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Sacred Geometry  Page 2

Sacred geometry has many diverse and extremely interesting aspects. Click on the links below to get to the place on this page that most interests you. We start with Pythagoras of Samos, who helped put sacred geometry "on the map", and then go logically through the process of creation into more specific areas of sacred geometry. You can also search this page for subject matter by pressing Ctrl+F on your keyboard to open the Find feature ((Apple+F for Mac). Enjoy!
Pythagoras.
Geometry and creation.
The Five Platonic Solids.
The Golden Ratio.
Phi and the Great Pyramid.
Visualizing crystal structures for higher consciousness.
Cymatics and Creation by Sound.
Pythagoras.
People have understood the spiritual significance of geometry for ages, from the ancient Egyptians to the Vedas in early India. These were studied by Pythagoras, who helped immensely in putting sacred geometry "on the map" i.e. making it more generally known. Born around 580 BC in Samos, this amazing Greek philosopher, mathematician, and spiritual leader formulated principles that influenced the thought of Plato and Aristotle and contributed to the development of mathematics and Western rational philosophy. He founded the Pythagorean brotherhood in Cortona, a mystical school developed as the archtype of a goldenage educational community.
They uncovered many of nature's secrets, and held that at its deepest level, reality is mathematical in nature. (Pythagoras once declared: "All is number." This theme has been repeated many times since, including by Bruce L. Cathie in his many books on harmonics, like The Bridge to Infinity.) They studied the mathematical laws behind the rhythm and harmony of music. They also pondered the intangible, not only by the logic of the mind but by the intuitive faculties of the heart. Pythagoras believed in life after death and reincarnation, and taught that the soul can transcend the cycles of rebirth and rise to union with the divine. Their community followed an ethical and moral code, and strove to put their spiritual knowledge into practical everyday use.
John Strohmeier and Peter Westbrook say the following on page 66 of their book Divine Harmony: The Life and Teachings of Pythagoras (Berkley Hills Books, 1999): "For Pythagoras, mathematics was a bridge between the visible and invisible worlds. He pursued the study of mathematics not only as a way of understanding and manipulating nature, but also as a means of turning the mind away from the physical world, which he held to be transitory and unreal, and leading it to the contemplation of eternal and truly existing things that never vary. He taught his students that by focusing on the elements of mathematics, they could calm and purify the mind, and ultimately, through disciplined effort, experience true happiness."
Unfortunately for Pythagoras (and for all of us since) there was much jealosy directed towards him and his community, plus resistance to his enlightened ideas. He and many of his followers were eventually killed, and his teachings largely destroyed. So much of what we know about him is from thirdhand accounts, some of it being aimed at discrediting him. But his legacy and genious live on, and many after him have been inspired to study and elaborate on this wonderful subject.
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Geometry and creation.
The role of geometry in the process of creation is covered well by Robert Lawlor, in his fascinating book Sacred Geometry: Philosophy and Practice (Thames and Hudson, London, 1982). In it he says the following: "The perspective of volume offers yet another metaphor for the original and evercontinuing creative act of the materialization of Spirit and the creation of form. The very ancient creation myth coming from Heliopolis in Egypt gives an example of this mode of envisioning. Nun, the Cosmic Ocean, represents pure, undifferentiated spiritspace, without limit or form. It is prior to any extensive, any specificity, any god. It is pure potentiality. By the seed or will of the Creator, who is implicit within this Nun, the undifferentiated space is impelled to contract or coagulate itself into volume. Thus Atum, the creator, first creates himself or distinguishes himself from the undefinable Nun by volumizing, in order that creation might begin."
The Five Platonic Solids.
"What form, then, might this first volume have? What indeed are the most essential volumetric forms? There are five volumes which are thought to be the most essential because they are the only volumes which have all edges and all interior angles equal. They are the tetrahedron, octahedron, cube, dodecahedron and the icosahedron, and are the expressions in volume of the triangle, the square and the Pentagon: 3, 4, 5. All other regular volumes are only truncations of these five. These five solids are given the name 'Platonic' because it is assumed that Plato has these forms in mind in the Timaeus, the dialogue in which he outlines a cosmology through the metaphor of planar and solid geometry."
In this dialogue, which is one of the most thoroughly 'Pythagorean' of Platos works, he establishes that the five elements of the world are earth, air, fire, water and aether (prana). Lawlor continues: "Plato's fabricator of the universe created order from the primordial chaos of these elements by means of the essential forms and numbers. The ordering according to number and form on a higher plane resulted in the intended disposition of the five elements in the physical universe. The essential forms and numbers then act as the interface between the higher and lower realms. They have in themselves, and through their analogues with the elements, the power to shape the material world.
"As Gordon Plummer notes in his book The Mathematics of the Cosmic Mind, the Hindu tradition associates the icosahedron with the Purusha. Purusha is the seed image of Brahma, the supreme creator himself, and as such this image is the map or plan of the universe. The Purusha is analogous to the Cosmic Man, the Anthropocosm of the western esoteric tradition. The icosahedron is the obvious choice for this first form, since all the other volumes arise naturally out of it."
Icosahedron:
 The Hindu tradition associates the icosahedron with Purusha, the seed image of Brahma, the supreme creator himself. 
Dodecahedron:
 The icosahedron generates the dodecahedron, representing Prakriti, the feminine power of creation of the matter (mother) universe. The star born within it's pentagon (below) produces the Golden Ratio, the energy of resurrection and symbol of the Christ. 
The star born within the pentagon:
Patterns created from the moving 3D Icosahedron lattice above:
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The Golden Ratio.
The golden ratio is the mathematical proportion 1:1.6180339...ad infinitum. It's mathematical name is phi, from the Greek alphbet. It is also known as the Divine Proportion, the Golden Section and the Golden Mean. It is found abundantly in nature and in the works of man. Find it in the pine cone, the daisy, the seashell, the sunnflower, the great pyramid and the Parthenon. Or way out in the macrocosm in the spiral nebulae as described above. It is so integrated into life that many philosophers, artists, mathematicians, and scientists have see it as an essential aspect of beauty and perhaps of life itself. The Egyptians believed it was in the creative process and the fire of life. Plato considered it key in the understanding of the cosmos.
The golden ratio is found by dividing a line (AC) at a particular point to give two unequal sections, where the smaller one (AB) is proportionate to the larger one (BC) as the larger one (BC) is to the entire line (AC). The ratio is expressed as AB/BC = BC/AC. There is only one such point (B) on any line. It is easy to produce this ratio geometrically. One of many ways is the fivepointed star, as drawn within a pentagon above. Every outside line on the pentagon is in a phi ratio to any line of the star. Also, every line on the star has 3 segments. One side segment is in a ratio of 1:1.618 to the rest of that line. This ratio is what was used in the architecture of the great pyramids of Egypt, and results in the focusing of universal energy (a.k.a. prana) at a particular point, which helps in rejuvenation of life (or the preservation of dead bodies). This resurrection energy and ratio is the mark of the Christ, the victor over death and rejuvenating principle in nature.
Another aspect of the golden ratio is the Fibonacci Series. It is a series of numbers discovered by Leonardo Fibonacci. Each number in the series is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, ... The amazing thing is that the ratio of each successive pair of numbers is approximately 1.618 (phi). The higher the numbers, the more accurate they get. 8 divided by 5 is 1.60, but 89 by 55 is 1.61818... Once again, this series is found abundantly in nature, like the way that plant stems branch out.
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Phi and the Great Pyramid.
In the diagram above, the big triangle is the same proportion and angle of the Great Pyramid, with its base angles at 51 degrees 51 minutes. If you bisect this triangle and assign a value of 1 to each base, then the hypotenuse (the side opposite the right angle) equals phi (1.618..) and the perpendicular side equals the square root of phi. And that’s not all. A circle is drawn with it’s centre and diameter the same as the base of the large triangle. This represents the circumference of the earth. A square is then drawn to touch the outside of the earth circle. A second circle is then drawn around the first one, with its circumference equal to the perimeter of the square. (The squaring of the circle.) This new circle will actually pass exactly through the apex of the pyramid. And now the “wow” part: A circle drawn with its centre at the apex of the pyramid and its radius just long enough to touch the earth circle, will have the circumference of the moon! Neat, huh! And the small triangle formed by the moon and the earth square will be a perfect 345 triangle (related to the Pythagorean Theorem). Using the 345 triangle is also the easiest way to draw a squared circle.
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Visualizing crystal structures for higher consciousness.
If you really want to get into the spirituality and personal growth aspect of sacred geometry, it is recommended to meditate on the geometric structure of crystals. David Tame, in his fascinating book The Secret Power of Music, (Destiny Books, 1984) says the following: "On the subject of crystals, modern esotericists have often recommended crystals, pictures of crystals, and models of the molecular structure of crystals as a subject for meditation. Contemplation upon their geometry is said to provide a route by which the consciousness of man can attune itself to the various qualities of the Consciousness of the Supreme."
Here is a related and rather challenging exercise. It is a message from the Ascended Lady Master Leto through the messenger Elizabeth Clare Prophet, April 15, 1976, found on an audio tape (#B7629). "You have been told .... to meditate upon the crystal and its structure. Now I ask you ... to select images
of crystals and then to diagram the molecular structure of crystals and to begin to memorize these structures and to endow them with flame, and to realize that your altar is a place of change .... And therefore consider that the crystal, especially of rare gems and their molecular structure  and I meditate now with you upon the structure of the emerald  is a means of contacting the plane of Spirit. For these emanations of the mind of God manifest in matter are parallel to the energies manifest in Spirit. They are timespace coordinates of the coalescing of energy in infinity.
"If you can therefore meditate upon, and endow your meditation with the flaming fire, this is what will take place: You will trace from effect to cause, to first cause, the very components of creation. And if you will study the treatise on alchemy by beloved Saint Germain, you will find that your visualizations will bring into manifestation elements of the Christ consciousness, abundance, the ability to expand a mandala for the community of the Holy Spirit."
(Quoted with permission. Copyright © Summit University Press. All rights reserved. Website: www.summitlighthouse.org)
It is not that easy to find the molecular structure of crystals. To help you on your way, here is a link to an excellent website by the Institut LaueLangevin (ILL) in France on The atomic structure of materials. This site allows you to obtain a 3D VRML model of structures by clicking on flagged links. If you do not have a 3D viewer on your computer, click the Help link at the top of their page. Once you have one, click on the “Gems and Minerals” link. Clicking on flagged links will allow you to navigate WITHIN and around these 3 dimensional crystal structures. Very cool! PS: If you have any ideas on how to “endow these structures with Flame”, please share them with us. Thanks.
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Cymatics and Creation by Sound.
Now we come to one of nature's greatest mysteries, the creation of matter by sound. "And God SAID, let there be light." "In the beginning was the WORD, ... and the Word was God." Jesus and his followers obviously studied the ancient Hindu Vedas, written long before his time. They state: "In the beginning was Brahman with whom was the Word. And the Word was Brahman." The sacred texts of India teach that sound holds the key to the mysteries of the universe, to the creation and constant changing of our world. They put much emphasis on mantras, spoken prayers and sacred music. We in the West, constantly being bombarded by discordant sounds and music, are largely unaware of its tremendous power.
The book Cymatics: A Study of Wave Phenomena & Vibration by Hans Jenny sums up the phenomena of the effect that sound has on matter. He developed techniques to show that every frequency and musical note has a particular geometric pattern that it forms, like this colour image on the book's cover.
He built on the work of Chladni, who created sand patterns on a metal disc through sound. You can find more on Cymatics at this AlphaOmega website. More insight into this subject is given by David Tame in his fascinating book The Secret Power of Music mentioned above. In his chapter The Physics of the Om, he says: "Complex and meaningful patterns are even more apparent in Jenny's soundaffected substances when viewed at the microscopic level. Then are revealed beautiful and mathematicallyprecise mandalastructures looking like groupings of microscopicallyviewed snowflakes. The stressinteractions created in substances by their exposure to sound frequencies always result in formations replete with meaningful numerolical, proportional and symmetrical qualities."
He goes on to discuss the works of several researchers who prove that the wholenumber ratios of musical harmonics correspond to an underlying numerical and geometric framework existing in nature and matter. This sheds light on Goethe's statement "Geometry is frozen music." The book also covers research on the effects of sound and music on the human body and psyche. In general, the findings show that classical type music promotes physical and mental health, whereas rock music has the opposite effect. However, this subject is beyond the scope of this website, but well covered in The Secret Power of Music book. A more spiritual study of sound, mantra and prayer is covered in The Science of the Spoken Word and The Creative Power of Sound (pocket book) by Elizabeth Clare Prophet (Summit University Press).
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