Quantum Naughts and Crosses 11

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(continued from here)

In this module we will engage in some play of the imagination with the xz-planes. In the post to follow an xz-plane with all of its resident hexagrams will be presented. (1) Note first that the tetragrams shown above are composed from lines 1, 3, 4 and 6 of their hexagrams. Lines 2 and 5 which determine the vertical y-axis are constant throughout each of the two xz-planes (yang for y = +1; yin for y = -1) and are suppressed here to demonstrate the tetragrams more clearly. The * symbol is used here as a placeholder for a Cartesian coordinate, in this particular case the y-coordinate value.

There are 16 possible tetragrams (2^4 = 16). All appear in each of the six faces of the mandalic cube but are formed from from different lines of the hexagrams and occur at different locations in their respective planes except for complementary faces in which the formation and placement are identical in both planes. That is why tetragrams of complementary faces can be shown with a single diagram using placeholders for the two lines that do differ in corresponding hexagrams of the two planes.

If in the vertex hexagrams we were to suppress lines 4 and 6 also, replacing the placeholders at line 3 by yang lines, we would then see the resident trigrams of the xz-plane having y = +1. If instead we replace that same placeholder with yin lines we would see the resident trigrams of the xz-plane having y = -1. These are the trigrams that interact to compose the hexagrams of the xz-planes, four different trigram types in each of the xz-planes. The trigrams complementary to the four in one of the xz-planes are of course found in the other.

If we exchange yang lines for the placeholders in lines 2 and 5 above we arrive at the hexagrams of the xz-plane having y = +1. If instead we exchange yin lines for these same two placeholders we get the hexagrams of the xz-plane having y = -1. This is just like Cartesian coordinates but with an important twist, namely the Cartesian forms are derived from the more inclusive (2) mandalic forms by compositing. (3) From the higher perspective of mandalic geometry Cartesian dimensions are composite dimensions actualized from mandalic dimensions of potentiality.

There is clearly a fixed unpremeditated but necessary order (4) to placement of the hexagrams and all their components (i.e., lines, bigrams, trigrams, tetragrams, pentagrams.) Still, the holistic interrelated dynamic contextual placement of hexagrams allows for a considerable number of different spatiotemporal changes or processions. The large number of degrees of freedom present at all steps (quanta) along the way is quite naturally related to the perceived arrow of time. More about this will be said in future posts. (5)

At this point it would be good to recall that a hexagram represents a specific manner of intersection of six different dimensions of potentiality. It is like an evanescent will-o’-the-wisp, having only a fleeting existence but recurring persistently in spatiotemporal terms. A hexagram is in essence a snapshot of frozen potential spacetime. Its existence is not a fundamental but a derivative one, changing as its component parts coalesce, dissociate and commingle once again.

(continued here)

 

 (1) The xz-planes in the context of the mandalic cube are the two planes directly below when the cube is viewed from above. In order to preserve Cartesian coordinate convention the plane with y = +1 (yang) is to be viewed from outside the cube whereas the plane with y = -1 (yin) is best viewed from a conceptual vantage point within the cube. Of course the conceptually adept can view this plane from above and outside the cube as well but for the rest of us the intervening xz-plane with y = +1 may interfere with adequate visualization, even in dealing with a non-solid transparent cube as we are, at least initially until one grows accustomed to doing so.

(2) More inclusive in terms of both information content and number of spatiotemporal possibilities or degrees of freedom permitted.

(3) The term “compositing” as used here refers to the mechanism by which a composite Cartesian (or Cartesian-like) dimension (dimension of actuality) is derived from two or more higher mandalic dimensions (here referred to as “dimensions of potentiality”.) Take note also of the related more general usages of the term as in graphics and video terminology

(4) The term “order” here does not refer to any particular linear sequence or arrangement of succession of hexagrams as none such exists in the multidimensional probabilistic context of mandalic geometry. Rather it refers to a condition of methodical arrangement among component parts such that proper functioning is achieved by both parts and whole in a synchronous manner.

(5) For the time being suffice it to say that our linear notion of time is as faulty as are our linear notions of space and number.

© 2014 Martin Hauser

              Leibniz on Necessary Being
   
What, asks Leibniz, sufficiently explains a contingent being? “Contingent” here simply means something that could have been other than it is. The entire composition of the universe, the total aggregate of substances throughout space and time, are all contingent. There are other possible things, he reasons, but there are also other possible universes that could have existed but did not. All the many contingent things in the universe, he concludes, do not sufficiently explain themselves. There must be, Leibniz insists, something beyond all the contingent things which explains them, something which is itself necessary and therefore requires no explanation other than itself.
God, Leibniz concludes, is the necessary being which provides the sufficient explanation of all contingent things–why the universe is the way it is rather than otherwise. To this point in Leibniz’ reasoning, God’s necessity is the only aspect proffered (there is little religious or theological signification in this initially unembellished metaphysical concept). God as a being may be necessary, but if the contingent universe were a mere random or arbitrary act of God, then God would not comprise the required explanation of all things.  
God must not only be necessary but also the fount of the intelligibility of all things. It must be possible, therefore, to inquire into the reasons God had for creating this universe that actually exists rather than some other. And if we are to explain the lucidity of the universe by God, then God must have such comprehension that God could be said to know what it is that is being allowed to exist–that is, God must be able to grasp whole concepts and to see at once the total demonstration of things. God so far is therefore (1) a necessary being, (2) the explanation of the universe, and (3) the infinite intelligence.
[paraphrased from www.iep.utm.edu/leib-met/#H5]

              Leibniz on Necessary Being

 

What, asks Leibniz, sufficiently explains a contingent being? “Contingent” here simply means something that could have been other than it is. The entire composition of the universe, the total aggregate of substances throughout space and time, are all contingent. There are other possible things, he reasons, but there are also other possible universes that could have existed but did not. All the many contingent things in the universe, he concludes, do not sufficiently explain themselves. There must be, Leibniz insists, something beyond all the contingent things which explains them, something which is itself necessary and therefore requires no explanation other than itself.

God, Leibniz concludes, is the necessary being which provides the sufficient explanation of all contingent things–why the universe is the way it is rather than otherwise. To this point in Leibniz’ reasoning, God’s necessity is the only aspect proffered (there is little religious or theological signification in this initially unembellished metaphysical concept). God as a being may be necessary, but if the contingent universe were a mere random or arbitrary act of God, then God would not comprise the required explanation of all things.  

God must not only be necessary but also the fount of the intelligibility of all things. It must be possible, therefore, to inquire into the reasons God had for creating this universe that actually exists rather than some other. And if we are to explain the lucidity of the universe by God, then God must have such comprehension that God could be said to know what it is that is being allowed to exist–that is, God must be able to grasp whole concepts and to see at once the total demonstration of things. God so far is therefore (1) a necessary being, (2) the explanation of the universe, and (3) the infinite intelligence.

[paraphrased from www.iep.utm.edu/leib-met/#H5]

Six-Dimensional Boolean Lattice of Hexagrams
 
It is my intention in this blog to demonstrate that inevitability in the structure of the universe such as described by Brian Greene in the preceding post is factual, that it is based on the essential necessary geometry of space, time, matter, and energy, and that it can be effectively described by a simple algebraic geometry with relatively few elementary rules of symbol manipulation using a very old form of notation in a new way.
The image above [click image to enlarge] demonstrates the 6-dimensional combinatorial analysis of the 64 I Ching hexagrams in terms of Boolean algebra. Our goal here will be to demonstrate these same relationships in simple geometric form and show how doing so results in a mapping of the elementary particles (or strings) and the four forces of nature (strong, weak, electro-magnetism, and gravity.) The structure arrived at is internally self-consistent and unique (in the sense that no alteration is possible without destroying its efficacy.) Additionally, this mapping could result in a revolutionary new model of the atom, one consistent with the current findings of particle physics and superstring theory but in some ways going beyond both.
Note - A two dimensional projection of the Boolean lattice is shown here. However, the structure itself should be understood as a 6 dimensional hypercube.
Click image above to enlarge.
Image credit:  Dr Andreas Schöter         [http://www.yijing.co.uk/downloads/LoA.pdf]
© 2012 Martin Hauser

Six-Dimensional Boolean Lattice of Hexagrams

 

It is my intention in this blog to demonstrate that inevitability in the structure of the universe such as described by Brian Greene in the preceding post is factual, that it is based on the essential necessary geometry of space, time, matter, and energy, and that it can be effectively described by a simple algebraic geometry with relatively few elementary rules of symbol manipulation using a very old form of notation in a new way.

The image above [click image to enlarge] demonstrates the 6-dimensional combinatorial analysis of the 64 I Ching hexagrams in terms of Boolean algebra. Our goal here will be to demonstrate these same relationships in simple geometric form and show how doing so results in a mapping of the elementary particles (or strings) and the four forces of nature (strong, weak, electro-magnetism, and gravity.) The structure arrived at is internally self-consistent and unique (in the sense that no alteration is possible without destroying its efficacy.) Additionally, this mapping could result in a revolutionary new model of the atom, one consistent with the current findings of particle physics and superstring theory but in some ways going beyond both.

NoteA two dimensional projection of the Boolean lattice is shown here. However, the structure itself should be understood as a 6 dimensional hypercube.

Click image above to enlarge.

Image credit:  Dr Andreas Schöter         [http://www.yijing.co.uk/downloads/LoA.pdf]

© 2012 Martin Hauser