A new quantum logic: Symmetry and asymmetry - I

 

Isfahan Lotfollah mosque ceiling symmetric

(continued from here)

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.

(continued here)

Image: The ceiling of the Sheikh-Lotf-Allah mosque in Isfahan, Iran. By Phillip Maiwald (Nikopol) (Own work) [GFDL or CC-BY-SA-3.0], via Wikimedia Commons

 

 (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

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. :)

Mandalic geometry and polar coordinates - V

(Continued from here.)

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.

                              image

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© 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.

       image

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.

 

                               image

Of particular note here, however, is the fact that imathematics, the root system for the simple Lie group G2 is in the form of a hexagram [6-pointed star].

                             

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© 2013 Martin Hauser

An introduction to geometric group theory
   
This PDF may be helpful to some who want to delve more deeply into the subject of geometric group theory. I caution you though, the material presented here is not for the faint of heart, among whom I number myself without hesitation.
From the author’s Preface:

These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as several staff members. I therefore tried to present a logically coherent introduction to the subject, tailored to the background of the students, as well as including a number of diversions into more sophisticated applications of these ideas. There are many statements left as exercises. I believe that those essential to the logical developments will be fairly routine. Those related to examples or diversions may be more challenging.
The notes assume a basic knowledge of group theory, and metric and topological spaces. We describe some of the fundamental notions of geometric group theory, such as quasi-isometries, and aim for a basic overview of hyperbolic groups. We describe group presentations from first principles. We give an outline description of fundamental groups and covering spaces, sufficient to allow us to illustrate various results with more explicit examples. We also give a crash course on hyperbolic geometry. Again the presentation is rather informal, and aimed at providing a source of examples of hyperbolic groups. This is not logically essential to most of what follows. In principle, the basic theory of hyperbolic groups can be developed with no reference to hyperbolic geometry, but interesting examples would be rather sparse.
[www.math.ucdavis.edu/~kapovich/280-2009/bhb-ggtcourse.pdf]

An introduction to geometric group theory

 

This PDF may be helpful to some who want to delve more deeply into the subject of geometric group theory. I caution you though, the material presented here is not for the faint of heart, among whom I number myself without hesitation.

From the author’s Preface:

These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as several staff members. I therefore tried to present a logically coherent introduction to the subject, tailored to the background of the students, as well as including a number of diversions into more sophisticated applications of these ideas. There are many statements left as exercises. I believe that those essential to the logical developments will be fairly routine. Those related to examples or diversions may be more challenging.

The notes assume a basic knowledge of group theory, and metric and topological spaces. We describe some of the fundamental notions of geometric group theory, such as quasi-isometries, and aim for a basic overview of hyperbolic groups. We describe group presentations from first principles. We give an outline description of fundamental groups and covering spaces, sufficient to allow us to illustrate various results with more explicit examples. We also give a crash course on hyperbolic geometry. Again the presentation is rather informal, and aimed at providing a source of examples of hyperbolic groups. This is not logically essential to most of what follows. In principle, the basic theory of hyperbolic groups can be developed with no reference to hyperbolic geometry, but interesting examples would be rather sparse.

[www.math.ucdavis.edu/~kapovich/280-2009/bhb-ggtcourse.pdf]

Lie Groups and Particle Physics
   
Lie groups are involved in the description of a large number of physical phenomena and see broad application throughout the sciences. In particle physics of the present day, Lie groups play a primary role. But it wasn’t until Murray Gel-Mann’s proposal in 1961 of his Eightfold Way, an application of Lie groups to the categorization of hadrons into octets (alluding to the Noble Eightfold Path of Buddhism), that physicists realized the importance of group theory to particle physics. 

The Eightfold Way may be understood in modern terms as a consequence of flavor symmetries between various kinds of quarks. Since the strong nuclear force affects quarks the same way regardless of their flavor, replacing one flavor of quark with another in a hadron should not alter its mass* very much. Mathematically, this replacement may be described by elements of the SU(3) group. The octets and other arrangements are representations of this group. For a more detailed explanation of this fact, see particle physics and representation theory.
[en.wikipedia.org/wiki/Eightfold_Way_(physics)]

Although Gel-Mann humorously labeled his theory ‘Eightfold Way’ as a hat tip of sorts to the Noble Eightfold Path of Buddhism he never thought the two actually related in any true manner and was reportedly somewhat sensitive to remarks suggesting any such relationship. Ironically, both his Eightfold Way and the structure of the I Ching (not a Buddhist work but related to Taoism) are built upon the mathematical edifice of group theory and are therefore intimately related. The I Ching, of course, predates the development of modern group theory, as well as the Eightfold Way, by a few millennia.
   
*Although the quark flavors of the three generations do differ considerably in mass these differences prove negligible in the hadrons which are much heavier than quarks and derive most of their mass from the kinetic energy of their component quarks rather than from the rest masses of the quarks.

Lie Groups and Particle Physics

 

Lie groups are involved in the description of a large number of physical phenomena and see broad application throughout the sciences. In particle physics of the present day, Lie groups play a primary role. But it wasn’t until Murray Gel-Mann’s proposal in 1961 of his Eightfold Way, an application of Lie groups to the categorization of hadrons into octets (alluding to the Noble Eightfold Path of Buddhism), that physicists realized the importance of group theory to particle physics. 

The Eightfold Way may be understood in modern terms as a consequence of flavor symmetries between various kinds of quarks. Since the strong nuclear force affects quarks the same way regardless of their flavor, replacing one flavor of quark with another in a hadron should not alter its mass* very much. Mathematically, this replacement may be described by elements of the SU(3) group. The octets and other arrangements are representations of this group. For a more detailed explanation of this fact, see particle physics and representation theory.

[en.wikipedia.org/wiki/Eightfold_Way_(physics)]

Although Gel-Mann humorously labeled his theory ‘Eightfold Way’ as a hat tip of sorts to the Noble Eightfold Path of Buddhism he never thought the two actually related in any true manner and was reportedly somewhat sensitive to remarks suggesting any such relationship. Ironically, both his Eightfold Way and the structure of the I Ching (not a Buddhist work but related to Taoism) are built upon the mathematical edifice of group theory and are therefore intimately related. The I Ching, of course, predates the development of modern group theory, as well as the Eightfold Way, by a few millennia.

 

*Although the quark flavors of the three generations do differ considerably in mass these differences prove negligible in the hadrons which are much heavier than quarks and derive most of their mass from the kinetic energy of their component quarks rather than from the rest masses of the quarks.

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.

Bruce A. Schumm (2004). Deep Down Things: The Breathtaking Beauty of Particle Physics.  Johns Hopkins University Press. pp. 143–144. ISBN 0-8018-7971-XOCLC 55229065.

Group Theory - Axioms
   
Not every set with an associated operation on its elements is a group. To form a group the set and operation (or rule of combination) must satisfy the following four axioms:
1-Closure
The combination of any two elements (according to the rule of combination of the set) must itself be an element in the set.
2 -Associativity

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider, for instance, the following equations:

Consider the first equation. Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the value of the expression was not altered. Since this holds true when performing addition on any real numbers, we say that “addition of real numbers is an associative operation.”
Associativity is not to be confused with commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not.
en.wikipedia.org/wiki/Associativity]

3-Identity
The set must contain an element which, when combined (according to the operation of the set) with any other element, results in that second element. For example, for the set of whole numbers under the operation of multiplication the identity element is the number 1 since any number multiplied by 1 returns the same number.
4 - Invertibility
For each element in the set there must be one and only one element also in the set which, when combined (according to the operation of the set), results in the identity element.

In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can ‘undo’ the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.
[en.wikipedia.org/wiki/Inverse_element]

Group Theory - Axioms

 

Not every set with an associated operation on its elements is a group. To form a group the set and operation (or rule of combination) must satisfy the following four axioms:

1-Closure

The combination of any two elements (according to the rule of combination of the set) must itself be an element in the set.

2 -Associativity

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider, for instance, the following equations:

(5+2)+1=5+(2+1)=8 \,5\times(5\times3)=(5\times5)\times3=75 \,

Consider the first equation. Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the value of the expression was not altered. Since this holds true when performing addition on any real numbers, we say that “addition of real numbers is an associative operation.”

Associativity is not to be confused with commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not.

en.wikipedia.org/wiki/Associativity]

3-Identity

The set must contain an element which, when combined (according to the operation of the set) with any other element, results in that second element. For example, for the set of whole numbers under the operation of multiplication the identity element is the number 1 since any number multiplied by 1 returns the same number.

4 - Invertibility

For each element in the set there must be one and only one element also in the set which, when combined (according to the operation of the set), results in the identity element.

In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can ‘undo’ the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.

[en.wikipedia.org/wiki/Inverse_element]

Group Theory - Introduction
   

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy four conditions called the group axioms, namely closure, associativity, identity and invertibility. 
Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.
The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.
[en.wikipedia.org/wiki/Group_(mathematics)]

Group Theory - Introduction

 

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy four conditions called the group axioms, namely closure, associativityidentity and invertibility.

Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

[en.wikipedia.org/wiki/Group_(mathematics)]