A new quantum logic: Symmetry and asymmetry - I
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Physics considers symmetries quite important as they often carry in concealment laws of nature. Asymmetries are important too. These are at times broken symmetries which physics also makes much of and which can instruct us in important ways as well. We will look at what appears to be one of these broken symmetries and attempt its demystification. We will be describing what must surely be one of the most basic asymmetries of dimensional numbers, one which involves the most fundamental polarity of nature. Could this be one of the earliest broken symmetries? And could dimensional numbers be the fount of symmetry, asymmetry and broken symmetry alike?(1)
The way mandalic geometry views symmetry differs slightly from the way physics does. This must be explained before we attempt the broken symmetry demystification mentioned above. To accomplish this we will need to discuss a number of other topics, including that of the rotation group of the cube, composite dimension, isotropic and anisotropic space, variant perspective, and invariance under Planck scale interchangeability. We have a long road ahead of us before we reach the fabled Fortress of Broken Symmetry and attempt to penetrate its formidable encasement.
(1) Within mathematics symmetry in geometry is very closely linked to group theory. Mandalic geometry at its current stage of development is concerned principally with the symmetries and asymmetries inherent in the cube, sphere, and their extrapolations to six dimensional space, or rather to the composited 3D/6D space of the mandalic cube and mandalic sphere. Wikipedia has an excellent introductory article on groups in mathematics which can be found here. An equally excellent article dealing with both symmetry and symmetry breaking appears in The Stanford Encyclopedia of Philosophy. For a discussion of the relation between groups and graphs I think you can do no better than this original tumblr post.
© 2014 Martin Hauser
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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.
© 2013 Martin Hauser
Mandalic geometry and polar coordinates - IV
(Continued from here.)
LEVELS 1 and 5
9-Six hexagrams reside at Level 1 and six at Level 5. These occur in groups of two at each of three points in Level 1 and in groups of two at each of three points in Level 5. The hexagrams in Level 1 all contain a single positive (yang) line and the hexagrams in Level 5 all contain a single negative (yin) line. In both Level 1 and Level 5 the three resident groups are situated 120° from one another. The three groups of Level 5, however, are phased 60° from those of Level 1. All these points and hexagrams are found in Shell 2 of the spherical mandala. In the Cartesian form of the mandala these points and hexagrams are the cube edge centers closest to the maximum yin and maximum yang hexagrams or vertices. There are six other points, each with two resident hexagrams, in Shell 2 but these will all soon be seen to lie at Level 3.
10-Connecting the points of the three groups in Level 1 or Level 5 results in an equilateral triangle. If one were to look down (or up) at these two triangles superimposed upon one another one would see a 6-pointed star. This of course hearkens back to a lot of intellectual history through the ages having historical, religious and cultural contexts. The 6-pointed star, for example, has significance to sacred geometry as it is the compound of two equilateral triangles, the intersection of which is a regular hexagon.
© 2013 Martin Hauser
Concerned only with the ethereal world of abstraction, mathematicians are free to introduce this notion of a group and to require that anything that is a group satisfy these criteria. These axioms are neither right nor wrong; they are simply the rules that mathematicians have chosen to require of something that they have decided for some reason or other to call a “group.”
The wonder of group theory is that its relevance to the disciplines of both mathematics and natural science far exceeds the self-contained boundaries within which it was first developed.—