## 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.

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© 2013 Martin Hauser

## Mandalic geometry and polar coordinates - IV

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**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.

** Of particular note here**, however, is the fact that in mathematics, the root system for the simple Lie group G_{2} is in the form of a hexagram [6-pointed star].

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© 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.

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