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

                                                                         (Continued here.)

© 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].

                             

                         image

                                                                        (Continued here.)

© 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)]

1-How can I access the initial page of the blog directly?
Congratulations on your perspicacity! You’ve detected the strangely counter-intuitive manner of page labeling of this blog which attempts (unsuccessfully, I might add) to reverse time. Whereas we commonly label years and the days, weeks, and months of the year in ascending chronological order, within the confines of this blog pages are numbered in reverse chronological order. So what is called ‘page 1’ is the current (most recent) page and its content is always changing over time. The initial page of the blog chronologically, though its content is unchanging, has a page label which changes (increases) over time. This was not by my own choice, but since I know so little HTML we are stuck with it. And would you expect less of a blog dealing with quantum mechanical weirdness anyway? 
The way around this is as follows: 
At the bottom of the page which greets you when you first access the blog you will see Page 1 of x where x is the then current total number of pages. To access the true (as opposed to the spurious) Page 1 directly, change the address in the address bar to blindmen6.tumblr.com/page/x and hit ENTER (or refresh the page). The electrons will take care of matters from there.
2-Whenever I attempt to use the Search function I get either no results or only a single page of results? Is something amiss here?
You bet your strange quarks there’s something amiss here. And I don’t know how to fix it. Frankly, the Search by tag function works better than the Search function. If your Search yields no results there likely are none. But if it gives you one page of results there are likely more. 
Here’s what to do:
In your Search results (assuming there are any) find a tag which matches or most closely matches your search term. Click on that tag and you will get all the posts that have been so tagged. I try my best to tag posts exhaustively but the terms I’m thinking in may not always match those you are thinking in. Even the electrons can’t help in that case.
3-You call your blog ‘QUANTUM GEOMETRY’ but I’ve seen almost no geometry and, for that matter, not all that much quantum. I have seen a lot of art, modern and otherwise. This is really an art blog in disguise, isn’t it?
In fact, no. The images attached to the verbal content of the posts are carefully selected for their relevance as well as their visual appeal. But they are basically hooks by which to grab your attention. Most every post is related to geometry and/or quantum mechanics in some manner though this may not initially be clear. If you continue to follow the blog, this will become increasingly clear over time. We are still (as of June 28, 2012) building a common foundation for future discussions. In time more posts will appear more directly related to geometry and the quantum world. The art stays though. If I can’t woo you with ideas, I’ll wow you with visual imagery. Also, art is a reminder that ideas at their inception are rarely greeted with open arms and mind.
4-What are your qualifications for discussing physics, mathematics, and Chinese philosophy?
Glad you asked. I do have a postgraduate degree but not one in physics, mathematics, philosophy, or sinology. My background is more in the fields of chemistry and biology. Both of those fields are intimately connected to the fields of physics and mathematics, less so to the fields of philosophy and sinology. Frankly, I don’t think it much matters. This is an interdisciplinary blog. What I lack in comprehension of the advanced mathematical equations employed by theoretical physicists I make up for by bringing a fresh new perspective to the field. My interest in Chinese philosophy, the philosophy of science, and philosophy in general dates back more than half a century. I’ve been involved with studies of the I Ching, particularly of its mathematical underpinnings, for nearly that long. Where I disagree with the prevailing views in physics and sinology I will plainly state so. But I believe anyone with a questioning mind has the right to use it as s(he) sees fit. Half a century ago ‘quark’ was a soft creamy acid-cured cheese of central Europe made from whole milk. It wasn’t until 1964 that Murray Gell-Mann proposed this name for his newly dreamed up particle which at that point he still regarded as just a convenient mathematical device to explain the then prevalent particle zoo of hadrons. His new particle and the ideas behind it were not initially universally well-accepted. The proof of the pudding is in the eating.
5-Isn’t the I Ching just a discredited guide to divination?
Don’t tell that to Carl Jung. (May he rest in peace.) The I Ching has held an exalted place in Chinese thought for over two millennia. It is one of the oldest Chinese classic texts. The book indeed contains a divination system and it is still widely used for this purpose. But it is also a historical document, a compendium of philosophical and ethical thought, a work relating linguistics to visual imagery which predates written language, and most important to our usage here, a treatise on mathematics, particularly combinatorics and group theory. One of the alternate names for the I Ching is The Book of Changes. Both the I Ching and modern physics are intimately involved with symmetry, space, time, and change. The I Ching approaches these matters from a very different perspective than does physics. Therein lies its unique value.
6-Why do you use so many metaphors in your posts?
And you’re just referring to the obvious metaphors. There are many more hidden from view. Use of metaphor is one of the behaviors that makes humans human. It is the ability to see that one thing can represent, or stand in for, another. It is the ability to make use of symbols. All language is fundamentally metaphor. Language has its very origin in metaphor. The I Ching in its earliest formulation predates written history and arose from a stratum of mind that did not yet universally separate wholes into parts or distinguish between right brain- and left brain-activities. Poetry, philosophy, science, art and imagery of all kinds were not yet specialized categories of thought. They were not yet mentally speciated, to borrow a term from biology. But all languages retain elements of metaphor. Physics and mathematics are special forms of language. When Einstein states that mass and energy are interchangeable he is using metaphor. E = mc^2 is metaphoric in form because the equals sign both separates and unites two objects or events that are distinguishable (different) but interchangeable (somehow the same.) In thinking metaphorically like this Einstein changed the course of history and the way we view the world. All equations are similarly metaphoric in form. So mathematics is rife with metaphor. Metaphor is everywhere. We cannot escape it. Looking into a mirror we recognize the image as that of ourselves but different. Viewing the universe is like looking into a mirror - - - it reflects back our own image but somehow changed. We only know the universe as metaphor. All else lies unrevealed behind the veil. 

1-How can I access the initial page of the blog directly?

Congratulations on your perspicacity! You’ve detected the strangely counter-intuitive manner of page labeling of this blog which attempts (unsuccessfully, I might add) to reverse time. Whereas we commonly label years and the days, weeks, and months of the year in ascending chronological order, within the confines of this blog pages are numbered in reverse chronological order. So what is called ‘page 1’ is the current (most recent) page and its content is always changing over time. The initial page of the blog chronologically, though its content is unchanging, has a page label which changes (increases) over time. This was not by my own choice, but since I know so little HTML we are stuck with it. And would you expect less of a blog dealing with quantum mechanical weirdness anyway? 

The way around this is as follows: 

At the bottom of the page which greets you when you first access the blog you will see Page 1 of x where x is the then current total number of pages. To access the true (as opposed to the spurious) Page 1 directly, change the address in the address bar to blindmen6.tumblr.com/page/x and hit ENTER (or refresh the page). The electrons will take care of matters from there.

2-Whenever I attempt to use the Search function I get either no results or only a single page of results? Is something amiss here?

You bet your strange quarks there’s something amiss here. And I don’t know how to fix it. Frankly, the Search by tag function works better than the Search function. If your Search yields no results there likely are none. But if it gives you one page of results there are likely more. 

Here’s what to do:

In your Search results (assuming there are any) find a tag which matches or most closely matches your search term. Click on that tag and you will get all the posts that have been so tagged. I try my best to tag posts exhaustively but the terms I’m thinking in may not always match those you are thinking in. Even the electrons can’t help in that case.

3-You call your blog ‘QUANTUM GEOMETRY’ but I’ve seen almost no geometry and, for that matter, not all that much quantum. I have seen a lot of art, modern and otherwise. This is really an art blog in disguise, isn’t it?

In fact, no. The images attached to the verbal content of the posts are carefully selected for their relevance as well as their visual appeal. But they are basically hooks by which to grab your attention. Most every post is related to geometry and/or quantum mechanics in some manner though this may not initially be clear. If you continue to follow the blog, this will become increasingly clear over time. We are still (as of June 28, 2012) building a common foundation for future discussions. In time more posts will appear more directly related to geometry and the quantum world. The art stays though. If I can’t woo you with ideas, I’ll wow you with visual imagery. Also, art is a reminder that ideas at their inception are rarely greeted with open arms and mind.

4-What are your qualifications for discussing physics, mathematics, and Chinese philosophy?

Glad you asked. I do have a postgraduate degree but not one in physics, mathematics, philosophy, or sinology. My background is more in the fields of chemistry and biology. Both of those fields are intimately connected to the fields of physics and mathematics, less so to the fields of philosophy and sinology. Frankly, I don’t think it much matters. This is an interdisciplinary blog. What I lack in comprehension of the advanced mathematical equations employed by theoretical physicists I make up for by bringing a fresh new perspective to the field. My interest in Chinese philosophy, the philosophy of science, and philosophy in general dates back more than half a century. I’ve been involved with studies of the I Ching, particularly of its mathematical underpinnings, for nearly that long. Where I disagree with the prevailing views in physics and sinology I will plainly state so. But I believe anyone with a questioning mind has the right to use it as s(he) sees fit. Half a century ago ‘quark’ was a soft creamy acid-cured cheese of central Europe made from whole milk. It wasn’t until 1964 that Murray Gell-Mann proposed this name for his newly dreamed up particle which at that point he still regarded as just a convenient mathematical device to explain the then prevalent particle zoo of hadrons. His new particle and the ideas behind it were not initially universally well-accepted. The proof of the pudding is in the eating.

5-Isn’t the I Ching just a discredited guide to divination?

Don’t tell that to Carl Jung. (May he rest in peace.) The I Ching has held an exalted place in Chinese thought for over two millennia. It is one of the oldest Chinese classic texts. The book indeed contains a divination system and it is still widely used for this purpose. But it is also a historical document, a compendium of philosophical and ethical thought, a work relating linguistics to visual imagery which predates written language, and most important to our usage here, a treatise on mathematics, particularly combinatorics and group theory. One of the alternate names for the I Ching is The Book of Changes. Both the I Ching and modern physics are intimately involved with symmetry, space, time, and change. The I Ching approaches these matters from a very different perspective than does physics. Therein lies its unique value.

6-Why do you use so many metaphors in your posts?

And you’re just referring to the obvious metaphors. There are many more hidden from view. Use of metaphor is one of the behaviors that makes humans human. It is the ability to see that one thing can represent, or stand in for, another. It is the ability to make use of symbols. All language is fundamentally metaphor. Language has its very origin in metaphor. The I Ching in its earliest formulation predates written history and arose from a stratum of mind that did not yet universally separate wholes into parts or distinguish between right brain- and left brain-activities. Poetry, philosophy, science, art and imagery of all kinds were not yet specialized categories of thought. They were not yet mentally speciated, to borrow a term from biology. But all languages retain elements of metaphor. Physics and mathematics are special forms of language. When Einstein states that mass and energy are interchangeable he is using metaphor. E = mc^2 is metaphoric in form because the equals sign both separates and unites two objects or events that are distinguishable (different) but interchangeable (somehow the same.) In thinking metaphorically like this Einstein changed the course of history and the way we view the world. All equations are similarly metaphoric in form. So mathematics is rife with metaphor. Metaphor is everywhere. We cannot escape it. Looking into a mirror we recognize the image as that of ourselves but different. Viewing the universe is like looking into a mirror - - - it reflects back our own image but somehow changed. We only know the universe as metaphor. All else lies unrevealed behind the veil.