden Mean In Encyclopedia The number f turns up frequently in geometry, in particular in figures involvingpentagonal The Doctrine of the golden mean (Zhong1 Yong2, ), http://encyclopedia.laborlawtalk.com/Golden_mean

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law The golden ratio is a number , approximately 1.618, that possesses many interesting properties. It was studied by ancient mathematicians due to its frequent appearance in geometry . Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean golden section golden number or divine proportion It is usually denoted by the Greek letter (phi). Contents showTocToggle("show","hide") 1 Origin of name 2 Definition of the Golden Ratio 3 Properties 4 Alternate forms ... 11.3 Other Origin of name The name "golden ratio" first seemed to have been used in the form sectio aurea , "golden section", by Leonardo da Vinci Mark Barr and taken from the first Greek letter in the name of the Greek sculptor Phidias , who was long believed to have used the golden ratio in his designs.

57. Storm's Nest - The Golden Logos Sacred geometry The golden Logos. The golden Proportion, expressed in Robert Lawlor on the golden mean - A proportion is formed from ratios, http://home.earthlink.net/~johnrpenner/Articles/GoldenLogos.html

To see the world in a grain of sand, and heaven in a wildflower. (William Blake) Sacred Geometry - The Golden Logos The Golden Proportion, expressed in equation form: A:B = B:(A+B) This is the formula of the celebrated golden section , a uniquely reciprocal relationship between two unequal parts of a whole, in which the small part stands in the same proportion to the large part as the large part stands to the whole . (Gyorgy Doczi, The Power of Limits Exerpts from Parabola Magazine, Winter 1991 Parabola: The Golden Mean , Winter 1991, Vol. XVI, No 4, ISSN: 0362-1596. - Astrid Fitzgerald The Golden Proportion in Your Hand Mathsoft - Mathematics of Constants Golden Links ... SUBMIT AN ARTICLE posted: september 25, 2000 updated: september 27, 2004

58. Golden Mean golden mean in Web Design and Color in web site design. Finding a satisfactoryformat for a square canvas; Sacred geometry The golden mean Spiral is the http://www.saunalahti.fi/jawap/link/goldenmean.html

Principles of Web Design : Golden Mean design Table width=700 height=433 BLUE : RED : IMAGE : GOLDEN MEAN The Golden Mean is considered to be the most aesthetic and pleasant division of line and space. In mathematical terms, the Golden Mean is a ratio of the number 1.618*size or size*0.618. Artists and Architects are using the proportions of the golden mean. Learn how to: Find a product to sell Accept credit cards online Build your own website Write copy that sells links: Golden Mean design. Fibonacci Numbers and The Golden Section in Art, Architecture and Music Fibonacci, golden section, golden mean, golden ratio, divine proportion, Phi, golden rectangle, Leonardo da Vinci, poetry, symmetry, patterns, Stradiviari, violin design, Mozart, Beethoven, Bartok, Debussy, Satie, Schubert, Bach, art, architecture, music, geometry, design, structure, beauty, recreational maths, combinatorics, number theory, discrete mathematics The Archdome - Golden Mean Design Golden Mean - The Archdome enfolds the golden spiral (also known as the Fibonacci spiral). Scientists have studied the dome and have come to a number of conclusions about the energy patterns it creates Golden Mean Home Page The Golden Mean Group provides Architectural Design, Strategic Planning, and Facility Consulting services to private, commercial, and institutional clients throughout the United States.

59. Design And Proportion - The 'Golden Mean' The golden mean is merely a mathematical ratio usually discerned by the nor do I slavishly design my paintings or canvases to follow its geometry. http://www.geocities.com/~jlhagan/lessons/design2.htm

DESIGN 2 THE GOLDEN MEAN or GOLDEN SECTION The 'Golden Mean' is merely a mathematical ratio usually discerned by the painter as the ratio of the larger side of a rectangle as it relates to the shorter. Derived by the ancient Greeks it can be constructed geometrically or expressed as a simple ratio, namely 1:1618... Like "pi", the number 1.618... is an irrational number. Both the ancient Greeks and the ancient Egyptians used the Golden Mean when designing their buildings and monuments. The builders of Paestum used the Golden Mean in their temples. Artists as diverse as Leonardo da Vinci and George Seurat used the ratio when constructing their paintings. In classical architecture it was thought this particular ratio was the most pleasing to the eye and its extrapolation into a spiral could be found replicated in nature in such diverse things as pine cones and sea shells or the curve of a fern. I see no particular theological significance in the golden mean, nor do I slavishly design my paintings or canvases to follow its geometry. Well that noted then how is this 'golden mean' found using a ruler and a compass?

60. ArtLex On The Golden Mean The golden mean / section defined with images of examples from throughout have been instructed in drawing, geometry, symmetry and the golden section, http://www.artlex.com/ArtLex/g/goldenmean.html

G olden Mean or Golden Section golden mean or golden section - A proportional relation ( ratio ) obtained by dividing a line so that the shorter part is to the longer part as the longer part is to the whole. Another way to describe this: a proportion between the two dimensions of a plane figure or the two divisions of a line, in which the ratio of the smaller to the larger is the same as the ratio of the larger to the whole: a ratio of approximately 0.618 to 1.0 The Golden Mean produces a harmonic effect called eurythmy found in nature as well as in a wide variety of works of art and design Artists of various periods and cultures have found that dimensions determined by this formula are aesthetically appealing. Other images of the Golden Mean: The Parthenon , Athens, superimposed with a diagram demonstrating the use of the Golden Mean in the Parthenon's design. See architecture and Greek art Architect Charles E. Jeanneret, known as "Le Corbusier" (French, 1887-1965), often used golden rectangles in his designs for buildings. One of these is the United Nations building in New York. The upright part of the L has sides in the golden ratio, and there are distinctive marks on this taller part which divide the

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