sacred-mathematics

Chaos Theory.

Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by Fractal Mathematics, which captures the Infinite Complexity of Nature. Many natural objects exhibit Fractal Properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit Complex, Chaotic behavior. Recognizing the Chaotic, Fractal Nature of our world can give us new insight, power, and wisdom. 

Principles of Chaos

The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. 

Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the world, accurate long-range weather prediction will always remain impossible.

Order / Disorder: Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.

Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. 

Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

The Law of Octaves (8)

The Law of Octaves was first suggested by Pythagoras in ancient Greece. Having observed that the eight notes of the conventional Occidental musical scale were governed by definite mathematical relationships, Pythagoras proceeded to create a whole cosmology based on 8s. In this octagonal model Pythagoras made numerous mistakes, because he was generalizing from insufficient data. However, his work was the first attempt in history to unify science, mathematics, art and mysticism into one comprehensible system and as such is still influential.

      In China, roughly contemporary with Pythagoras, the Taoists built up a cosmology based on the interplay of yang (positive) and yin (negative), which produced the eight trigrams of the I Ching, out of which are generated the 64 hexagrams.

      In India, Buddha announced, after his illumination under the Bodhi tree, the Noble Eightfold Path. Patanjali subsequently reduced the science of yoga to eight “limbs” or, as we might say, eight “steps.”

      The game of chess appeared, somewhere in the East, with a grid based on 8x8 (64) squares.

      Kepler discovered the laws of planetary motion serendipitously, while trying to make the planets fit into the Pythagorean octave.

      In the 1860s, English chemist John Newland showed that all the chemical elements fall into eight families. Since Pythagorean mysticism was unfashionable at that time, Newland was literally laughed at and rejected by the Royal Chemical Society. In the 1870s, with much more detail than Newland, the Russian chemist Mendeleyev proved once and for all that the elements do, indeed, fall into eight families. His Periodic Table of the Elements, an octave of hauntingly Pythagorean harmony, hangs in every high-school chemistry class today. (The Royal Society later apologized to Newland and gave him a Gold Medal.)

      Nikolai Tesla invented the alternating current generator which unleashed the modern technological revolution after a series of visions in which, among other things, Tesla “saw” that everything in the universe obeys a law of Octaves.

      Modern geneticists have found that the DNA-RNA “dialogue"—the molecular information system governing life and evolution—is transmitted by 64 (8x8) codons.

source: http://deoxy.org/eoctave.htm

The Theory of Everything - The E8 Lie group, a perfectly symmetrical 248-dimensional object and possibly the structure that underlies everything in our Universe.

Mathematics is the language of the Universe. Absolutely everything, from a plane crash to your skin pigment to the shape of a Sphere can all be expressed using mathematical equations. This last example is most important in pursuit of the Theory of Everything.

In the 19th century, the mathematician Sophus Lie created algebraic formulas to describe the shape of symmetrical objects. These are called Lie Fields. In the 1890s, Wilhelm Killing found a set of Lie fields that described perhaps the most complex shape in our Universe, the E8 Group. The E8 group, an interrelated 248-dimensional symmetrical object, is an extremely complex one.
This dense object is so complex, in fact, that it was plotted by computer for the first time in 2007. It took a team of 18 mathematicians four years to calculate and plot the formula for E8. So how can a symmetrical shape be the key to the Universe? First remember that a geometrical shape is merely the graphic representation of mathematical formulae. It is a pattern that is expressed in math and forms a shape when plotted. In this sense, the E8 could be the framework into which everything - all forces and particles - fits in our Universe.

Physicists generally held that gravity couldn’t be expressed mathematically in the same way as electromagnetism and strong and weak nuclear forces could.
But Garrett Lisi had heard about a mathematical way of expressing gravity, called MacDowell-Mansouri gravity. Using this expression, Lisi can use mathematical expressions to plug gravity into E8, along with electromagnetism, and weak and strong nuclear forces. All four of the forces in the Universe create a distinct effect on all of the most basic subatomic forms of matter - called elementary particles. When these particles interact with force carriers (called bosons), they become different particles. For example, when one of the most basic quantum particles - the lepton - encounters a weak-force boson, it becomes a neutrino. A lepton interacting with a photon (a boson that carries an electromagnetic charge) becomes an electron. So while there are limited numbers of the most basic particles, when they encounter the different forces, they change to become other, distinct particles. What’s more, for every particle, there is an equally distinct anti-particle, for example an anti-quark or anti-neutrino. In total, these make up the elementary particles, and there are 28 of them. Each of these distinct elementary particles has eight quantum numbers assigned to it, based on the charges each particle has. This brings the number of distinct particles to 224. These numbers helped Lisi make the particles fit into the E8 model. While the E8 is expressed as a 248-dimensional object in one way, it can also be expressed as an eight-dimensional object with 248 symmetries. Lisi used E8 within eight dimensions for his calculations. For the remaining 24 places unfilled by distinct known particles, Lisi used theoretical particles which are yet to be observed.

Take another look at E8, and notice how the lines radiate from each point:
Lisi assigned each of these 248 points to a particle, using the eight numbers based on their charges as coordinates within the eight dimensions. What he found was that, like the symmetries in the E8 group, quantum particles share the same relationship within the symmetrical object. He has hope that he has figured out a way to crack the Theory of Everything, because when he rotated the E8 filled with the force-influenced (including gravity) quantum particles, he found patterns emerging between particles and forces - photons interacting with leptons, for example, created electrons. The connections shown within points on the E8 match up to real, known connections between particles in our physical world.

4 (the) 1(st post of) 2011

well I didn’t want to just haphazardly start a blog on any random day. and I needed some time to organize my thoughts and try to build a little hype. why today? today’s date 4/1/2011 reduces to 9

4 + 1 + 2011 = 2016 => 2 + 0 + 1 + 6 = 9

follow me?

the number 9 is very significant in the Numerology of lots of cultures all over the world all throughout history. some of the awesome qualities that 9 boasts are being the greatest single digit and being composed of three 3’s, with the number 3 representing a Divine Trinity in many schools of spirituality. so 9 is the ultimate! there are also a whole bunch of other unique magical numerical properties of the number 9. remember how you learned your “times tables” for 9? the sum of the digits of all two-digit multiples of 9 is 9:

9 x 2 = 18 , 1 + 8 = 9
9 x 3 = 27 , 2 + 7 = 9
9 x 4 = 36 , 3 + 6 = 9
9 x 5 = 45 , 4 + 5 = 9
9 x 6 = 54 , 5 + 4 = 9
9 x 7 = 63 , 6 + 3 = 9
9 x 8 = 72 , 7 + 2 = 9
9 x 9 = 81 , 8 + 1 = 9

remember? pretty cool, right? I’m a bit of a math nerd, so stuff like this has always really excited me. anyway, welcome to the blog. welcome to 4/1/2011. and welcome to the number 9. read more about it next!

Logarithmic Spirals of the Nautilus Shell.

The spiral is a common element of Sacred Geometry as well as to all natural development. Spirals in nature tend to follow the Golden Ratio (Phi) or Fibonacci Sequence in their rates of expansion. The key to Sacred Geometry is the relationship between the progression of growth and proportion. Harmonic proportion and progression are the essence of the created universe and is consistent with nature around us. The natural progression follows a series that is popularized in the West as the “Fibonacci Series” where the first two numbers in the series are added to create the third number for a series of number that begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…and goes on ad infinitum. The ratio of the numbers gains great importance as the series continues. By dividing one number by the previous number, the answers result in or come closer to phi: 3/5 = 1.6666, 13/8 = 1.6250, 233/144 = 1.6180. These numbers can be demonstrated with the spiral of the Nautilus. 

The Golden Ratio (phi) is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. The ratio links each chamber of the nautilus to the new growth and symbolically, each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage. This geometry of the Nautilus can be found in the spiral patterns of cauliflower, the placement of the leaves on most plants, the arrangement of pattern on a pine cone. The ratios can be retrieved from the shape of our DNA and the measurement of distant galaxies as the Sacred Geometry demonstrates the blueprint of the sacred foundation of all things and the interconnectedness of all the various parts of the whole. 

The Signs as Types of Witches

Disclaimer: All information on types of witches came from http://www.wildwitchacademy.com/different-types-of-witchcraft.html and http://www.spiritwalkministry.com/traditional_witchcraft/types_of_witchcraft . I apologize if any of this seems incorrect to practicing witches.

Augury Witch: (Virgo, Pisces) Augury witches help provide direction and guidance on spiritual quests by interpreting signs and/or symbols along the way. They have the gift of prophesy, not divination, and read omens during travel.

Ceremonial Witch: (Capricorn, Taurus, Leo) Ceremonial witches combine witchcraft and ceremonial magic. They also combine tradition disciplines with scientific principles and sacred mathematics, and may call a variety of spiritual entities to them that represent archetypal manifestations of desired energies.

Eclectic Witch: (Gemini, Sagittarius, Aries, Aquarius) Eclectic witches pick and choose traditions, religions, and practices to create their own personalized form of witchcraft. They base their systems off what they are capable of and what their intents are, and study many different paths.

Green Witch: (Taurus, Libra) Green witches focus mostly on the natural world, such as natural items and places. They seek to achieve magic through communion and closeness with Mother Nature.

Hedge Witch: (Virgo, Aquarius) Hedge witches focus on an earth-based spirituality, and represent spirit flight the boundary between our physical realm and the spiritual realm. They make powerful midwives and healers.

Hereditary Witch: (Capricorn, Leo, Cancer) Hereditary witches are born into the traditions that their family practices, and are taught the older ways of doing things. They grow up learning things the ways their ancestors did.

Kitchen/Cottage Witch: (Libra, Cancer) Kitchen, or Cottage, witches use the tools they have close at hand in order to make their homes more sacred and secure. They are concerned with the practical side of religion, magic, and the elements, and they look for sacred moments/objects in everyday life.

Solitary Witch: (Scorpio, Aries, Gemini) Solitary witches practice alone and without tradition. They are said to have developed skills in previous lifetimes and they will practice almost any magical system but are among natural witches.

Wicca: (Sagittarius, Libra) Wiccans are highly religious in practice, but maintain a good balance between ceremonial/religious rituals and nature-based rituals. They typically believe in a equal pair of God and Goddess, or a ruling Goddess with a God taking the role of assistant.

Wild Witch: (Pisces, Scorpio) Wild witches use natural tools for divination and do not believe in harming anyone or anything, and their knowledge often comes from family. They often live away from people and share their home with wildlife of all kinds.

Composite Dimension:
Bringing It All Together - XXXV
Dimensions of Potential - II




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The mandalic cube consists of four distinct orbitals or shells, numbered from innermost (Orbital 0) to outermost (Orbital 3).[1]  This numbering of shells is based on the  radius[2] of the sphere,  the surface of which contains all hexagrams (points) resident in a particular orbital. For Orbital 0, the radius equals zero (0). For Orbital 1, the radius is 1 or square root of 1. For Orbital 2, the radius is square root of 2. For Orbital 3, the radius is square root of 3. The radius of Orbital 1 is a given[3]  and the values of the radii of Orbitals 2 and 3 are easily derived by using the Pythagorean theorem.  Although some of these orbital distances involve irrational numbers,  the shortest route between any adjacent discretized points of adjacent orbitals is uniformly equal to one (1) as the crow flies, which is to say, following a rectilinear path in three-dimensional terms.

The outermost hexagram shell (Orbital 3) holds eight hexagrams. These are the eight hexagrams having doubled trigrams that reside at the vertices of the mandalic cube. The hexagram containing six yang Lines is referred to as HEAVEN, and the hexagram with six yin Lines, as EARTH, just as are the trigrams which compose them. All hexagrams composed of doubled trigrams share the same name as their component trigrams.  The daughter trigrams, with one yin Line, form hexagrams with two yin Lines and  four yang Lines,  while the son trigrams,  each having two yin Lines, form hexagrams with four yin Lines and two yang Lines in the outermost orbital with doubling of their Lines.

So all eight hexagrams in this group occupy a single orbital but they are distributed  among four different degrees of electric charge or balance of negative and positive,  yin and yang.  Both orbital number and electric charge  can be understood  in terms  similar to quantum states of particle physics. The number eight determines the full complement of hexagrams which comprise a complete outer shell,  as it does for valence electrons of chemical elements in another, very different, context.  Here,  the number eight arises naturally, as related to the number of octants of 3D Cartesian coordinates and 6D/3D hybrid mandalic coordinates. In many seemingly unrelated contexts the number eight presents, representing wholeness or a plateau of completeness.  This is intimately related to group theory and the nature of symmetry. Also, of course, to sacred geometry.

After the following short detour to assuage the rational mind we will consider the hexagrams in Orbital 2, which is somewhat more complex in composition than is Orbital 3 just discussed.

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Notes

[1] Previously, in Hexagramium Organum before the split off of this blog, the orbitals were numbered in the reverse direction,  from 1 to 4 with 4 at the center. I think the numbering schema presented here makes much more sense because it better relates the cubic and spherical presentations of the hexagrams and clarifies the relationship of orbital radii to the Pythagorean theorem.  In short,  this seems to be a more natural numbering system. A salutary incidental consequence is that this brings the numbering of the mandalic orbitals into alignment with that of the atomic orbitals (1,2) - all, that is, except for Orbital 0 which physics forbade. To be fully consistent,  physics should also forbid spontaneous pair production  and virtual particles.  To do that, though,  would be tantamount to denial of quantum mechanical uncertainty and quantum fluctuation. (And there goes the subatomic neighborhood!)

[2] That is to say, the radius in three-dimensional terms.

[3] The radius of Orbital 1 is determined by the length of the side of the cube (=2), which is integral to the system of mandalic geometry and which is also the diameter of Orbital 1. (Since r = d ÷ 2 , r = 1)

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

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