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We will now discuss the nonagon and yet more nines.
Nonagon – 1260º
Interior angle = 40º
Each exterior angle = 140º
Sum of angles = 1260º
Area of a Nonagon
A = (9a2)/4 x cot(180º/9) where a = side length
Constructing the Nonagon
Like the heptagon, a precise version is impossible to construct with the geometer’s tools. A version can be approximated.
Reference Construction Lesson #77: Nonagons & Nonagrams
Numbering the 9 points around a circle with 9 at the top and moving clockwise:
Keith Critchlow writes, “We can make a linear path around the nine points by missing ever other point and traveling in the following sequence: 9, 2, 4, 6, 8, 1, 3, 5, 7 and back to 9. This results in a ‘blunt’ nine-pointed star (in red). If we miss two points in the sequence we do not get a continuous flow, but rather three intersecting equilateral triangles (in blue). The fourth version becomes the most interesting because it contains all the other versions with itself (in green). This star is created by following the sequence: 9, 4, 8, 3, 7, 2, 6, 1, 5, 9.”
Here they are singly:
Nine-fold Division of a Circle
A 2/9 80º rhomb can be used as a template for the ninefold division of a circle.
This is seen in the Bronze Age Bush Barrow gold breastplates uncovered near Stonehenge.
If the rhomb is placed on a circle with the 80º angle at the center if would divide the circle exactly into two ninths.
See Time Stands Still by Keith Critchlow for more details.
Three that are Nine
Using three equally sized interlocking circles arranged along a vertical axis (see pg. 86 Time Stands Still)
Keith Critchlow writes, “It can be seen that the three becomes nine if each realm is described in terms of its relationship to each other:
- earth relates to itself, mankind and heaven, its source;
- mankind takes the central role of linking heaven, the source and archetypal world, to both itself and the earth;
- heaven is self-sufficient but also relates to its two major creations, mankind and earth.
Hence the three are essentially nine by reflection.”
The Vesica Piscis as the ‘Form Generator’ & the Nine Interlocking Circles
Reference Construction Lesson #10: The Vesica & Nine Interlocking Circles.
Four interlocking circles atop five interlocking circles creates a series of Vesica Piscis from which the regular polygons emerge. Credit: Bruce Lyons
The succession of polygons as they arise out of the scission of unity.
As unity, the circle, self-divides, its center becomes the duality of points A and B.
The line AB unfolds into the equilateral triangle.
“All things being dual by nature are 3 by principle.”
The equilateral triangle unfolds outwards to:
- define the sides of the square (4)
- the pentagon (5)
- the hexagon (6)
- the octagon (8)
- the decagon (10)
- the dodecagon (12)
The vesica represents the seed.
From its germination arises the 7 circles (the root)
and the polygons (the germ, giving rise to branches).
Credit: Bruce Lyons
Lawlor writes, “The 3 contained within the Vesica Piscis is the formative power giving rise to the polygonal ‘world’.”
The Platonic Solids & the Nine Concentric Circles
See page 107 in Robert Lawlor’s Sacred Geometry: Philosophy & Practice
He writes, “In this demonstration the regular polyhedra are determined by nine concentric circles whose pattern gives all the necessary information for the construction of these forms. Each volume is in a simple harmonic relationship to the others, and it is the same transcendental function, √2, √φ, and φ that make up these patterns of relationships. As in the previous drawing, all the volumes emerges simultaneously. But in this case if one of the concentric circles is removed then the pattern cannot yield the remaining volumes. This is an image of the great Buddhist idea of the co-dependent origination of the archetypal principles of creation.”
The Enneagram in this context was introduced by the mystic, philosopher, and composer Georges Gurdjieff (1860’s?-1949). He claimed to have learned it in a monastery in Central Asia.
“It is the greatest mistake, to think that man is always one and the same. A man is never the same for long. He is continually changing. He seldom remains the same even for half and hour.” ~ Gurdjieff
It is a way of looking at whole events with hidden inner connections that weaves the principles of the Monad, Triad and Heptad into one diagram.
The Monad represents the cycle of the whole process (the outer circle).
The Triad provides structure and balance (the red triangle).
The Heptad guides the process but is devoid of balanced polarity (the green lines & top point).
The Enneagram binds them together (red & green).
The inner lines can help us to think ahead at any time.
“You can apply the structure of the enneagram to any cyclic whole event in your lie if you can determine its inner Triad and the intervening stages that make the process a cycle. The Triad and Heptad are complements, each containing what the other is missing and comprehended in the nine-stage whole of the ennead.”1
The Enneagram as a Model of Personality Types2
|Type||Characteristic Role||Ego fixation||Holy idea||Basic fear||Basic desire||Temptation||Vice/ Passion||Virtue|
|Goodness, integrity, balance||Hypocrisy, hypercriticism||anger||serenity|
|2||Helper||flattery||Freedom, will||Being unloved||To feel love||Deny own needs, manipulation||pride||humility|
|3||Achiever||vanity||Hope, law||worthlessness||To feel valuable||Pushing self to always be ‘the best’||deceit||Truthfulness authenticity|
|4||Individualist||melancholy||origin||Having no identity or significance||To be uniquely themselves||To overuse imagination in search of self||envy||Equanimity, emotional balance|
|Mastery, understanding||Replacing direct experience with concepts||avarice||non-attachment|
|6||Loyalist||cowardice||faith||Being without support or guidance||To have support and guidance||Indecision, doubt, seeking reassurance||fear||courage|
|7||Enthusiast||planning||Wisdom, plan||Being trapped in pain and deprivation||To be satisfied and content||Thinking fulfillment is somewhere else||gluttony||sobriety|
|8||Challenger||vengeance||truth||Being harmed, controlled, violated||self-protection||Thinking they are completely self-sufficient||lust||innocence|
|Wholeness, peace of mind||Avoiding conflicts, avoiding self-assertion||sloth||action|
Sri Yantra – Nine Interlocking Triangles
The Nine Yantras
The Sri Yanta is the most revered of all Tantric diagrams. It is called the “Mother of All Yantras”.
It is considered the ‘abode’ of Supreme Knowledge.
It is a very ancient symbol with an unknown origin.
It images all the necessary functions active in the universe through its nine interlocked triangles.
It is drawn from nine triangles, four pointed upward and five pointed downward, thus forming 42 (6 x7) triangular fragments around a central triangle.
The Sri Yantra – “Mother of all Yantras”
Robert Lawlor says, “There is probably no other set of triangles which interlock with such integrational perfection. To immerse oneself in such a geometric diagram is to enter into a kind of philosophic contemplation.”
The yantra represents the primordial sound of AUM.
It can represent a place or process of spiritual pilgrimage.
It can also represent the cosmos – micro and macro.
The nine-interlocking triangles surrounded by two circles of lotus petals encased in the ‘earth citadel’.
The interlocking triangles are in perfect distance from each subsequent set of triangles based on the ratio of 3.14 (pi).
The Elements of the yantra include:
- Center point (bindu) – unity, the origin, principle of manifestation and release of supreme consciousness
- Circle – extension of the point – creation of individualized consciousness out of cosmic consciousness
- Triangle – principle of movement; harmony, balance
- Square – material needs – stability, solidity and contentment. This contains a line with 36 right angles.
- Lotus petals – eightfold manifested energy – earth, fire, water, air, ether, mind, intellect, ego
- Concentric circles – three gunas – satwa – harmony, rajas – action, tamas- inertia
The Isotropic Vector Matrix (IVM) and the Sri Yantra
If you look at a 3D IVM (pictured below) from the North Pole downwards towards the South Pole you will see the 9 interlocking triangles of the Sri Yantra.
An image of the IVM. Looking down from the North Pole reveals the triangles of the Sri Yantra.
This beautifully combines an ancient spiritual lineage with a modern scientific one.
Reference Construction Lesson #78: Constructing the Sri Yantra.
- Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
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