PRIME MOVERS
Number and the multi-dimensional Circle

Pamela A. F. Crane
DFAstrolS,DMSAstrol ,DPCCH

Work in Interdimensional Astrology provokes an interest in dimensions generally and their effect on numbers.
What is a "dimension"?
Esoterically, we tend to think of a dimension not just as a single quantitative measurement but as an experientially (or at least conceptually) real spatial and/or temporal environment which is distinct in quality from others that may coexist with it. Such dimensions, as they refer to human life, appear to exist as increasingly rarefied states of matter and/or spiritual energy. These may interpenetrate each other yet still be unknown to each other, and may only interact where the dense becomes refined and the fine is able to condense its energies sufficiently. These dimensions, as suggested elsewhere, may possibly be described by a "Fractal" astrology, refined to the Dwad and the Superdwad (similar but different in phase to the 12th and 144th harmonics.) Ideally one would go even further into levels of refinement corresponding to the 1728th and then the 20736th!... but our data can never be precise enough.
Mathematically, one dimension gives us a line, two dimensions extend this into a plane. three develop a 'solid' form - and these are the dimensions with which we are all familiar, as they shape our everyday existence. What we call 'Time' is generally accepted as our fourth dimension. But in modern mathematics and particle physics, the description of ultra-refined material phenomena demands calculation in four, ten, or even twenty-six dimensions which are then required to compactify into experiential 3-D space.
Setting aside the theoretical mechanisms of such compactification, it seemed worthwhile to follow curiosity into an area that may so far be unexamined by the scientists, as it rests on assumptions that only the astrological mind would fully endorse. These assumptions are (1) that as most - if not all - phenomena are cyclic, the properties of the circle should engage us more than those of the line, and (2) that the division of a circle into 360 discrete degrees recognises a natural vibrational sensitivity to the first twelve harmonics and their products, except the seventh and eleventh. To increase subdivision of the circle to include the 7th and 11th would immediately refine it beyond the Dwad, and so leave the dimension of familiar material forms.

The diagrams and tables which follow ( to be found in the collection of images at the very bottom of the page of posts - I am unable to include them in the text) are the result of expressing all 360 degrees of the Circle in successively higher dimensions by the simple formula {N to the x power mod360} - ie, the degree-number is raised to the required power (multiplied by itself the required number of times) and multiples of 360 subtracted from the result where necessary. The remainder is the new degree on the circle generated by this process.
EG: 11 Aries in the 2nd dimension, hence squared: 112 = 121

= l Leo
23 Virgo in the 3rd dimension, hence cubed: 1733 = 5177717;
5177717 mod 360 = 197 = l7 Libra
7 Taurus in the 5th dimension: 375 = 69343957; mod 360,

= 37 =7 Taurus!
( Normal Harmonic calculation operates on the same principle, except there the formula is the degree-number multiplied by the given harmonic before removing multiples of 360.)
It is worth mentioning at this point that Degrees/Minutes/Seconds can of course be raised to the same powers... in theory! But in practice you will find that this introduces large margins of error in the resulting degree-positions once you go beyond the second dimension, where the margin of error on 1 second of arc squared is already 10 minutes of arc. So in this study, we are dealing solely with whole degrees, for the reasons given above.

One tantalising coincidence, before we look at the diagrams, is that the whole circle of 360 degrees raised to the tenth power gives us 3.65615844E+25 in scientific notation, which, divided by 1023, closely approximates the number of Earth's axial rotations (days) in one complete orbit of our Sun - 365.6158 - approximating to the exact figure of 365.2422. If one kilometer is multiplied by 1024 it takes us to the limits of the known universe, therefore so does 10km23. 10km is 6.2 miles. Perhaps this measure has some significance?

The following text refers to the DIAGRAMS at the 'foot' of the blog.

SQUARING THE CIRCLE -

by squaring the one-dimensional circle, the line as a closed loop, It becomes geometrically a surface, perhaps a Möbius strip. But when we square each zodiacal degree-value In turn, a sequence of fresh degrees is generated which in the '2nd-Dimensional' diagram draws a helical progression of larger and larger angles from 1 Aries to 30 Gemini and then reverses the same sequence from l Cancer to 30 Virgo. This repeats again for the second half of the zodiac.
It is therefore ASYMMETRIC, and oscillates about the Cardinal Points.
Just 35 of the 360 degrees form the pattern.

CUBING THE CIRCLE takes the loop into three dimensions - and raising each degree-value to the third power creates an entirely different diagram from the foregoing.
This sequence of generated degrees (76 of the 360) introduces SYMMETRYabout one axis, l Aries-Libra. The sequence does not reverse, and repeats three times in the circle from 1 Fire to 30 Water. The degrees of the natural pentagram come into strong focus - 30 Pisces/0 Aries, l2 Gemini, 24 Leo, 6 Scorpio and l8 Capricorn. Upon it can be superimposed the figure of Leonardo's man, his head in two Mars decans, his hands in four Mercury decans (two out- stretched on the Gemini-Capricorn cross-bar) and his spread legs following the flow of the pattern down to the feet at junctions of Mercury and Mars. Sun-lines and Moon-lines cross each other on the spine at the major chakras of Gonads, Solar Plexus and Heart, also above the head, over the Crown.
We have moved into the world of form with the action of Trinity, and are looking upon the idea of Man - united with his Cross.

THE FOURTH DIMENSION reduces the number of degrees to 16 - 1, 15 and 24 of Fire, 10 of Earth, plus 3 Pisces and three 'rogue' degrees - 21 Gemini, and 6 and 15 Scorpio, which upset the potential symmetry about the 21 degrees Gemini-Sagittarius axis.
Like 2-D, this is an oscillating sequence, reversing at 15 of Fixed signs, and repeating four times over the circle. It is a numerical parabola, which you can see clearly in its table, and which mirrors our
experience of duration and patterns of change more credibly than a repeated linear sequence. Life rises, peaks, and falls away.

With the FIFTH DIMENSION we are entering mysterious territory; there are no popular concepts of what a fifth dimension may comprise. It is a most beautiful pattern; five axes of perfect symmetry contain a lacework of interconnections which intersect to form concentric rings of gathered energy. There are seven foci of energy, including the centre and periphery, and six intervals. It is thought-provoking tofind a suggestive resemblance to electron shells in a pattern generated by the Creative number 5, and the same number of intervals as there are electrons in the Carbon atom. Carbon is, in our world, the stuff of life. (For more on Carbon, see "The Hidden Splendour", available from the author.)
Once again, the degrees of the natural pentagram are strong, and form the five axes of symmetry.

The pattern for the SIXTH DIMENSION is a curiosity: its sequence is very short, picking out just 12 different degrees which reverse after15, with twelve repeats of the oscillation, throwing the cusps and mid-points of each astrological sign into prominence. Four points of the natural pentagram appear - but the fifth, 12 Gemini, invisibly marks the one axis of symmetry. The shape itself is irregular, until its mirror-image 'shadow' is drawn in, as shown.

THE SEVENTH DIMENSION is amazing... a seventh heaven? Like the third, it forms about the single axis of 0 Aries-Libra and contains similar patterns. But it is a far more complex and delicate tracery, taking the full 360 degree circle to complete. Like the fifth, it is beautiful; but its patterns remind one of neural networks. Perhaps It has some relation to the angelic kingdom, or to Man in his spiritual state.

In the EIGHTH DIMENSION 'rogue' degrees from 6 and 15 Scorpio again bring asymmetry into an otherwise balanced pattern pivoted about 21 Gemini - Sagittarius, comprising the actual sequence of 16 Fire and Earth degrees plus its 'shadow' - as we found with the Sixth. The pattern repeats four times in the circle with reverses after each 45 degrees.

THE NINTH DIMENSION reminds us of the idea of Nine as a number of perfection: this displays another most lovely five-fold pattern, again focussed on the natural pentagram. It is essentially a radiant star, suggestive of no organic form except the flower in its perfection, and shines three times from the circle.

A TENTH DIMENSION re-introduces a contrasting and definite asymmetry in a 24-degree sequence that repeats after each 90 and reverses after each 45 The structure is based on the grand trines of 1,16 and 24 of Fire, 10 and 19 of Earth, and 4 of Air, but it is upset by 8 Aries, 21 Gemini, 6 and 15 of Scorpio. Like all the other Base-2 dimensions, it is a pattern of oscillation.

Comparison with the ELEVENTH DIMENSION could not be more dramatic; its delicate networks are very similar to those of the Seventh, but not identical. Like Three and Seven, it produces symmetry about the 0 Aries-Libra axis and the idea of life-form - and to me it is reminiscent of the bird ( often a Dove ) as symbol of the animating and inspiring spirit, particularly when reversed to emphasise a 'descending' form. It may be from these primal geometries that the collective unconscious draws its enduring symbols, clothing them - like the stars - in recognisable forms.

In the TWELFTH DIMENSION we have a return to utter simplicity, eight key degrees that lay the ground-plan for the beautiful traceries we saw in Three, Seven and Eleven. It repeats twelve times around the circle, and reverses every 15 degrees, a simple sequence with a shadow. It combines the properties of symmetry and oscillation.

It would be tempting to end the sequence at that point - but the apparent requirement of twenty-six dimensions for satisfactory structures in particle physics suggests that we should go further. I have not attempted all the powers between twelve and twenty-six, but offer the 26th itself, and the 13th.

The THIRTEENTH DIMENSION turns out to belong with the FIFTH & NINTH, and like the 5th is a harmonic web of five-fold symmetry. It differs, however, in its constructional sequence, so that where the 5th is made mainly of trines, sextiles and inconjuncts ( all plus 1 degree of course) the 13th has a preponderance of challenging squares and oppositions. This is a fascinating thing to find associated with numbers that are part of fortune's folklore - five being the number of the pentagram or Lucky Star, and thirteen traditionally the Unlucky Number! The 'shells' are there, as in the 5-D pattern - but because of the different chords the energy distributions are different.

With TWENTY-SIX we are back with the asymmetry now familiar in dimensions that are multiples of 2, and no further comment seems necessary.

The SEQUENCE TABLES

A table-by-table analysis Is needless, as the phenomena they produce can be summarised quite briefly.

As one would expect, raising any multiple of 30o+1 to any power, mod360, gives a multiple of 30+1 - ie 1 degree of a Sign - as its answer; and any multiple of 30 to any power mod 360 results in 0.
Immediately, by doing maths on our circle, the fiducial 0 (0 Aries) and the pulse of 1 degree of a 'Sign' every 30 are naturally emphasised.

Every sixth dimension - 1st, 7th, 13th, 19th - actually emphasises the beginning of each Sign, as the power of 1 is 1, the power of 31 is 31,that of 61 is 61, and so on (all modulo 360.)

Every fourth dimension - 1st, 5th, 9th, 13th, 17th - the natural sequence progresses by multiples of 30+1, drawing in the circle those relationships that we call Aspects; each plus an orb of 1 degree, the margin within which we know planetary and angular directions to beactive. These are the dimensions with 5-fold symmetry, the creative Star.

Every fourth dimension from the third - 3rd, 7th, 11th, 15th, 19th - the single axis of symmetry emerges from an unaesthetic linear sequence, which however still gathers its numbers into astrologically recognisable groupings across the page. One finds lines of Fire, Earth, Air and Water grand trines, Cardinal, Fixed and Mutable Grand Crosses, perfect semi-sextile or inconjunct sequences, and repeated natural pentagram degrees.

The 9th dimension elegantly combines its aspect sequence with cross-page lines of Qualities and Elements - the latter in their correct order. Even the four columns of computation yield respectively a Fire, Earth, Air and Water emphasis, converted back to the language of astrology. Nine Is quite perfect!

And the final oddity is that any multiple of 30, +6, raised to any power greater than 2, mod 360, always results in 216 - astrologically 6 Scorpio. In other words, the first multiplication giving this persistent degree is 6 x 6 x 6 . .! Does this corner of the natural pentagram have a darker significance than even its Scorpio qualities imply?
Tantalisingly, when the human form is spread on the natural pentagram - so closely approximated by Leonardo's figure - 6 Scorpio becomes the place of the left foot, and invokes another package of symbolism: the idea of Left as symbol of Wrong in its opposition to Right, the idea of mankind possessing an "Achilles Heel" which is a fatal flaw, and the image out of Genesis of humanity crushing the Serpent under his heel.

This accumulated imagery is so powerful that it is worth looking more closely at the other four points of the Creative Star. The right foot, then, would be set at 24 Leo - in the sign of the Sun... or the Son.. .archetype of the spiritual light that balances and challenges Scorpionic darkness. Both feet stand in decans of Mars; but the right is placed in Leo's Aries decan, referring back to the source degree of 0 Aries that crowns the head and symbolises the Father of whom Leo is the Son. When the pentagram is drawn, the first line is the chord of 144 degrees from Aries to Leo. The left foot is in the Scorpio decan of Scorpio, relating to no other point on the star, only to itself.

Carrying these ideas further, we find that on the right hand is 12 Gemini, the sign most closely linked to Breath and the Word, and positive like Aries and Leo. On the left hand is Capricorn 18 - a Sign like Scorpio popularly associated with images of evil or misfortune. We project our own wrongs onto the Scapegoat, sheep are to be
separated from the goats, and the devil himself, Satan/Saturn, appears horned and goat-hoofed in folk imagery. The left foot is linked to the left hand, being in the Capricorn Dwad; both right hand and right foot are in Venus Dwads, Libra and Taurus respectively, embracing justice and kindness. There is one link between Left and Right - the left hand in Capricorn's Leo Dwad suggesting that as in the book of Job, the Tempter may be constrained to do the work of God!.. working through the negative and earthy to bring out a buried light.

This can be read another way, too. The story of Genesis is told by drawing the pentagram on the zodiac circle. In this, the line Aries-Leo shows God the Father creating Adam, the first Man, from the clay of the Taurus dwad. Leo to Capricorn's Leo dwad takes the 'rib' of Adam the Man to form the Woman from the Taurus decan and lead humanity 'into temptation' through ambition and sensuality. The Capricorn-Gemini line begets the brothers Cain and Abel, and Cain, moving next between Gemini and Scorpio, experiences mankind's first jealousy and hatred and commits the first murder. But there is a last line to be drawn, from Scorpio to Aries, promising the hope of return to the Father by the path of suffering and redemption. The whole drama of Darkness and Light is contained in the natural pentagram, which must not be inverted, for then the Father is displaced, giving ascendancy to Adam and the Tempter, and the left-hand path is presented as if it were the right; Scorpio 6, 6x6x6 in the Third Dimension, usurps the place of Gemini 12, the Beast standing in the place of the Holy Spirit. This is what happens when we stand true values on their head.
This is the imagery and the message of St.John's Revelation.

It is notable that the degrees of the natural pentagram raised to any power, will only generate themselves or each other; and that the fourth dimension is ruled by Scorpio 6, the only point on the star to manifest apart from the first degree of Aries. If the fourth dimension is that of Time, then it is existence in Time that gives such
power to the dark forces, against which we men and women are pitted during incarnation after incarnation to temper and purge and prove ourselves before our return to the Father. Time is the dimension of the "Fall".
Aries 1 is pitted only 8 times against 12 of Scorpio 6. Our hell is here.

In the beginning... there is just the circle of vibration. The First Dimension, the Source of All. The Second Dimension winds the numbers into a helix, and something is created out of "nothing". In the Third Dimension, space is created and within it eternal symmetries, the primal forms and ideas; a star is born! But with the Fourth the innocence of eternal life is cast into the agonies of Time, symmetry spoiled, the fire of the spirit in the Fire Grand Trines embattled by the forces of darkness from Scorpio. But rising Into Five we enter a realm of utter harmony and beauty, a taste of heaven. Dimension Six is an enigma - in form a spread pentagon, focussing only12 of the 360 degrees, drawn asymmetrically, yet symmetrical when its 'shadow' is drawn in; repeating every 30 degrees; and emphasising not Scorpio 6 but Aries 1. Here, then, the idea of the Father, the Source,the Alpha/ Omega point, is dominant. The axis of symmetry of this imperfect pentagram is the unmarked, 'invisible', 12 Gemini point of the Word or Breath. My feeling is that this may be the dimension of judgment where the soul becomes aware of its incompleteness and must account to its Maker... before rising again to the Seventh Heaven with its delicate and beautiful vibrations, its sense of holiness of being. Beyond this plane comes another level fraught with disharmony in Eight, a domain of darkness, which leads to the utter perfection of ascended spirit in the Ninth Dimension, where Man achieves Mastery and the Christ-light shines forth as a dynamic creative force. A further hurdle to the spirit is presented in Ten; then the Eleventh Dimension fills with the choirs of angels before the throne of the Twelfth. Here, power belongs to the Father; every one of the significant Primes - 1,7,11,13,17,19,23,29 - raised to the 12th power resolves to 1. There is no pentagram here, only the First and Last Great Ideas of the Source (Aries 0/1), Mind and Matter ( Gemini 21 - Aquarius dwad - and Capricorn 10 - Taurus dwad ), the forces of Light In Leo 16 and 24, and those of Darkness in Scorpio 15 and 6.
If there is truth in these conjectures, then Twelve is the power, the dimension, of the Creator of the knowable universe, and hence a holy number. All the argument for the preservation of duodecimal measures amid the influx of decimals is entirely justified. Retaining a 12-base symbolises an adherence to the Created Order, while the spread of base-10 is symptomatic of its violation and the dethroning of the sacred by the secular.

It is in the powers of the 'significant primes' listed above that the patterns of form and harmony are principally displayed, hence the title of this piece, "Prime Movers". The study began as a hunt for the pattern of prime numbers which has become a most popular sport amongst number-crunchers of all nations. That hunt is still on, and is best left to the well-equipped; but it has led us on a most interesting and thought-provoking journey from which we may have more to learn.

MIND IN THE COSMOS

A couple of days ago I was watching, fascinated, the film 'Dangerous Knowledge', re-screened on BBC2. It concerned the obsessions of mathematicians and the consequences for their sanity.

Georg Cantor spent his life trying to get a mathematical handle on (of all things) infinity. He got to the point where he was even adding 'infinities' together and making them jump through conceptual hoops. It drove him mad.

With a south node in Aquarius I must be careful where I send my thoughts!... but it did get me pondering. A number of questions arose in my mind which may or may not make much sense, but here they are for what they are worth.

Cantor was tantalised by the number of possible points on the circumference of a circle ( and we astrologers deal with circles Great and small all the time.) Take a circle, draw radii from its centre to the edge until there is no room left on that edge - and the area will be black with lines - and you have a finite construction. But as soon as you widen that circumference, the original radii will of course fan out, leaving spaces on the new perimeter! So the number of radii in a circle is infinite, assuming the potential circumference is infinitely far away.
My question is - why start from the centre? Wherever the outer limit of any Great Circle is (and if it is Infinity, so be it, because it is beyond the conception of a finite mind) it can be completed with as many nano-nano points as it has room for, and then they can form the origin of radii to the circle's centre.
Maybe a silly thought, but let's play with ideas!

This whole conundrum arises because mankind insists on dividing things. So another question is, why assume division is appropriate? Why not try the other assumption that some phenomena function as unified Wholes?
As Astrologers we work with wholes in the form of Ideas that are grouped in twelve sections of a circular heaven. The Great Circle is the Ecliptic, and the 12 sections are what we call the Zodiac. These can be subdivided (yes!) into 144 subsets known as the Dwads; and there are a number of differing great-circular frameworks that place these Ideas in different contexts of human life and earthly existence. We also deal in Degrees - 360 of them in each circle - and in minutes - 60 to each degree; and degrees themselves carry individual meanings. But small as these are, detailed as they become, they are still Wholes, and we don't need to mess with them too much! We also work with the geometry of the circle to assess the harmony or lack of it between zodiacal Ideas that are focussed by the presence of a planet, or the Moon, or a key spatial node such as the Ascendant, the Midheaven, or the Nodes of the Moon (there are others.) But there is little to be gained in understanding by straying beyond the kind of geometries that would produce recognisable music on a taut string. So beyond 12ths or 24ths not many 'aspects' are used; 11ths are, but odd geometry is tricky to understand and work with. So why not work with Wholes elsewhere instead of constantly dividing? Maybe lines spread out into membranes? surfaces into mobius loops and toroids? Waves, particles, strings and superstrings all fall into this category, I daresay. And as astrology is more wave-like and rhythmic and musical than particulate, my feeling is that it is in this category of cosmic phenomena that the astrologer and the physicist/chemist/cosmologist will eventually meet.

And then there's Pi.
Following on from that last argument, why are folks not content with the elegant fraction, 22/7, and forever going on about Pi not resolving to a neat decimal place?
1/3 doesn't either - it's 1.3333333333333 on into good old infinity again, and yet we recognise a third of something when we see it. It doesn't have a fuzzy edge that wafts us away into hyperspace (unless you insist on examining the quantum conditions around your slice of short-cake - in which case there may be no hope for you.)
I say, let's get back to fractions. Decimals are all very handy but they do throw up this infinity problem again, as does any BaseN arithmetic. Yes it's very entertaining, and people spend their lives pushing Primes and ratios and divisions to their impossible limits - but where is understanding in all of this? (Except to recognise, forcefully, our own limitations as an incarnate family!) Look at Phi. Beautiful. Not because it's 1:1.61803...(etc) but because of the proportion itself that is built organically into so many living and wonderfully architectural things.

Somewhere in all this maths and cosmology there has to be the place where Mind Is. When you have knocked around the worls for 65 years as I and many of you have, and spent 40 or so of them as an astrologer, healer, and and general mixer with friends unseen, you know the spiritual dimensions are a reality, that they interpenetrate the apparent world, and Science will always fail until it allows for that reality in testing its own propositions. It tends on the whole not to. I've met quite a few scientists in various fields and even if they profess a belief in spirituality, or in God, I don't really see this in their work. It gets parked just outside while they go in the lab for the day to study multi-dimensional theory. One day the penny will drop, and some bright young person will get the most deserved Nobel Prize in history.

This ubiquitous Mind - consciousness - meaning - purpose - is ultimately the basis for our Astrology. The Universe is permeated with it. It is our Matrix. We are the Cinderella of Science. Look - astrologers recognised millennia ago the 'sensitive dependence on initial conditions' that is integral to modern Chaos Theory. Astrology is likewise fractal in its nature (numbers 12 and 60 being key to our work with the intelligent cosmos.) We are manifest Thought.

I have a paper that I wrote a few years back, on the properties of the Circle and muliple dimensions, that I may put up here later on if I can get the diagrams to go on the blog with it. Watch this space.

That'll do for now. Have to get tea at some point this evening!