41. On Geometry And Advanced Energy Conversion This true golden mean Spiral has very little in common with the former oneabove. 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?

42. ENC This Week Classroom Calendar golden Section in Nature and The golden Sectionthe Number and Its geometry. ; uk/Personal/R.Knott/Fibonacci/fibInArt.htmlRead about the golden mean and the http://www.enc.org/thisweek/calendar/unit/0,1819,152,00.shtm

43. Golden Ratio -- From MathWorld From the definition of the golden mean combined with the Markowsky, G. MisconceptionsAbout the golden Ratio. College Ogilvy, C. S. Excursions in geometry. http://mathworld.wolfram.com/GoldenRatio.html

Number Theory Constants Continued Fraction Constants Number Theory ... Lambrou Golden Ratio 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"). is also known as the divine proportion, golden mean, and golden section and is a Pisot-Vijayaraghavan constant . It has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers Given a rectangle having sides in the ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle , and successive points dividing a golden rectangle into squares lie on a logarithmic spiral . This figure is known as a whirling square . The legs of a golden triangle are in a golden ratio to its base and, in fact, this was the method used by Pythagoras to construct From the definition of the golden mean combined with the above figure

44. Links To Other Resources And Information A Little geometry. Sacred geometry Home Page The unique numbers in math. Life. Mathematics.Fibonacci - The golden mean - Using trigonometry, limits, etc. http://www.goldenmuseum.com/links_engl.html

Links to Other Resources and Information Fibonacci Series and Golden Section - General Links Dr. Ron Knott's Fibonacci Numbers and the Golden section Site - One of the best and most complete sites on the Internet on this topic, and a winner of many awards. Fibonacci Quarterly Home Page - Research Magazine. The life and numbers of Leonardo Fibonacci The Golden Mean Fascinating Fibonaccis - Mystery and Magic in Numbers The Golden Mean ... Golden Ratio - The 'Phi-Nest': Source to the golden section, golden mean, divine proportion, Fibonacci series and phi Relationship to Divine Creation The Evolution of Truth (Main Site to the this site) Art and Architecture Fibonacci - The LIX Unit - In Astronomy and the Ancient Pyramids Fibonacci numbers and the Golden Section in Art, Architecture and Music Golden Section in Art, Architecture and Music The School of Pythagoras The School of Pythagoras ... Secrets of the Great Pyramid DNA and Phi Dan Winter on Super DNA and Physics of Phi Geometry A Little Geometry Sacred Geometry Home Page - The unique numbers in math Life Fibonacci - The Numbers of Life - An Overview Fibonacci Number in Human Growth and Development Mathematics Fibonacci - The Golden Mean - Using trigonometry, limits, etc.

45. Geometry In Art & Architecture Unit 2 geometry has two great treasures one is the theorem of Pythagoras, the other thedivision of a line into mean and extreme ratios, that is , the golden mean. 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.

46. The Geometry Junkyard: Pentagonal Geometry And The Golden Ratio Penrose tilings and the golden mean, K. Wiedman. Perplexing pentagons,Doris Schattschneider, from the Discovering geometry Newsletter. http://www.ics.uci.edu/~eppstein/junkyard/pent.html

Pentagonal Geometry and the Golden Ratio This page includes geometric problems defined on regular pentagons, involving pentagonal angles, or based on the golden ratio (the ratio of diagonal to side length in a regular pentagon). A Brunnian link . Cutting any one of five links allows the remaining four to be disconnected from each other, so this is in some sense a generalization of the Borromean rings. However since each pair of links crosses four times, it can't be drawn with circles. Constructing a regular pentagon inscribed in a circle, by straightedge and compass. Scott Brodie. Also described by M. Gallant Cut-the-knot logo . With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon. Digital Diffraction , B. Hayes, Amer. Scientist 84(3), May-June 1996. What does the Fourier transform of a geometric figure such as a regular pentagon look like? The answer can reveal symmetries of interest to crystallographers. The downstairs half bath . Bob Jenkins decorated his bathroom with ceramic and painted pentagonal tiles. Equilateral pentagons . Jorge Luis Mireles Jasso investigates these polygons and dissects various polyominos into them. Fibonacci spirals , Ned May.

47. The Geometry Junkyard: Penrose Tiling Links in the geometry Junkyard.Category Science Math Recreations Tessellations...... The geometry Center's collection includes programs for generating Penrose tilings Thepenrose tile and the golden mean towards hyperdimensional intergeometry. http://www.ics.uci.edu/~eppstein/junkyard/penrose.html

Penrose Tiles Penrose was not the first to discover aperiodic tilings , but his is probably the most well-known. In its simplest form, it consists of 54- and 72-degree rhombi, with "matching rules" forcing the rhombi to line up against each other only in certain patterns. It can also be formed by tiles in the shape of "kites" and "darts" or even by deformed chickens (see the "perplexing poultry" entry below). Part of the interest in this tiling stems from the fact that it has a five-fold symmetry impossible in periodic crystals, and has been used to explain the structure of certain "quasicrystal" substances. The Art and Science of Tiling . Penrose tiles at Carleton College. Duane Bailey's color postscript Penrose tiler Cellular automaton run on Penrose tiles , D. Griffeath. See also Eric Weeks' page on cellular automata over quasicrystals Clusters and decagons , new rules for using overlapping shapes to construct Penrose tilings. Ivars Peterson, Science News, Oct. 1996. Dr. Matrix' programming challenge asks for a Windows Penrose tiler. This page also includes background material on tiling and aperiodicity as well as some of the theory of Penrose tilings. Five-fold symmetry in crystalline quasicrystal lattices , Donald L. D. Caspar and Eric Fontano.

48. Loving Unified Field Geometry - Mountain Man's UseNet Archive looks like Betsy Ross's pent stars within stars, created by 10 golden mean spiralsin a pent nest circle. (Top down view DNA). Waves in this geometry can add http://www.magna.com.au/~prfbrown/news96_2.html

Mountain Man's UseNet Archive Loving Unified Field Geometry for Softies by Dan Winter Web Publication by Mountain Man Graphics, Australia in the Southern Summer of 1995 Loving Unified Field Geometry for Softies by Dan Winter Date: 3 Jan 1996 From: Dan Winter To: sci.physics Subject: Quanta and Consciousness The Heart, Fractals and the Universe >> The quanta of consciousness is recursion.. >> as is the branching, the quantum collapse to center >> called gravity occurs only in the presence of >> fractal pattern compression, optimized by golden mean/phi >> (idealized translation vorticity).. >> fractality defines/limits spin density/ mass >> awareness-distributed as spin-onsets with recursion in harmonic >> system, dodeca(dna/earthgrid/zodiac) permits cross scalar >> info transfer in as above/so below, pressure fractal.. >> if time is spin measured and the time spiral is fractal(mayan), >> then when does it become self aware? "You Couldn't Get a Better Squeeze".. Loving Unified Field Geometry for Softies. by Dan Winter 12/24/95 On the side of the box containing a well designed crank operated Tomato Juicer, is printed the motto of today's inspiration: "You Couldn't Get a Better Squeeze". The news for preservers of sweet memories is juicy. Properly applied pressure, sorts waves into preserveable order. The key operating component of the Tomato juicer is a human operated turning spiral helix engraved on a cone vortex. What emerges from the apex of the cone in the shade of living blood, is a harvest of sweetness.

49. Sacred Geometry The design of TARA chapel is based on the laws of sacred geometry. It'sstrongest components are harmony and the golden mean proportions. http://homepages.paradise.net.nz/rainersh/sacred_geometry.htm

Sacred Geometry at Tara Sanctuary (The ideas on this Web Page came from a paper called: "The Tara Chapel at Mana is based on Sacred Geometry of Bio Harmonic Architecture" written by the architect of Tara Sanctuary Reinhard Kanuka-Fuchs) The design of TARA chapel is based on the laws of sacred geometry. It's strongest components are harmony and the golden mean proportions. With it's purpose of practising unity in Meditation, Song, Dance and Prayer the harmonic proportions are essential. From the outside harmony is expressed in a multitude of equilateral triangles touching various important points of the building, connecting them. The unity principle can also be demonstrated by a series of concentric circles with the centre of gravity in the centre of the small "soul eye" windows above the annexes The centre of the chapel is a point of gravity, union and essence and it radiates symmetrically out in all directions touching annexes and octagon corners. In the same way bells radiate sound in all directions from the gravity centre of the bell framing. The law of the golden section as expressed in the beauty of the human form, can be seen in various aspects of nature. It is present in the harmonious properties of the human voice as brilliances appearing as high frequencies in the golden mean (Fibonacci) proportions of 3000, 5000, 8000 Hertz. Throughout the building the golden mean ratio of 1:1.618 can be found, pleasing to the eye and soul, our sense of harmony and beauty. It is creating a space of divine acoustics, where the sound seems to ascend. Drawing IV accentuates examples of the golden ratio in various parts of the chapel and bell tower.

50. Zausner Golden Mean - 1/18/97 A book on the golden mean by Pacioli was published in Venice in 1509. The divisionof Arithmosophy that deals in visual order is called Symbolic geometry. 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)

51. Golden Triangle And Spiral The golden mean Spiral, the seven toroid cascade from the heart's cardiorythm at theGreeks, and the K'mer's of Cambodia, one universal geometry and number 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.

52. Mathematics - Immanuel College Library Resource Centre Calendars, dates and time, Chance and data. Fibonacci numbers and golden mean,geometry. Humour, Logs and exponentials. Measurement, Middle School. Number, Space. http://www.schools.ash.org.au/immanuel/htm/maths.htm

Immanuel College Library Resource Centre Mathematics General links Algebra Business maths Calculus ... Mathematicians General links Online Interactive Mathematics (Teaching Treasures) Awesome Library Elementary School Math ClarisWorks Tutorials Sketchpad Online Activity Guide The Mathematical Association of SA ... Back to top Algebra TEAMS grant algebra lessons Exercises in maths readiness Business maths You may also like to visit the Business studies page and the Economics page for more business/government information. Investor information BT Funds Management Australian Stock Exchange Australian Securities Commission Australian Tax Office ... Back to top Calendars , timetables, dates and time Today's Date Calendar Zone 2000 Calendar Any day in history ... Public Transport Timetables Chance and data AFL tipping game Custom sunrise sunset calendar Sunrise, sunset and twilight times Transport SA ... Back to top Fibonacci numbers and Golden Mean The Fibonacci numbers Fibonacci numbers and the Golden section Sample problems The Golden Mean ... Back to top Logs and exponentials The precalculus page The decibel scale Logarithms and modelling Middle School METeorology on the net Middle School Maths Internet sites Middle School Student Center A+ Math ... Back to top Number

53. Mathematics Links and golden Spiral. The golden mean 3 Another investigation of the algebraand geometry of the golden mean. The golden Section Constructing http://www.msad51.org/GHS/mathlinks.html

athematics inks Curves Golden Mean Fractals Number Theory ... Geometer's Sketchpad Return to Mathematics athematical urves Famous Curves Applet Index Use Java applets to experiment with the graphs of different curves Xah: Visual Dictionary of Special Plane Curves A visual dictionary of special plane curves Famous Curves Index Witch of Agnesi A really nice discussion of this special parametric curve. Witch of Agnesi A gif animation of the curve known as the Witch of Agnesi. Return to Mathematics Links olden ean The Golden Mean, Rectangle, ... The Golden Mean #1 A nice summary of the golden ratio and its derivation. The Golden Mean #2 Some "golden geometry", Penrose tilings, and Golden Spiral. The Golden Mean #3 Another investigation of the algebra and geometry of the Golden Mean. The Golden Section Constructing the Golden Section The Golden Rectangle Constructing the Golden Rectangle and Golden Spiral The Golden Rectangle Animated Animations illustrating the Golden Rectangle and Golden Spiral The Fibonacci Numbers The Fibonacci Numbers and Golden section in Nature An interesting and well illustrated site investigating the occurrance of the golden ratio in nature.

54. Golden Spiral April 21, 1999. Experiencing Sacred geometry without IntellectuallyUnderstanding It. Although the golden mean Spiral is Page http://www.tenisciler.com/cherokee-indian-dictionary.htm

55. Science & 'Physics Of Consciousness' Based On Sacred Geometry Plus or Minus Charge. The Areas of Mathematical Synthesis Between Chaos Theory(Complexity), Fractal geometry and the golden mean .by Nigel Reading http://soulinvitation.com/sbscience.html

Thematic Exerpts from Sacred Geometry Site return to sitemap Intro Glossary About Dan Winter ... Global Sacred Geometry Master Links Directory Magazine format Winter of 2001 Summary: Implosion Group - Secret to the Heart's Music- Key to Implosion and Fusion in the Heart Start Here... (from the Sacred Geometry Basics sections: Laymens Intro Syllabus THE PHYSICS OF Phi, Compression, Implosion, Gravity, Time, and Love Phi-Lo-Tactics: Finding Corpo-REAL Heart! In business the Lo Phi way. New animation of perfect compression / pics of Phylotaxis / new technology literature exerpts:" maximum Complexity is found via self-organised criticality at the edge of Chaos, which is epitomised by the Golden Mean, as the emergent geometric manifestation of the principle of least action: therefore its full temporal/ spatial action is analogous to creation itself." Fusion-Phi Phenomenon-Unlocking Ultimate PHIre. Fusion & New Energy Sources: is PHI (Golden Mean) a key to ALL of it's meanings? Along with "The Phi Phenomenon" key to VISUAL Fusion, is the possibility of FUSION at the Atomic Level, intimately linked to the pure principle of Golden Ratio based PHI Embedding, with all the Implications of creating self-organizing/self-aware wave systems this implies..? Is this an ultimate way to unlock the key to maintaining the PHIre of life? Getting Feeling Centered in A Gravity Lens: A Visual Understanding of Gravity Implosion, Solar Flare Peak, Gravity Lensing, and Relationship between Emotions Gone Shareable and Flux Lines Focused in to Sorting for the Shareable.

56. Storm's Nest - The Golden Logos The golden mean rectangle may be increased or decreased by the additionof a square, which is symbolic in sacred geometry of the creation. http://home.earthlink.net/~johnrpenner/Articles/GoldenLogos.html

"To see the world in a grain of sand, and heaven in a wildflower. (William Blake) The Golden Section "I came across a defintion of the Golden Mean which states that it is the unity that includes both what is perceived as well as the perceiver." "There are two ways [and the Golden Proportion is the only number for which this is true] mathematically, by which you can place the Golden Mean between those two points. The lesser in relation the the greater in the Golden Mean can signify that the outer is the greater and the inner is the lesser, or it can signify that the inner is the greater and the outer is the lesser. But the 'mean' is a flucutation in between those two things." So the Yin and the Yang are as the proportions of the Golden mean unto each other, and thus chase each other dynamically in asymetrical equilibrium. (Ibid. p.30-31, The Golden Proportion: A Conversation between Richard Temple and Keith Critchlow). Extra Links: Astrid Fitzgerald The Golden Proportion in Your Hand Mathsoft - Mathematics of Constants Golden Links ... BACK TO STORM'S JOURNAL to submit your article, send eMail to: johnrpenner@earthlink.net

57. The Golden Mean A particularly elegant manifestation of the golden mean is in other also determinedby the golden ratio Robert Lawler, Sacred geometry, Thames and Hudson, 1982. http://home.earthlink.net/~klogw/TheGoldenMean.htm

The Golden Mean A Pictorial Essay Scholars of the occult have speculated that the Golden Mean was of supreme importance to the ancient Pythagoreans as a key to unlocking the mysteries of the universe. One of these mysteries is that from the Golden Mean the Pentagram can be constructed by means of compass and straight edge alone. The following pictorial essay shows the classical construction used to obtain the segment length which is in golden ratio proportion to a segment of unit length. The Pentagram is then easily constructed from these two lengths. Can you see how? An interesting application of the Golden Mean is the construction of the logarithmic spiral. The logarithmic spiral is found throughout nature from conch shells to the pattern of seeds in sun flowers to the pattern of lines in the feathers of peacocks. The spiral is constructed by starting with two golden mean rectangles and successively adjoining squares to the preceding construction. Can you see how? A particularly elegant manifestation of the Golden Mean is in the design of Japanese sand gardens. As can be seen in the diagram below three rocks are laid out collinear on a line such that the line segments they define are in golden ratio proportion to each other. Although not shown in the diagram, the two lines of rocks are at an angle to each other also determined by the golden ratio. All around the rocks sweeping to the coping tile around the garden is elegantly raked sand, almost giving the rocks the appearance of islands rising up out of tranquil waters. The entire effect is that of correct proportion achieving balance and tranquility.

58. Geometry The golden mean (or golden Section), represented by the the Fibonacci series and thegolden Rectangle carefully 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

59. Spiritual Structure Traditional Sacred geometry. The Fibonacci series can be used to defineboth the golden mean and Pi, the two main irrational numbers. http://www.dimensional.com/~ahm/matrix/SpSt/geometry.htm

Human Spiritual Structure: Sacred Geometry Notes from Nothing: This is supposed to be the basic geometry of the universe, the underlying patterns of all existance. The following derivation is said to parallel the creation of the world given in Genesis. First there is the void. Then spirit reaches out in six directions, up, down, back, forth, right and left, creating three axis' which allows measurement and movement. Filling out the axis's results first in a square and then in an octogaon, two base to base pyramids. (In planar figure this also gives the hexagon, and by reprojecting into 3 dimensions the cube.) Straight lines are male, curves female, so spining the octagon to create a sphere generates female from male. Spirit moved to the surface of the sphere and then began to make more spheres. In planar form six more can fit around the circle (Spirit rested on the seventh day). In space perhaps there are also six, one centered out each axis? These are all interlocking, i.e. the center of each is on the circumferance of the others. By rotation this generates a tube torus? which appears to be a torus with a zero inner radius, e.g. a donut without a hole. This is used to generate a figure inside a tetrahedron which in turn generates the various Hebrew letters, by projection back to a plane. [Stan Tenen]. The Greek and Arabic letters can also be formed by a similar process.

60. Geoman Text adds two of the three fundamental polygonal shapes in plane geometry, the equilateraltriangle, and the pentagram, both in the golden mean relationship to the 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.

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