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- 1. Sacred Geometry What it is and a case for why it should be included in elementary education By Eileen McKusick
- 2. Better Living Through Geometry “There is a pattern in the heavens, where those who want to can see it, and establish it in their own minds.” Socrates “Geometry existed before the creation.” “No state can find happiness unless the artist drawing it uses a divine pattern.” Plato
- 3. Sacred Geometry- What is it? • Sacred Geometry is the blueprint of the natural world and the basis 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 the Universe organizes itself. • On every scale, every natural pattern of growth or movement conforms inevitably to one or more geometric shapes.
- 4. What makes it “Sacred”? (and isn’t that a bad word in our secular culture?“) • Sacred means ‘holy’ or spiritual, a term left by science to religion- meaning it has no place in a secular education. • Secular mathematics is what is taught in school- a focus on the quantity aspect of numbers (that turns children into ‘pocket calculators’) and gives no attention to the underlying connection of numbers and geometry to nature, the universe, or the human psyche. • Geometry and numbers are sacred because they codify the hidden order behind creation • This lack of awareness of underlying connective orderly patterns creates a cosmology or universal perception of separation.
- 5. What makes is Sacred (con’t) “Symbolic and sacred mathematics encode subtle experiences whose purpose is different from secular math. They can invigorate, refine and elevate us to discover the inherent proportion, balance , and harmony in the patterns that construct the universe and ourselves… 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”. Michael Schneider “man is the measure of all things”
- 6. “Official” Contemporary Perspective • “A contemporary usage of the term sacred geometry describes assertions of a mathematical order to the intrinsic nature of the universe. Scientists see the same geometric and mathematical patterns as arising directly from natural principles.” Wikipedia ??? What is the difference???
- 7. Patterns, patterns, everywhere • Everything has a pattern, and that pattern is the key to creating a specific effect – these patterns create everything in the material world. • So, what are the patterns and where do we see them? And what do they mean?
- 8. Nature
- 9. More Nature
- 10. Art
- 11. Architecture
- 12. Math
- 13. Science
- 14. The Human Body
- 15. A voyage from 1-10 • Next we will look at each of the numbers from 1-10 • Each number was seen as expressing an archetype of nature • “the original ten patents for designs found throughout the universe” • Tells the story of Creation from the perspective of a geometer.
- 16. The geometer’s tools
- 17. One- The Monad The number 1 is symbolized by the circle- the shape from which all other shapes ariseIt breathes into the void and gives rise to all other numbers 111111111 x 111111111= 12345678987654321 The area within the circumference which manifests between nothingness – the zero point inside- and everything – the infinite points around the circumference. Nothing exists without a center around which it revolves. When the center does not hold, the entire affair collapses. When something is pointless, it has no center holding it together.
- 18. The ouroborus • This is a symbol representing the monad, or one, that has arisen on every continent • The eternal unity of all things • The cycle of endless cycles repeating themselves • Universe: One Turn
- 19. Two- Dyad The marriage of two circles gives birth to the line. Also the overlapping space between the two circles is known as the vesica piscis or cosmic womb From this womb, the rest of the numbers and their respective shapes emerge. •The principle of the dyad is polarity •The paradox is that opposite poles remember their source and attract each other in an attempt to merge and return to a state of unity •Dyad simultaneously divides and unites, attracts and repels, separates from unity and craves to return to it. •A line creates both a boundary that divides and a line that binds.
- 20. It takes two to tango • Only by acknowledging both poles in the pair as inseparable can we overcome relative duality and get to their common source in the monad. In our deepest self, we are beyond all polarity. • For anything to manifest, however, there must be polarity. • Monad and Dyad were considered to be the parents of numbers. Dyad is the doorway between the one and the many. • Hydrogen, the first atom to emerge, has a positively charged proton and a negatively charged electron. It is the door between the unknown unity (monad) and the subsequent elements, which are built by fusing hydrogen.
- 21. Three- Triad Triad- the firstborn- Triangle is the first shape to emerge from the vesica piscis •Natural to divide a whole into three parts : Beginning, middle and end •An inherent tripartite wholeness of the universe •Trinity- tri-unity – three as one. Triangle- preeminent symbol of divinity •Opposites are balanced by a 3rd mediating element •Ancient mathematical philosophers referred to the triad as prudence, wisdom, piety, friendship, peace, harmony, unanimity, and marriage. •Fundamental structures of the world are braced by triangles- a triangle is self-sufficient: it encloses the smallest area with the greatest perimeter
- 22. Some examples of triads • • • • • • • • • Birth- growth –decay Creating-maintaining-destroying Brahma- Vishnu- Shiva Osteoblat – osteocyte – osteoclast Executive- legislative- judicial Father – son – holy spirit Light-energy -mass Spirit – soul –body Compass- straight edge- paper/pencil
- 23. A lesson from the triad Two forces can be opposites without resolution, like lawyers without a judge- but introduce a properly chosen third factor and it can induce a relationship that unifies them and brings them to a whole new level. A minimum of three elements is required to weave into a whole (think of a braid)- the principle of the triad can teach us how to resolve conflict. No enduring resolution of any kind is possible without three aspects, two opposites and a neutral, binding, balancing, arbitrating presence (think executive, legislative, judicial). Knowing how to choose the third factor means the difference between a conflict’s resolution and its perpetuation. The third element, which, if chosen properly achieves a previously unknown level of balance and completion.
- 24. Four- Tetrad The square, cube, and three dimensional space emerge from the vesica piscis next •4-ness/square associated with the earth – solid, stable earth •4 directions 4 winds 4 seasons 4 corners of the globe • 4 footed creatures are earthbound •Mater- Latin for Mother – Matter •Pater- Latin for Father- Pattern •Natural forms arise from mating of pattern and matter •The arising of these natural patterns in 3-D where they can be clothed in the four states of matter •4 elements or states of matter•Solid, liquid, gas, plasma / earth, water, air, fire
- 25. The five Platonic Solids or “regular polyhedra” • Fall in the domain of the number 4 because this is the birth of 3-dimensionality • These are Forms that fit perfectly within a sphere connected by identically shaped surfaces, edge lengths, and angles – an identical view in all directions Dodecahedron: Heaven above Tetrahedron: Fire/Plasma Octohedron: Air/Gas Icosahedron: Water/Liquid Hexahedron (cube): Earth/Solid
- 26. Classical view of these shapes • The Greeks taught that these five solids were the core patterns of physical creation. • Four of the solids were seen as the archetypal patterns behind the four elements (earth, air, fire, and water) • The fifth was held to be the pattern behind the life force itself, the Greeks' ether. This fifth solid, the dodecahedron, was in fact kept a closely guarded secret in the Greek school of Pythagoras and even Plato spoke little of it • They feared that this pattern could cause tremendous destruction if misused.
- 27. The Aether • The aether energy can arrange itself into basic geometrical wave patterns that were named after Plato, the Platonic solids to form matter. Almost 2,500 years ago, Plato wrote that the physical world was constructed from the Platonic solids. The Platonic solids arrange themselves in what chaos theory calls fractal patterns weaving a matrix in space interconnecting atoms with the stars. The scales of the Platonic solid shapes are different but the ratios between them are still the same (following the Hermetic principle as above so below).
- 28. Five- Pentad The introduction of life itself Pentagonal symmetry is the supreme symbol of life, universally appealing •Quintessence- the life that infuses the four elements that they cannot create on their own •Stardom conveys a deep seated association with excellence and brilliance and power. •Pentagons abound in the animal and plant kingdoms •Flower of every edible fruit has five petals •Seeds, holders of life itself, arranged in a star pattern in fruit •Look at leaves, flowers, fruits – see the invisible energy web of the archetype made visible as a pattern of living cells. •Pentad holds the principles of the geometry of regeneration. Regenerates within and without.
- 29. The Fibonacci Series • Fibonacci series begins with two “terms” • Zero plus unity, nothing and everything, the unknowable and the manifest monad • To find each next term, just add the latest two terms together. This produces an endless series: • 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610 … • Series grows by accruing terms that have come from within itself, from its immediate past • Each term may be traced back to its beginning as unity in the monad which itself arose from the “incomprehensible mystery of zero” • The average number of petals on each type of flower will be Fibonacci numbers, same with pine needle clusters.
- 30. 0,1,1,2,3,5,8,13,21,34,55,89, 144,233, 377,610….. • Divide each fraction by the term before it: …8/5, 13/8, 21/13, 34/21, 55/34, 89/55… • And the numbers get closer and closer to 1.6180339… or Phi • The Golden Mean, golden ratio, divine proportion, golden cut, etc : the ideal balance in life • Phi is a greek letter and it was named as such in this last century to honor Greek sculptor Phidias who used it to proportion his designs, from the Parthenon to the famous statue of Zeus
- 31. Golden Rectangle/Phi Spiral
- 32. The Spiral • The purest expression of moving energy. • The universe moves and transforms in spirals, never in straight lines – for example, plant growth • Clashing opposites resolve in spiral balance – balance in motion made visible • Archimedian spiral, like a coiled rope or Fibonacci spiral (nautilus) • Spirals have a calm eye that corresponds to zero at the beginning of the Fibonacci sequence. • Calm eye is the unmanifest, spiral is the manifest • Without the eye there would be no spiral expansion or dissolution, no whirling, no balance, no life • Calm center exists within all living things, including self- not as a concept but as an actual part of the pattern and dynamic process that is “Self”.
- 33. More about spirals • Spirals grow by self-accumulation – and shrink by dissolution • Matter precipitates where positive and negative spiral rhythms synchronize. • Energetic lines of force that the more dense substance of the “world mother goddess” or “mater” precipitates- descending from light (plasma) to gas to liquid to finally solid expression.
- 34. Fractals- “as above, so below” • • • • • The repetitive patterns of the Platonic solids that fit into each other are fractals. A fractal is a repetitive pattern that can be scaled to any size. The scale may change but the ratio is held constant. Now the fractal patterns that shape the atom, according to Daniel Winter also shape our planets and stars, in fact the universe. A fractal has self-similarity at all scales, it’s the same geometric pattern repeated. The inner structure in a fractal is reflected in the outer structure. Fractal means fraction of the all, indicating that each piece is part of the whole. It is the basic idea of a hologram and this is why according to Daniel Winter the universe at large is just a super-hologram. The fractals of electromagnetic energy interconnect everything with everything within the universe and are the basic building blocks of this hologram. The fractal repetitive structures of electromagnetic energy weave a giant cobweb throughout the universe. The wavelengths of planets and stars are huge in comparison with the wavelengths of the atom, however when their wavelengths fit into the Fibonacci sequence of the Golden Mean, they interfere non-destructively; they implode and form a fractal attractor that we call gravity! This is how the planets and stars are connected by means of imploding electromagnetic waves that we experience as gravity. So if the universe is in essence a hologram and there is only one principle that shapes everything from atoms to planets, stars, and the universe at large, we should find proof of these vortexes, Platonic Solids, and doughnut structures in all parts of the universe.
- 35. Six- Hexad Structure-function-order Space-power-time • 6 days of activity around a central day of rest • Hexagram star- Seal of Solomon/Shield of David/Mark of Vishnu • Hexagon works in situations requiring strength and stability (hex nuts, for example) • Best use of material with the least amount of weight : in beehives1.5 ounces of wax holds 4 pounds of honey! • Circles and the hexagons formed by them are said to tessellate • Hexagonal tessellation recurs endlessly in nature and in the human body – lungs muscles the eye and more • Six sided forms in nature are often non-living – ie. snowflakes • 3 corner 120 degree joints- common in the realm of molecules and crystals • Quartz is ordered, timekeeper
- 36. Twelve- Six doubled • Dodecagon- Twelve – The framework number – used in teepees and lodges in native Americans and other societal structures • Sunlight- the sun king – surrounded by 12Jesus, King Arthur and others • 12 constellations of the zodiac 12 part epic myths 12 step programs 12 disciples 12 tribes of Israel 12 notes in the chromatic musical scale Acupuncture- 12 meridians around one central channel • weights and measures exhibit 12-ness - Metric system makes calculations easy but dissociates us from a framework system linked with the cosmos. • 12 around one within our deeper psyche
- 37. Seven- Heptad • The heptagon cannot be born through the vesica piscis – it cannot be constructed by the compass, straight edge and pencil accurately. • Cannot be entered (divided) and cannot produce other numbers within the decad- this is why it is considered a virgin or holy number • Nature does not produce heptagons – nothing in the mineral world is 7 sided • 7 creates a familiar rhythm- days of the week, notes in the musical scale, 7 years til every cell in the body is replaced, information stays on your credit report for 7 years. • Temples to Athena, who sprang from the head of Zeus, were designed and constructed around the number 7 • 7 chakras – white light prisms through the body to separate out into these energy centers like the colors of the rainbow. • 7 sets of endocrine glands
- 38. 7 and 12 in music and elsewhere • 7 notes in scale plus 5 sharps/flats = 12 note chromatic scale • Musical scale was designed to play the harmonies of the heavens, the music of the spheres, allowing higher principles to enter our lives through our sense of hearing and emotion. • Music was seen as having great power for producing heaven on earth • Music bypasses the intellect with the power to manipulate passions and emotion- in this way, music was used to heal. • Ancients sought to tune or purify themselves so that the unseen divinity deep within them could better play upon their instrument in the world. • “We are the 7 stringed lyre of Apollo, whose music is heard by both god and humans” • 12 has relations with all the numbers in the decad, 7 has none • Hidden, unborn, eternally elusive side of 12 but they are often together (12 meridians, 7 chakras)
- 39. 7 notes/centers in the body •7 chakras – white light prisms through the body to separate out into these energy centers like the colors of the rainbow. •Correspond to the 7 sets of endocrine glands
- 40. Eight-Octad • • • • • Octave- brings you back to the beginning Chess 8x8 I Ching 8x8 Peace and Stability Also relates to the moon Octad displays wholeness, cycles, polarity and manifestation as the doubling of 4 • Mitosis- 8 step process • Of particular significance and symbolism in China
- 41. Nine - Ennead •3x3 x3 trinities •The ultimate expression of something •Whole nine yards- go all the way •Dressed to the nines •Nine innings in baseball •Nine months gestation •Nine justices in the supreme court •Boundary between mundane and infinite
- 42. Ten-Decad tetraktys •All encompassing perfection •Completion of a journey and a return to the origin •A journey into limitlessness •Two parent 1-2 7 children 3-9 Portrait of a whole family of numbers gathered together •Comprehends and harmonizes all numbers below it •Generative power to the infinite •Fulfillment- new beginnings
- 43. Seed of Life/Flower of Life/Tree of Life
- 44. In Education • At every scale of existence, from the subatomic to the galactic, the same key patterns bring all things into existence. • These patterns can easily be taught to and shared with children – in doing so, we foster in them an awareness of the underlying patterns and harmony inherent in the natural environment. • This can be a key component in creating a more balanced and harmonious future. • “Better living through geometry” • (maybe we can call it “classic geometry“)
- 45. Classroom projects
- 46. Resources • Bangs, Herbert. The Return of Sacred Architecture: The Golden Ratio and the End of Modernism. Inner Traditions, Vermont, 2007 • Doczi, Gyorgy. The Power of Limits: Proportional Harmonies in Nature, Art and Architecture. Shambhala, Boston, 2005. • Ghyka, Matila. The Geometry of Art and Life. Dover Books, New York, 1946. • Michell, John. How the World is Made: the Story of Creation According to Sacred Geometry. Inner Traditions, Vermont, 2009 • Michell, John. The Dimensions of Paradise: Sacred Geometry, Ancient Science, and the Heavenly Order on Earth. Inner Traditions, Vermont, 1971 • Schneider, Michael S. A Beginner’s Guide to Constructing the Universe. Harper, New York, 1994. • Skinner, Stephen. Sacred Geometry: Deciphering the Code. Sterling Publishing, New York, 2006.

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