A Common Geometry © John C. Gonsowski - 6/29/01
[Editor's comment - The illustrations referred to in this paper - figures 1 through 9 - are large and complex. So we have made it possible for you to view them in an auxiliary window.
Abstract
Psychologist Carl Jung studied a spiritual quaternity (Fonda, 1997, June)
which consisted of: 1 - Adam; 2 - Eve; 3 - Second Adam (Son of God); 4 -
Second Eve (Mary). This quaternity is often depicted geometrically as a
square formed by the four endpoint directions of an x-y coordinate system
(Fudjack & Dinkelaker, 1999, February, 2). This Jungian quaternity is also
analogous to the first four points of the nine point Enneagram of Process
personality model (Fudjack & Dinkelaker, 1999, February, 2). The square
coordinate system is also used to represent the x-y-z-t of spacetime
(Smith, 2000, July, 1). For the Jungian quaternity, Adam and Second Adam form
opposites as do Eve and New Eve on the square coordinate grid. For
spacetime, the two transverse (perpendicular to line of travel) space
dimensions form opposites as do time and longitudinal space (in line with
travel). If four dimensions were enough to describe everything there would
be no need to go beyond this description in order to show the same geometry
being used by personality, physics and spirituality. However, it is easy to
see that more is needed. Spacetime has material objects that require their
own dimensional representation and the Enneagram has nine points not four.
There are many more not so easy to see extra points. As for how the extra
points relate to the four square ones, a first guess might be that the
extra points form up in third and higher dimensions some more two
dimensional, four point squares that project exactly down to the original
two dimensional, four point square. However, if things were that easy
everyone would have agreed on a theory of everything long before now.
The actual geometry (Fudjack & Dinkelaker, May, 1999) is: liminocentric
(boxes inside boxes where the innermost one is "equivalent" to the
outermost one); strange-attractor-like (not all projected but containing
distortions based on direct projections); and fractal (the same general
patterns can be found in various places). The geometry that does this is
the A-D-E series (Smith, 2001, February, 1). The fractal-strange attractor
property of this geometry can be seen in two dimensional projections where
various distortions of Z and double-V shapes occur. These two shapes are
related to the heavenly and temporal order I-Ching divination device
movements (Fudjack & Dinkelaker, 1999, March). The fractal-liminocentric
property of the A-D-E series geometry can be seen in the physics model
based on the A-D-E series. At the highest energy level, this model is
described like a giant computer where every bit of information is next to
every other bit (Smith, 2001, March, 4), kind of the ultimate in
liminocentricity. Dropping down a level to string theory worldlines through
the many worlds, one finds closed string loops where the radius of the loop
being zero is physically equivalent to the radius of the loop being
infinity (Smith, 2001, June, 6). Dropping down a level to a single
universe, one finds a triality supersymmetry (Smith, 2001, June, 3) where
spacetime can rotate into matter which can rotate into antimatter which can
rotate back to spacetime. Dropping down another level one finds a spacetime
where the dimensions are mathematically complex numbers rather than real
numbers causing each spacetime dimension to be like a circle that starts
and ends at the same point (Greene, 1999). Dropping down one more level one
finds gravity and other forces that all are described as having wrap back
on itself torus-like geometry (Smith, 2000, April). The progression from
forces to spacetime to universes is mimicked spiritually by a Tree/Wheel of
Life to Enneagram to Sri Yantra progression (Fudjack & Dinkelaker, 1999,
April, 1) and (Fudjack & Dinkelaker, 1999, April, 2) and mimicked
personality-wise by an MBTI to Enneagram to Full Circumplex theory
progression (Gonsowski, 1999, April).
The A-D-E series described in this paper has two types of geometries
and three types of algebras associated with it. The paper primarily
discusses an eight dimensional geometry (Smith, 2001, February, 1)
that itself is not a space in the physical sense but is a geometry that
relates physical spaces to each other. The other type of geometry is for
the physical spaces themselves. One obvious physical space is spacetime
itself and other physics entities that get treated as physical spaces
include force particles, matter/antimatter and quantum strings. There are
three types of algebras associated with the A-D-E series, Lie algebra,
Jordan algebra, and Clifford algebra. Lie algebra uses (Smith, 2001, January)
what are called bracket products to take a times b to (1/2)(ab - ba). Since
ab-ba is a measure of commutitivity, this algebra in general is rather
different than the real and complex (real plus imaginary) commutitive
algebras that most people are familar with. If you non-rigorously think
(Smith, 2001, April, 2) of the Lie algebras as rotating spheres then you
can non-rigorously think of the Jordan algebras as mapping the spheres into
ellipsoids. Lie algebras define the conventional rotation actions of force
particles while the Jordan algebras define the spacetime reconfiguring actions
of quantum strings. Jordan algebras can thus be thought of as universe
curving actions that can occur due to things like Einstein relativity and
Jungian synchronicity (Smith, 2001, March, 3). Clifford algebra is an "information
theory" algebra. You can think of it as the information bit processing that
takes place under the covers for the geometries and other algebras.
In (Gonsowski, 1999, April) I mapped MBTI personality dimensions (Myers & McCaulley, 1985) to various circumplex models. The MBTI circumplex related to the Enneagram of Process (Fudjack & Dinkelaker, 1999,February, 1) is shown below:
The points of the circumplex above do not form a circle. Circumplexes as the name implies are usually presented as circles. My previous work had circular circumplexes also but Figure 1 is more geometrically correct. Figure 1 has the F points directly above the T points and this makes more sense if you think about it than the circular circumplex which has F points sometimes to the upper left and sometimes to the upper right of the T points. This geometry is part of what is called the A-D-E series (Smith, 2001, February, 1). Specifically, the Figure 1 circumplex is a two dimensional projection of D3 root vector geometry. The 3 in D3 means three axes (SN, PJ, FT) and the D means all the points are bivectors (two factor types in personality terms). For those that are interested, this projection is formed by starting with all points having coordinates for the three axes consisting of two coordinates that are +1 or -1 and one coordinate of zero. Next rotate the 1-2 axes by 45 degrees in their plane then rotate the 2-3 axes by the arccosine of the square root of 18/19ths. The first two coordinates become the circumplex coordinates. Z patterns Figure 1 has a red line (squished Z) and a blue sideways Z that show where the Enneagram of Process first and second quaternities are on the circumplex. The Z patterns also show up at the 5th axis of the A-D-E series and show up in the form of ideal symmetrical Zs at the 7th and 8th axis level. A two dimensional projection of this appears below:
The red Z represents where two 7th axis and four 8th axis first quaternity Zs are superimposed. The blue Z represents where two 7th axis and four 8th axis second quaternity Zs are superimposed. The pink and purple Zs represent where 5th axis first and second quaternity Zs are located. The yellow "squished Z" line represents where the three axis circumplex first quaternity Z shows up on this projection and the green Z is where the three axis circumplex second quaternity is located on this projection. Notice I used 1 and 2 to represent the first and second quaternity ends of the 5th axis. The 4th axis as you might expect has the MBTI IE factors for its ends. The 6th, 7th and 8th axes are perpendicular to the plane so you can think of them as axles going through a wheel while the first five axes are like spokes of the wheel. The A-D-E series ends at the 8th axis. The Figure 2 projection is formed by starting with all the points where two axis coordinates are +1 or -1 and the other six axis coordinates are zero plus all the points where all coordinates are +.5 or -.5 and the number of minus signs for the 8 coordinates is an odd number. Next rotate the 6-7 plane by 45 degrees then rotate the 6-8 plane by the arccosine of the square root of 2/3rds then rotate the 7-8 plane by the arccosine of the square root of 3/4ths. Next rotate the 2-3 axis plane by 45 degrees then rotate in sequence the 1-3 and 2-4 planes by the arccosine of the square root of 2/3rds. Next rotate the 1-4 plane by 45 degrees then rotate the 2-5 plane by the arccosine of the square root of 3/4ths and finally rotate the 1-5 plane by 45 degrees. One does have to make sure to rotate in the correct direction (clockwise vs counterclockwise) and assign the positive and negative ends of the five axes correctly. Spacetime and Gravity So how are the MBTI 3 axis and Enneagram quaternity 5th axis Zs related? One place to look for clues is in particle physics where A-D-E geometry is also used. In the A-D-E physics model, the 5th axis D5 adds a spacetime with 4 complex inner dimensions and 4 complex outer dimensions (Smith, 2001, June, 1). Being complex means that one end of the 5th axis is mathematically imaginary and the other end mathematically real. The Enneagram could be said to have an “imaginary” first quaternity and a “real” second quaternity. Complex dimensions are circular rather than linear (Greene, 1999) and the Enneagram as suggested by its points arranged in a circle adds this circular aspect to personality space. The Enneagram is analogous to the inner spacetime dimensions. The yellow straight line (squished Z) on Figure 2 may be related to why the inner spacetime dimensions are curled up so that we physically only notice the four outer spacetime dimensions (Greene, 1999). The outer imaginary first quaternity and real second quaternity Zs of the 5th axis along with their 3rd, 7th and 8th axis counterparts are shown below: The Figure 3 two dimensional projection is the same one as for Figure 2, Figure 3 just shows the non-Enneagram “outer” Zs instead of the Enneagram “inner” Zs. Figure 3 also uses the same color scheme as Figure 2 for differentiating the Zs. Notice for Figure 3 that the 1-4 points of the “imaginary” first quaternity Zs form parallel lines with the 1-4 points of the “real” second quaternity Zs. The 2-3 points also form parallel lines. For the Figure 2 "inner" Enneagram Zs, the 1-3 "imaginary" first quaternity Z lines are parallel to the 1-2 "real" second quaternity Z lines and the 2-4 "imaginary" first quaternity Z lines are parallel to the 3-4 "real" second quaternity Z lines. These parallel lines may be related to the transverse-longitudinal structure of spacetime (Smith, 2000, July, 1). What 3 axis thing is the 5th axis spacetime mapping to? In physics, D3 represents the gravity-Higgs field (Smith, 2001, February, 1). The D3 gravity-Higgs field can be thought of as describing the links connecting neighboring D5 spacetime vertices (Smith, 2000, July, 1). Thus in some sense personality is determined by MBTI roads connecting Enneagram locations. The Five-Factor Model Since I’ve described the Enneagram as a 5th axis thing, one is tempted to wonder if the Five-Factor Model (FFM) is more valid than the four factor MBTI (McCrae & Costa, 1989). Indeed if one looks at the Figure 2 projection, the 1-2 5th axis seems to match with an NFJ-STP scale which has been linked to FFM Neuroticism (Gonsowski, 1999, April). Neuroticism is the one FFM scale considered not to match with an MBTI scale (McCrae & Costa, 1989). Even though it does seem the FFM has a basis in A-D-E series geometry, there is a math rule under which the MBTI can be seen as more fundamental. The rule is called Clifford 8-fold periodicity (Smith, 2001, June, 2). This rule says that the 8 ends of the first 4 axes are enough to describe all the higher axes. For example, the formula Cl(10) = Cl(8) X Cl(2) says that the 10 ends of the first 5 axes are equivalent to the tensor product of 4-axes ends with the 2-axes ends. Does this mean that one does not need the Enneagram quaternity and FFM 5th axis stuff? Well no the Enneagram quaternity and FFM 5th axis is still valid, it just happens to already exist in some more difficult to understand way using just the 4 axis stuff. This difficult to understand way may be related to the concept of MBTI pure types (Dinkelaker & Fudjack, 1998, July). Three Axis Spirituality For the twelve D3 gravity-Higgs points, eight are gravity and four are Higgs and for the three D3 axes, two are gravity and one is Higgs (Smith, 2001, February, 1). The eight points plus 2 axes equals ten gravity dimensions have been linked to the spiritual geometry of the ten Kabbalah points (Smith, 2000, July, 2). The Kabbalah is described as having torus-like geometry (Fudjack & Dinkelaker, 1999, April, 1). Gravity also has torus-like structure (Smith, 2000, April). The Majesty Kabbalah point is the first axis (the idea of majestic perception provides a link to the MBTI S-N perceiving first personality axis). The Endurance Kabbalah point is the second axis (the idea of no closure enduring provides a link to the P-J closure second personality axis). The Kabbalah’s Understanding and Wisdom match with NP theorists and NJ gurus and are related to N abstract human knowledge. The Kabbalah’s Power and Kindness match with SP environment mastering and SJ guardian care and are related to S concrete Godlike knowledge. The first S-N Majesty axis thus seems to have God and humanity at its endpoints. The SP Power to NP Understanding S to N God to human transference seems to be an Old Testament thing while the SJ Kindness to NJ Wisdom God to human transference seems to be a New Testament thing. Thus the ends of the second P-J Endurance axis seem to be Old Testament and New Testament. Old vs. New Testament also hints at the time road of the gravity second axis (Smith, 2000, July, 1). The Kingdom-Foundation-Beauty-Crown Kabbalah central pillar seems to match with SF nurturing in the womb, NF forming ideals, FP peaceful harmony with others and FJ “public service” adult. The Fs thus seem to describe the ideal mortal existence. Also notice the FJ crown sits in the middle of a crown shaped group of points at the top of the Figure 2 (and Figure 3) projection. Extending the Kabbalah Since the Kabbalah is like gravity-Higgs without the Higgs and like the Figure 1 circumplex without the Ts, we have to leave the Kabbalah to find out what is spiritually at the other end of the third axis. Given that one end represents an ideal mortal life, it is natural to assume that the other end might represent “immortality”. This would make the F-T judgment third axis kind of a “final judgment” mortal-immortal axis. ST dominance over death, NT transforming change, TP further progress, TJ new pure will could be one way to describe the “immortal” end of the 3rd axis. The 3rd axis Higgs field kind of adds an “immortal” aspect to spacetime by providing access from real spacetime to imaginary spacetime thus giving spacetime its never ending circular aspect (Smith, 2001, April). The F-T 3rd axis endpoints are part of both the “real outer” purple Z of Figure 3 and the “imaginary inner” pink Z of Figure 2 and this could be related to the Higgs field’s role in providing access to imaginary spacetime. Given that the Kabbalah is called the Tree of Life (Fudjack & Dinkelaker, 1999, April, 1), it kind of makes sense that the “immortal” stuff is left out of the Kabbalah. Trees are mortal. The Buddhist Wheel of Life on the other hand has all twelve D3 points (Fudjack & Dinkelaker, 1999, April, 2) and is thus able to represent life as a wheel with an “immortal” cycle of rebirths. So the Kabbalah with a Wheel of Life “immortal” extension would seem to act as three axes roads for the Enneagram’s 5th axis quaternity locations. These three axis roads seem to be God to humanity, Old Testament to New Testament, and mortal to immortal. 5th Axis Spirituality The 5th axis Enneagram spiritual locations can be seen in terms of Jungian Christian quaternities. Jung’s most studied quaternity matches with the Enneagram’s first quaternity (Fudjack & Dinkelaker, 1999, February, 2). This quaternity in Christian terms would be Adam-Eve-New Adam (God the Son)-New Eve (Mary). So things start out in the Adam location with humanity lonely and lacking understanding; at the Eve location Adam is no longer lonely but humanity still lacks understanding. At the New Adam location, humanity begins to have God the Son inspired hope for better understanding and at the New Eve location, the hope for better understanding is realized through Mary giving birth to Christ. The Enneagram’s second quaternity in Christian terms would be the baby Jesus, God the Father conscience, Jesus' Gospel life, and Jesus' Death on the Cross. Thus Adam’s realized hope for understanding continues to grow until full wisdom is achieved as represented by the Cross. The Enneagram 9th What is the Enneagram 9th? It actually is the 5th axis. The second quaternity points at one end of the 5th axis connect back to the first quaternity points at the other end of the 5th axis through the 5th axis itself. This fits with the one through nine clockwise ordered numbers on the Enneagram circle where 9 sits between 8 and 1 connecting the second quaternity back to the first. The idea of two quaternities each with Zish movement connected by going through the center (of the 5th axis) is known as “heavenly” order (Fudjack & Dinkelaker, 1999, March). Actually this heavenly order (of the I-Ching divination device) is more of a 4ish movement than a Zish movement. If you look at the “real outer” purple Z of Figure 2 and consider that for spacetime the transverse (first axis S-N) points should come first in an X-Y-Z-T spacetime then the Enneagram Z’s 1-2-3-4 order becomes 4-1-3-2. This 4-1-3-2 forms a 4ish pattern just like for the heavenly order. This Z vs 4 order difference may be related to the difference between the Enneagram’s path of renunciation and the Wheel of Life’s birth of ego orders (Fudjack & Dinkelaker, 1999, April, 2). Both the Z and the 4 are two dimensional paths on the four dimensional hyperoctahedrons that sit at each end of the 5th axis (Smith, 2001, June, 1). The 4 forms a crossing pattern that may be related to the Christian Cross. A four dimensional hypercross complete with 3 axis mapping and two dimensional projecting was depicted in a Salvadore Dali rendition of Christ’s Cross (Fudjack & Dinkelaker, 1999, June). The Enneagram 9th as Holy Spirit The Enneagram 9th has been connected to the Christian Holy Spirit (Fudjack & Dinkelaker, 1999, February, 2). The center of the 5th axis like all the axes is at the center of the Figure 2 projection. Notice the MBTI point FP is very close to the center of Figure 2. FP is the peaceful spiritual MBTI two factor type. The Enneagram 9th picks up its Holy Spirit relationship from FP. Thus spiritually the E8 death of Jesus on the Cross would be followed by the E9 Hoy Spirit reign that Jesus left to us. You could also think of E9 as Jesus' resurrected spiritual form. This E9 Spirit wraps back to E1 Adam (notice the TJ E1 point from the Figure 1 circumplex is also very near the center of the Figure 2 projection). The E9 Spirit wrapping back to E1 Adam kind of acts to cure E1 Adam’s “original sin” problems. The resurrected Jesus descending victoriously into hell symbolizes this nicely. This spiritual quaternity analysis is a rather simplified view of spirituality space. It’s like having a spacetime with a very small inner spacetime and no outer spacetime. A very small outer spiritual spacetime might consist of the Christian Trinity and human Jesus for an “imaginary” part and realms for a “real” part. The “outer imaginary” part maps to the same 3 axis points on Figure 3 that E3-E6-E5-E9 maps to on Figure 1 (with E9 as the FP Holy Spirit). The “real outer spacetimeish” realms could be thought of as the Buddhist Bardos (Smith, 2001, May, 1) or the Christian eden/hell, limbo/purgatory, earth and heaven. Even with a simplified Enneagram spirituality space one can see the Kabbalahish God to human, Old to New Testament, and mortality to immortality roads at work in the Enneagram. The 4th Axis There are roads I haven’t discussed yet, namely the 4th axis ones. The 4th axis is called in physics the U(1) physical propagator phase (Smith, 2001, February, 1) and it is used to keep track of time in situations where normal time breaks down (Smith, 2001, March, 1). For example suppose you were to travel back in time. Even though you went backwards in normal time, this backwards travel in normal time is still part of your personal future. Thus you can sort of think of this 4th axis time as personal time. What does this personal time mean in a spiritual sense? It is the time that takes over when one “becomes immortal”, where the normal old vs new testament kind of time doesn’t work any more. Without “becoming immortal”, the human mind may however be able to experience this 4th axis personal time during meditations. The idea of inward and outward meditation (Fudjack, 1999, September) could represent endpoints of this 4th axis time. An inward-outward meditation 4th axis would fit nicely with an A-D-E series personality model where MBTI introversion and extraversion are at the endpoints of the 4th axis. The idea that inward and outward meditation can get you to the same place is like the idea that the same point in spacetime can be found in both the past and future of your personal time. In the A-D-E series physics model, the ends of the 4th axis can also be noticed without normal time breaking down. At one end are negatively charged color/electroweak particles and at the other end are positively charged color/electroweak particles. Each end also contains some neutral particles (Smith, 2000, October). 4th Axis Roads vs. 3 Axes Roads While 3-axis gravity-Higgs roads described earlier can be reached from outer spacetime, the 4th axis color/electroweak roads are inner spacetime roads (Smith, 2001, March, 2). The 4th axis I and E endpoints being part of the “real inner” 2nd quaternity Z of Figure 2 may be related to the 4th axis providing inner spacetime roads. Going from a 3 axes D3 to the 4th axis D4 is like going from a “Star of David” shape to the shape on the outside of the Sri Yantra (Smith, 2001, February, 2). This can be seen in the shapes of the D3 and D4 projections below:
These two projections are made the same way as I described for the Figure 2 projection except that for the Figure 4 “Star of David” projection you leave out the rotations involving axes greater than 3 and for the Figure 5 “Sri Yantra” projection you leave out the rotations involving axes greater than 4. “Double V” Patterns The Figure 5 projection shows the “double V” patterns. The pink ideal symmetrical double V is at the introverted end of the 4th axis and the purple ideal symmetrical double V is at the extraverted end. The red double V of Figure 5 shows the 3 axis points mapped to by the introverted 4th axis double V and the blue double V shows the 3 axis points mapped to by the extraverted 4th axis double V. If you look at the circular order of the Figure 4 axes endpoints for the first three axes (F-S-P-T-N-J) and add a 4th axis endpoint (E or I) to each of the 3 axis endpoints then you get 4th axis points whose circular order is the same as the circular order of the 3 axis mapped to points on the Figure 1 circumplex. In other words,
Reassigning Enneagram E-I Biases What are the reasons for the Enneagram law of 7-law of 3 not matching with the MBTI E-I scale empirically? First from the point of view of the Z patterns on the circumplex the conventional Enneagram numbering is incorrect (Gonsowski, 1999, April). For the first quaternity Z, the introverted mystical conventionally numbered E4 should be an E3 law of 3 point (to match up with NF on the Figure 1 circumplex). This swap also makes the somewhat cold extraverted conventionally numbered E3 an E4 law of 7 point (to match up with NT on the Figure 1 circumplex). For the second quaternity, E5 swaps with E7 and since both points are law of 7 points, the E5 introversion problem transfers to an E7 introversion problem. This leaves 2 of 9 (E1 and E7) Enneagram types mismatching the introverted law of 3-extraverted law of 7 hypothesis. This 78% success rate isn’t bad when comparing the Enneagram and MBTI but I’d like to add some comments saying where these two errors may have come from. Theorists when making a personality test tend to want symmetry in their test and this symmetry would include the introversion-extraversion axis. This desire for introversion-extraversion symmetry is justified since for the Figure 5 fourth axis there are six introverted and six extraverted MBTI types. However when cramming this 12 point symmetry into the 9 Enneagram points a couple problems show up. First it is impossible to break 9 things equally in half. This may have lead to E1 losing its extraversion bias and becoming neutral with respect to introversion-extraversion. Actually E1 seems extraneous with respect to symmetry for all the MBTI axes (Dinkelaker & Fudjack, 1998, July). Theorists looking for symmetry may have split the remaining eight types up into four introverts and four extraverts when constructing a test which changes one extravert (E7 after renumbering) to an introvert when compared to the introverted law of 3-extraverted law of 7 hypothesis. The place to look for introversion-extraversion symmetry is up at the 6 vs 6 level rather than down at the 4 vs 4 level. Magic Squares The double V pattern is also seen in what are called magic squares. A magic square is a 3 by 3 matrix where all 8 rows, columns and diagonals sum to 15. This magic square double V pattern has been related to the I-Ching’s temporal order pattern (Fudjack & Dinkelaker, 1999, March). Temporal order fits nicely with the idea of the 4th axis being personal time. A pseudo magic square (Smith, 2001, February, 3) with the central row, central column, and diagonals summing to 15 also has a double V pattern described as 6 hands (from the two outer columns) around a still center (the middle column). It has also been related to the 6 days of creation plus a day of rest. The “still center” and “day of rest” would be symbolic of the 4th axis and the “6 hands” and “6 days” would be symbolic of the 6 points of the double Vs at each end of the 4th axis. This pseudo magic square has been shown to have a torus knot structure analogous to the torus-like liminocentric (Fudjack, 1999, September) 4th axis structure I discussed earlier for meditation experiences. This pseudo magic square has also been shown to have the structure of interpenetrating triangles of octonionic imaginaries (i-j-k and I-J-K). Interpenetrating triangles and double Vs are two dimensional projections of the three dimensional octahedrons (Smith, 1999, February, 1) formed at each end of the 4th axis. Interpenetrating triangles are also found in the interior of the Sri Yantra thus in addition to the outside shape, the Sri Yantra has interpenetrating triangle symbolism for the 4th axis. The Sri Yantra’s triangles as I will discuss later are symbolic of other axes too. Another type of pseudo-magic square also contains double V movements. These pseudo magic squares have 6, 7 or all 8 rows, columns and diagonals summing to multiples of 9. What are called Enneagram triads can be found in the rows, columns and diagonals of these pseudo magic squares. Enneagram triad patterns also seem related to the head-heart-gut (aka mental-conduct-wisdom) ways for dividing up the Hindu Chakras and the Buddhist 8-fold path. One specific one of these pseudo magic squares is known as the Celtic Enneagram (Dixon, 2000, September). The Celtic Enneagram has the number 9 in the center which is nicely symbolic of the Enneagram 9th being the 5th axis around which the Enneagram quaternities are placed. Divination Devices The Celtic Enneagram is also considered related to Tarot. While the Celtic Enneagram (and Enneagram of Process) quaternities are a 5th axis thing, Tarot is a 6th axis thing. The 78 Tarot cards are related to the 78 (including axes) E6 points (Smith, 2000, September). Another divination device, the I-Ching creates what is called a double bind relationship between the 4th axis temporal order double Vs and the 5th axis heavenly order Zs (Fudjack & Dinkelaker, 1999, March). While the 6th axis Tarot could get at the 4th and 5th axis temporal and heavenly orders as subspaces, the I-Ching and other divination devices like Wei Qi and IFA (vodou) get at the temporal and heavenly orders through the Clifford “8-fold periodicity” algebra (Smith, 2001, June, 4) and (Smith, 2001, June, 5). The first reference also relates the MBTI to Clifford algebra. As I mentioned earlier, Clifford 8-fold periodicity allows the full A-D-E series to be generated using 4 axes worth of information. The I-Ching’s double bind causes the heavenly order Zs to act as strange attractors for the temporal order double Vs (Fudjack & Dinkelaker, 1999, March). One way to visualize this strange attractor relationship is to notice that the ideal symmetric red Z of Figure 2 and Figure 3 is a top to bottom zig-zag while the ideal symmetric blue Z of Figure 2 and Figure 3 is a left to right zig-zag. The top to bottom and left to right Z zig-zags can be seen as a strange attractor cause of the top left to bottom right zig-zags of the Figure 5 double Vs. A specific Z vs double V strange attractor relationship can be seen by looking at Enneagram numbers (Fudjack & Dinkelaker, 1999, March). As mentioned earlier the first quaternity “imaginary outer” Z of Figure 3 maps to the E3-E6-E5-E9 Enneagram numbers. This 3-6-5-9 Z acts as a strange attractor for the 1-4-2-8-5-7 law of 7 double V at the extraverted end of Figure 5. It works as follows: the Z’s 3 causes the 1-4-2 double V movement; the Z’s 6-5 causes the 2-8-5 double V movement; and the Z’s 9 pulls the double V from 5 to 7. 6th Axis Personality Even if there wasn’t Clifford 8-fold periodicity, the five-factor model would not be the end of the story since we have more axes to add. The D series part of the A-D-E series does end at the 5th axis but the E series takes over. E6, E7 and E8 which complete the A-D-E series are not limited to two factor types like the D series. E6, E7 and E8 are able to have non-zero coordinates for all 6, 7 or 8 of their axes. Looking at E6 we find that it creates something called triality supersymmetry (Smith, 2001, June, 3). The triality term means that two additional circumplexes are created complete with real and imaginary Z patterns. The supersymmetry term means that the three circumplexes (and Z patterns) rotate into each other. In (Gonsowski, 1999, April) I identified the two additional circumplexes shown below:
One can rotate the Figure 1 circumplex into the Figure 6 one by rotating N to F, F to S, S to T and T to N (J and P stay the same). The 3 axis Z points created by doing this rotation to the Figure 2 and Figure 3 Zs are shown below together with the actual 6th axis points they map with: The 3 axis first quaternity “imaginary up” (up analogous to outer) Z is red, the 3 axis first quaternity “imaginary down” (down analogous to inner) squished Z (line) is yellow, the 3 axis second quaternity “real up” Z is blue, and the 3 axis second quaternity “real down” Z is green. The 6th axis first quaternity “imaginary up” Z is pink, the 6th axis first quaternity “imaginary down” Z is orange, the 6th axis second quaternity “real up” Z is purple, and the 6th axis second quaternity “real down” Z is brown. Black is used when the Zs overlap. This is the same projection as for Figure 2 and Figure 3 but with 6th axis points shown. One can rotate the Figure 1 circumplex into the Figure 7 one by rotating J to T, T to J, P to F and F to P (S and N stay the same). The 3 axis Z points created by doing this rotation to the Figure 2 and Figure 3 Zs are shown below together with the 6th axis points they map with: The projection and color scheme for Figure 9 is the same as for Figure 8. The sixth axis points for Figure 9 are at the opposite end of the sixth axis from the Figure 8 sixth axis points. Notice that the down and up Zs for Figure 8 and Figure 9 have the same parallel line properties as was described for the inner and outer Zs of Figure 2 and Figure 3. This is a result of the triality supersymmetry relationship the ends of the 6th axis have with the 5th axis. For reasons similar to why the Figure 2 fifth axis “inner” Enneagram is called the Enneagram of Process (Fudjack & Dinkelaker, 1999, February, 1), the Figure 8 “down” Enneagram can be called the Enneagram of Consulting and the Figure 9 “down” Enneagram can be called the Enneagram of Development. 6th Axis Physics and Spirituality In physics, the extra “spaces” added by E6 to create triality supersymmetry with D5 spacetime are matter and antimatter (Smith, 2001, February, 1). Antimatter kind of turns matter into “new” spacetime in the same way as (the Enneagram of) development turns (the Enneagram of) consulting ideas into “new” (Enneagram of) processes. Spiritually, the (Enneagram of) process involved is a “vine thru life” form to spirit (Adam to Holy Spirit) process. The (Enneagram of) consulting creates spiritually a “protégé to partnership” transition that helps put “fruit on the vine”. The (Enneagram of) development “sows seeds from the fruit of the vine” to create a new “vine thru life”. The (Enneagram of) development turns the unity of the (Enneagram of) consulting partnership (analogous to spacetime-matter unity) into blissful harmony (analogous to new empty space resulting from matter-antimatter mixing). Figure 8 and Figure 9 have interpenetrating pentagons shown. One pentagon represents where consulting/matter 6th axis points are located with no development/antimatter points and the other pentagon represents where development/antimatter points are located with no consulting/matter points. The rest of the 6th axis points are projected on to by both consulting/matter and development/antimatter. The 10 points of these two pentagons map to the 10 endpoints of the first 5 axes. This allows the 6th axis pentagon points to have the same perimeter order relationship with the Figure 1 circumplex as was described earlier for the 4th and 5th axes. The I-2-T-N-J order of the consulting/matter pentagon maps to SF-SP-TP-NJ-FJ on Figure 8 and the F-S-P-E-1 order of the development/antimatter pentagon maps to FJ-ST-ST-NT-NJ on Figure 9. The rest of the 6th axis points (not part of the pentagons) match radially with pentagon points so in some sense this ordering for the 6th axis pentagon points applies to all the 6th axis points. Earlier I mentioned that in mapping the 5th axis points to 3 axis points that four 3 axis points have to be reused since the 5th axis adds 16 points and there are only twelve 3 axis points. The reused points were NP-NF-SJ-TP. If you rotate these 5th axis mapped reused points into consulting/matter and development/antimatter 6th axis mapped reused points using the rotations described earlier then you end up with NP-SF-FP-TJ for consulting/matter and NP-FJ-NF-ST for development/antimatter. If you throw out the common NP point you end up with triangle points for the 5th axis that form a central triangle on the Figure 1 circumplex and triangle points for the 6th axis that form interpenetrating triangles on Figure 1. The interpenetrating 6th axis mapped triangles on Figure 1 are horizontally aligned just like the interpenetrating 6th axis pentagons on Figure 8 and Figure 9. Notice also that the interpenetrating triangle points represent points that are not common for consulting/matter and development/antimatter just like the interpenetrating pentagon points do. Matter and antimatter not having all common points may be related to the imbalance of matter and antimatter in real outer spacetime (Smith, 2001, May, 2). The interpenetrating pentagons/triangles being 180 degrees apart may be related to the idea of antimatter as a time reversed version of matter (Smith, 1993, June). Sri Yantra The central 5th axis mapped single triangle for Figure 1 (NF-SJ-TP) is related to the single triangle at the center of the Sri Yantra. The 6th axis mapped interpenetrating triangles for Figure 1 (SF-FP-TJ and FJ-NF-ST) are related to the innermost pair of interpenetrating triangles of the Sri Yantra. The lack of symmetry in the middle of the Sri Yantra (Fudjack & Dinkelaker, 1999, April, 1) is symbolic of how the Zs get "double bind" distorted at the 5th and 6th (and mapped to 3rd) axis levels as well as indirectly symbolic of the 4th axis double V interpenetrating triangles getting distorted at the 3 axis level. The perfect symmetry for the outer part of the Sri Yantra interpenetrating triangles is symbolic of the ideal symmetric 7th and 8th axis Zs of Figure 2 and Figure 3. The pair of Sri Yantra interpenetrating triangles just outside of the 6th axis innermost pair represents the 6th axis also but it represents the 6th axis in relation to the 7th and 8th axes (Smith, 2001, February, 2) instead of the 5th and 3rd axes. The 7th axis pair is just outside the 2nd 6th axis pair and the 8th axis outermost pair is just outside the 7th axis pair. The 8 complex dimensions of the 5th axis and the 16 complex dimensions of the 6th axis are depicted in the ring of 8 and ring of 16 lotus leaves on the Sri Yantra (Smith, 2001, February, 2). The two rings outside the lotus leaf rings symbolize the 7th and 8th axes. The use of a lotus leaf symbol could be related to the crown shaped pattern discussed earlier for the Figure 2 fifth axis projection. The crownish/leafish pattern on the exterior of the Figure 2 fifth axis projection contrasts with the boxy exterior pattern for the Figure 5 fourth axis (which is also the exterior pattern for the Sri Yantra). String Theory The 7th axis (E7) adds 54 points that form bosonic string world lines through a macrospace of many worlds (Smith, 2001, June, 6). Bosonic string theory is the oldest of the string theories (Greene, 1999). Sixteen of the 27 complex dimensions formed by E7 act to allow the bosonic string to address the 6th axis matter/antimatter spaces. These 32 E7 points project exactly to the 32 E6 locations on Figure 8 and Figure 9. Being in the same location as the 6th axis points, these 32 E7 points also form interpenetrating pentagons which is related to why E7 gets to have an interpenetrating triangle representation on the Sri Yantra just like E6 does. Eight more of the E7 complex dimensions as I mentioned earlier form the red first and blue second (imaginary and real) ideal symmetric Zs of Figure 2 and Figure 3. Each point of these two Zs as mentioned earlier is actually two E7 points (one for the inner spacetime Figure 2 points and one for the outer spacetime Figure 3 points). Thus 24 of the 27 E7 complex points are used to allow the bosonic string to address the spacetime, matter and antimatter of D5 and E6. Two more E7 complex dimensions form the bosonic string worldsheet itself. These four points are located two each at the TJ and FP locations on Figure 2. TJ and FP are the 3 axis E1 and E9 first and last Enneagram points. The 24 complex spacetime-matter-antimatter points plus the two complex string worldsheet points form the 26 dimensions of bosonic string theory. Going from the 26 dimensions of bosonic string theory to the full 27 complex dimensions is analogous to going from the 10 dimensions of superstring theory to the 11 dimensions of superstring M-theory thus the full 27 complex dimensions are referred to as bosonic M-theory (Smith, 2001, June, 6). The two points for this 27th complex dimension are located at the center of Figure 2. The 8th Axis and Beyond: A Conclusion Going from E7 to E8 is like going to a bosonic hyper-M-theory with 28 dimensions. These 28 E8 dimensions (Smith, 2001, February, 1) however are quaternionic (4 points) instead of complex (2 points). This means that E8 has 2 points (one at each end of the 8th axis) everywhere that E7 has one point. This also means that E8 like E7 and E6 gets an interpenetrating triangle representation on the Sri Yantra. E8 also adds four points at the center of Figure 2 that form the quaternionic 28th dimension. Two more points exist at the center with non zero coordinates only on the 8th axis. These two points are for mathematical bookkeeping having to do with the 8th axis points having a quaternionic instead of complex or real nature. So what does string theory worldlines through the many worlds mean? Well it’s the physics of Jungian synchronicity in the sense that the world (of many worlds) we live through has curved itself to our individual worldline selves. This string theory curvature of the universe is analogous to the curvature of spacetime by Einstein’s General Relativity (Smith, 2001, March, 3). From a personality point of view it kind of means that all the Enneagram locations connected by MBTI roads are curved based on the neighboring roads and locations. It may mean that what we perceive as our personality biases may be the result of many interfering personality worldlines encoded in the human mind. As all encompassing as the A-D-E series is, it does not represent all that is. The A-D-E series is a low energy realm that forms from the cooling off of a high energy realm known as simplex physics (Smith, 2001, March, 4). In simplex physics every location has roads connecting it to every other location. It’s like a giant computer where every information bit is next to every other information bit. Sounds kind of cozy. Many thanks to John Fudjack and Tony Smith for giving us the liminocentric many worlds. Is the world ready for it? References
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