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Tetryonics ~~ coming soon to your field of Consciousness ~~ K-EL ABRAHAM
If modern science cannot explain the very basics concepts of physics what hope is there that the most advanced explanations of the mechanics of our Universe is right?
+/- CHARGE the foundation of all mass, Matter and Force in our Universe..
Tetryonics
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Quantum Geometry: Causal Dynamical Triangulations
Mandalic geometry and polar coordinates - VIII
(Continued from here.)
LEVEL 3 (cont.)
12-Subgroups of Level 3 (cont.)
B-Concluding our survey of the spherical mandala we come to the second subgroup of hexagrams of Level 3, the level of charge density neutrality. These 12 hexagrams in groupings of two occupy six points in Shell 2 (heading from outer to inner) as do (together) the hexagrams in Levels 1 and 5 which have one solid (yang) line and 5 solid (yang) lines respectively. Along with the 8 hexagrams of subgroup A found in Shell 4 at the center of the spherical mandala these 12 hexagrams form the entire Level 3. Unlike those eight hexagrams discussed in the previous post, these twelve are each constructed from two trigrams which are not complements of one another while still retaining charge density neutrality. In some ways these six groups are the most mysterious of all the hexagram groups.
So to clarify, we are talking here about six equidistant points about the equatorial great circle of Shell 2 (from outer to inner) of the spherical mandala. That means the six groups are separated one from the next by 60°. We can anticipate that these consist of two sub-subgroups of points and hexagrams separated by 120° forming two equilateral triangles which are out of phase by 60°, creating still another 6-pointed star. Mathematics is so remarkably consistent. More to the point though, - - - because (and we are getting ahead of ourselves here) we are talking (or will be at some future time) not just about geometric points but about particles, force fields and strings as well - - - all these triangles, angles and 6-pointed stars give rise to various symmetries which in turn give rise to physics as it is.
LEVEL 3, GROUP B HEXAGRAMS
Of particular note here is the fact that the two hexagrams in row 1 and the two in row 4 above are unique in that the inverse of each of these hexagrams is identical to its complement. That means in some sense the anti-hexagram for each of these hexagrams dwells close within. There are only four other hexagrams out of the 64 that share this property. Those four also reside at Level 3 of the spherical mandala but are among the eight at the center point (Shell 4) in Group A. This attribute will come into play again in future posts when we address the subject of hexagrams as particle analogues and intersecting strings.
© 2013 Martin Hauser
Mandalic geometry and polar coordinates - VII
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LEVEL 3
12-That brings us to Level 3 of the spherical mandala, the final level to be discussed. It is in many ways the most interesting level and certainly the most complex. (Oh, but to be fair all the levels are intertwined and function in concert as a whole entity. The same may be said of the four shells and of all the hexagrams and hexagram groups.) This is the level of neutrality in terms of charge density. As in Levels 2 and 4 there are two subgroups. Complicating matters more it seems that each subgroup is made up of at least two sub-subgroups.
Okay, I confess, I'm still learning here myself. I'm recording all this information before I fully understand it so it won't be lost. As far as I know this body of knowledge doesn't exist anywhere else in this form - - - relating mathematics, physics, philosophy and ancient Chinese thought. When you see where we're going with this you'll understand why I believe it to be so important. My aim here is not so much to complete the work as to preserve it and give to specialists in the stated fields (mathematics, physics, philosophy, and sinology) a different frame of reference from the ones they have grown accustomed to. Hopefully it will be a useful and fertile one.
Subgroup A:
This consists of eight hexagrams which all share as a group the single point at the center of the spherical mandala (Shell 4). In the Cartesian form of the mandala they share the single point at the center of the cube, the origin in terms of Cartesian coordinates (0,0,0). As previously noted, this is the point of totipotency (in spite of its 3-dimensional labeling as zero.) Occuping a single point these eight hexagrams are necessarily confluent. Yet in some real sense they are also separated, one from another, by 90° through three dimensions. Ah, paradox - - - seems we can't escape it. Yes, I know the two statements together don't sound like they make a whole lot of sense. But it's that 3-dimension/6-dimension thing. In terms of three dimensions the 8 hexagrams occupy a single point, but in terms of six dimensions they occupy 8 different points which are interrelated in a very special way to one another and to all other hexagrams and groups of hexagrams in the mandala. [Think of the 3-dimensional point as existing in a fractal of sorts. At another scale - - - the scale of six dimensions - - - it presents as a dynamic ever-changing 8-vertex cube, with vertices arranged in two groups of four - - - two polestar vertices, each with three closely associated vertices which form an equilateral triangle and with the two triangles out of phase by 60°, resulting in yet another 6-pointed star.]
Although all 20 hexagrams in Level 3 have equal numbers of broken (yin) and solid (yang) lines, only these eight are composed of upper and lower trigrams which are complementary to one another. The other twelve are found in Shell 2 (heading outer to inner) and we will turn our attention to those in the next post.
If we were speaking here about matter and antimatter as currently widely understood (or misunderstood) we would perhaps expect only annihilation. According to the world view of the I Ching as well as that of modern physics, however, complements can create a world as well as destroy it. So this is the point of becoming as well as the point of unbecoming. What transpires depends upon the specific configuration empowered. For our purposes we can think of this as the vacuum point of modern physics where virtual particle pairs come into and pass out of existence. Or if you like, think of it as the still highly conjectural combined black hole/white hole of mathematics.
The pairings in the two groups above give unique insight into the very different worldview of the I Ching. For example WATER over FIRE (third from left in bottom row) is named AFTER COMPLETION. It pictures a time of perfect order. In the I Ching this is the penultimate hexagram (Hexagram 63). FIRE over WATER (third from left in upper row) is named BEFORE COMPLETION. It pictures a time of chaos when everything is out of order. In the I Ching, however, this is the final hexagram (Hexagram 64). The thinking here is that our world is a world of change and alternation. Order having reached its pinnacle leads inevitably to disorder, and disorder at its nadir turns back to a time of order. So the I Ching [the linear I Ching as handed down to us from antiquity] ends actually on a note of promise and hope.
(Continued here.)
© 2013 Martin Hauser
Mandalic geometry and polar coordinates - VI
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LEVELS 2 and 4 (cont.)
11-Subgroups of Levels 2 and 4
B-The second subgroup of Levels 2 and 4 contains hexagrams having 4 broken (yin) lines and 2 broken (yin) lines respectively, similarly to those in Group A but these hexagrams unlike those others are found in Shell 3 (counting outer to inner) rather than Shell 1. At both these levels there are three groups of four hexagrams, totaling twelve hexagrams at each level. The six points at which the groups are found correspond to the six cube face centers of the Cartesian form of the mandala. Note again that 4 hexagrams are found at each of the six points for a total of 24 hexagrams.
Once again the three groups at each of these two levels form an equilateral triangle with the groups at each level separated by 120° from one another. The two equilateral triangles so formed are again out of phase with one another by 60° and when superimposed form a third 6-pointed star.
LEVEL 4:
To conserve space and hopefully reduce confusion let's just say you can generate the B Groups and hexagrams of Level 4 by substituting solid lines for broken and broken for solid in all hexagrams seen above.
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© 2013 Martin Hauser
Mandalic geometry and polar coordinates - V
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LEVELS 2 and 4
11-Levels 2 and 4 of the spherical mandala can be divided into two distinct subgroups:
A-The first subgroup consists of the remaining six hexagrams of the outer shell (Shell 1). Three of these, each having 4 broken (yin) lines, occur at Level 2; three, each having 2 broken (yin) lines, at Level 4. At both levels the three hexagrams in this subgroup are 120° from one another, forming an equilateral triangle. These two triangles are out of phase with one another by 60°. Therefore when the two triangles are superimposed upon one another they form another 6-pointed star. These hexagrams are all cube vertices in the Cartesian form of the mandala. Together with the two polestar hexagrams they constitute all eight vertices of the Cartesian form of the mandala and the entire outer shell (Shell 1) of the spherical mandala.
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© 2013 Martin Hauser
Mandalic geometry and polar coordinates - III
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HORIZONTAL LEVELS
6-In addition to the four shells or energy levels the spherical mandala consists of seven horizontal levels (latitudes) of what can be considered charge density (or something akin to electromagnetism.) These will be referred to as Level 0 (bottom) through Level 6 (top). The progression from Level 0 (North pole) to Level 6 (South pole) is one from maximum negative (yin) to maximum positive (yang). Level 3 (equator), then, is the level of neutrality.
7-Level 0 corresponds to 90° North; Level 1 to 60° North; Level 2 to 30° North; Level 3 to 0°; Level 4 to 30° South; Level 5 to 60° South; Level 6 to 90° South.
LEVELS 0 and 6
8-A single hexagram resides in Level 0, the hexagram composed of 6 broken (yin) lines. Similarly, a single hexagram resides in Level 6, the hexagram composed of 6 solid (yang) lines. Both of these hexagrams occur in the outer shell (Shell 1) of the sphere, as the North pole and South pole respectively. Both are vertices in the Cartesian form of the mandala. There are six additional hexagrams in Shell 1, but as we will soon see, three of these reside in Level 2 and the other three in Level 4.
(Continued here.)
© 2013 Martin Hauser
Mandalic geometry and polar coordinates - I
THE SPHERICAL MANDALA
The I Ching mandala demonstrated in this blog has been derived using Cartesian coordinates. However the system works with the approach of polar coordinates in three dimensions as well. There are, of course, methods for translating from one to the other. In some ways the polar coordinate approach is more revealing. For one, it looks more like the kind of mandala most of us are familiar with (although in three dimensions.) For another, the various symmetry relationships are much more obvious.
A short derivation/description of the polar coordinate approach follows:
For reference see 1, 2, 3, 4, 5, 6, 7, 8, 9. These are all 2-dimensional representations necessitating mental compositing. Visualization becomes much easier with a 3-cube labeled appropriately with the 64 hexagrams. Rotate the cube so the hexagram with all solid lines is at the south pole* and the hexagram with all broken lines is at the north pole. These become the pole stars, so to speak, of the system.
Now imagine that the 64 hexagrams shown describe a single sphere consisting of four distinct shells (or alternatively describe four related concentric spheres.) We will refer to these as Shell 1, 2, 3 and 4 heading from outermost to innermost. The outermost shell is composed of the 8 hexagrams with doubled trigrams (1 at each of 8 positions) which were the 8 vertices of the cube seen in the Cartesian description. The next shell heading toward the center consists of the 24 hexagrams (2 at each of 12 positions) which were the cube edge centers in the Cartesian description. The third shell heading inward consists of the 24 hexagrams (4 at each of 6 positions) which were the cube face centers in the Cartesian description. The fourth shell is the single center point of the sphere viewed as a 3-dimensional structure. It consists of the 8 hexagrams which were at the central point or origin in the Cartesian description.
*Recall that in ancient Chinese cartography South was drawn where we show North and vice versa. This is purely convention, but an important one to the system described and we shall follow it here.
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© 2013 Martin Hauser
