**A new quantum logic: Inversion group of the cube**

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**This is an important topic which involves** the permutation group of the inversions of the vertices of the cube. This is not so much complicated as it is involved. Several posts will be aimed at introduction and development of the basics of the subject. This is fundamental material which is prerequisite for understanding the structure of the 6-dimensional mandalic cube and how its parts interact in the whole functionally.

**This subject is similar to the rotation group of the cube** but involves inversions or reflections through a point rather than rotations. The trigram vertices of the cube are multiples having split personalities. This is the case because, as has been stated many times before, the eight *points* which they specify and distinguish are fictions which result from intersections of various different dimensions. The inversions they engage in can therefore take place in a finite number of different ways, all of which must be taken into account in any consideration of the cube that aspires to a complete description and full understanding.**(1)**

**We have already met the list of characters.** The eight trigrams can be compared not so much to characters in a book, film or play as to the actors playing those characters. As sometimes occurs in theatrical works the actors here can play multiple roles. They must be conceived as having more than a single potential in terms of alternate character development, a requirement which issues from the fact that the *lines* and *planes* that intersect to form the unique vertex *points* do so functionally in such a manner that there are multiple ways the whole can be constituted from the parts and the parts from the whole.**(2)** We will look at all the inversion transforms of a cube that map it to itself.

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**Image:** Cubic crystal system, modified to demonstrate positioning of trigrams and Cartesian triads. Generalic, Eni. “Cubic crystal system.” *Croatian-English Chemistry Dictionary & Glossary*. 31 July 2014. KTF-Split.19 Aug. 2014. <http://glossary.periodni.com>.

** (1) This should call to mind the effects of measurement** on quantum systems. In the quantum world objects seem to exist natively in a state of totipotentiality. Until measured they exist in many places at the same time. Only when measured do they appear to us to have a definite location. The Copenhagen interpretation asserts that quantum mechanics rather than describing an objective reality deals only with probabilities of observing, or measuring, various aspects of energy quanta.

The act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible values. This feature of mathematics is known as wavefunction collapse. The essential concepts of the interpretation were devised by Niels Bohr, Werner Heisenberg and others in the years 1924–27. [Wikipedia]

**Mandalic geometry holds an alternative view.** It professes that the set of probabilities described by the Schrödinger equation and quantum mechanics does in fact refer to an objective reality, one having a logic of higher dimension and which coordinates space and time together though not in quite the manner Einstein envisaged. Within this higher logical patterning multiple outcomes are always possible and these we interpret as probabilities. Measurement does not collapse the wave function of the system but simply allows us to see some particular aspect of it uniquely as it happens to exist at a given moment in space and time. It asserts also that were we able to revisit that precise moment in space and time and reproduce all the conditions and the measurement made exactly as before, the result could well be other than what was found initially. Philosophically speaking, mandalic geometry adheres to a combined belief in both free will and determinism, but understood at a different dimensional level of geometrical reality than we generally interpret things.

**(2) **** This hints at the pluripotentiality** of subatomic particles which can take alternate forms and interact in various different manners depending upon context and with what specifically the interaction occurs. The same came be said about the chemical elements and other chemical entities in chemical reactions. Context is always of importance in determination of outcome. On the printed page this actuality is too often lost either in the confusion of breadth or in the overly narrow focus employed.

© 2014 Martin Hauser

**Please note -** This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a **TEXT** post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a **LINK** post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

**A new quantum logic: Back to square one**

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**A mystery now confronts us. **Have we uncovered an unexpected broken symmetry hidden in the multiplication operations of trigram patterns? In an attempt to answer this question we are going to use an indirect approach and revisit some basic themes. These themes include: the definition of multiplication; dimensional numbers and multiplication; magnitude, scalars and scaling; direction, vectors and sign; commutability; inversion and reflection; and the all-too-often-present conflation.**(1)** We can anticipate that it will take several posts at minimum to carry out this investigation.

**My suspicion is that the culprit here** is conflation and that what we have uncovered is not truly a broken symmetry but rather a clandestine error of conceptualization. The origin of our difficulties lies in the way we have been psychologically conditioned to view multiplication by that ubiquitous villain, conflation. Let’s see if we can unravel this labyrinthine deception.

**To begin we need to go back** to the fundamental polarity which defines two directions of a single dimension. *Positive* and *negative* lie at the root of our difficulties. Are we to view these two as asymmetric or symmetric? This is the basic question we must answer before addressing any of the other issues which grow out of this confusion. From a purely geometric perspective the two are symmetric. It is only when we attach names and numbers to the poles that they begin to appear asymmetric. This is a trick of algebra and language. It is, I think, a fallacy of mind.**(2)**

**And convention is undeniably an accessory to the fact.** We ourselves

*choose*names and other special distinguishing characteristics with which we label the directions and poles. Convention is humanly determined though nature has its own arbitrarily determined preferences as well.

**(3)**

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**Image: **Three classes of levers. By Pearson Scott Foresman [Public domain], via Wikimedia Commons

**(1) A seeming hodgepodge** but all significantly related to one another.

**(2) Even the ancients** had no difficulty balancing weights and objects about a fulcrum. They were masters of intuitive geometry and the building arts. Also one of the reasons the rationalists of the Age of Enlightenment had so much difficulty in accepting negative numbers may have been that their geometry seemed not require them despite the fact that their algebra clearly demanded them. Better the devil you know and all that.

**(3) Consider, for example,** its preference for the L-form over the D-form of amino acids in metabolism.

© 2014 Martin Hauser

**Please note -** This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a **TEXT** post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a **LINK** post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)