CURA : Home

Astral Matrix III : Matrix Geometry
by Patrice Guinard

9. Generalisation of the Rulers to the Dominion and the Cyclade
10. Matrix Geometry

Equipotence and Harmony of Structures (Astral Matrix I)
Logical Foundations of Rulerships (Astral Matrix II)

This text brings together the rearranged chapters 52 and 53 of my PhD thesis (1993). The second part, about geometry and planetary modelling exercises,
is an attempt to picture the astral Matrix under various orientations, while remaining open to new arrangements.


"The divine quaternary is the universal agent of all these marvels : it changes, it takes all shapes to fill all the voids ; but it keeps its immortal character forever." (Louis-Claude de Saint-Martin)

In addition to the disparate systems of "Houses" or Domiciles (oikoi) of Egyptian origin, and of Exaltations (hypsomata) of Babylonian origin, the Greeks were using various other modes of allocating the planets in the zodiac, which are of dubious practical use : for instance the Triplicities (trigona), the Borders or Terms (oria), which existed in an Egyptian form and a Chaldean form, to which Ptolemais added his own (Tetrabiblos, I 20-21), as well as Valens (Bouché-Leclercq, p.206-215), and the Decans (dekanoi) of Egyptian origin, transposing an astronomical tracking method organised according to the rising of stars and of constellations in the 36 decades of 10 days during the Egyptian civil year (around 2000 BC), which appeared in a meridional belt more or less parallel to the ecliptic (Bouché-Leclercq, 1899, p.232-233 ; van der Waerden, 1974, p.14-29 ; Neugebauer 1975, p.561). Here the astrological syncretism reaches its limits, and it borders on aberration and ridiculousness. The original idea of Houses turned into the fabrication of a plethora of artificial numerological correspondences. Bouché-Leclercq entertained himself with these in his time.

Because the matter of the Rulers is not so much the empirical observation of similarities between signs and planers, but rather the global isomorphism between groups, and first the planetary and the zodiac groups. More generally a single astral Matrix exists, divided into four structures, according to four conditional fields : energy and planets, space and houses, time and aspects, figures and cycles, structure and zodiac signs. Four frames for a single Matrix. By Rulers I mean the equipotence relationships of the frames, and, following, of the transversal equivalence of elements of each structure : Planetary, Dominion, Cyclade and Zodiac, which divide the same object, the astral Matrix (image of the psychic human potential), through its own mode. Uranian unification on the energetic frame equates Taurus assimilation on the structural frame and nightly objectivation on the spatial frame: as a first approximation, we are thus speaking of the same pattern through three different modes of the Matrix.

In this way, the concept of Rulers applies as well to the Dominion and even to the Cyclade: Individuation corresponds to the springtime zodiac signs, Alligation to the summertime signs, Participation to the autumn signs, Objectivation to the wintertime signs, and naturally the diurnal sectors to the diurnal (masculine) signs and the nocturnal sectors to the nocturnal (feminine) signs, as shown in the table below:

Individuation+ 3
Individuation- 8 Fame
Individuation+ 3 Situation
Alligation- 5
Alligation+ 2 Friendship
Alligation- 5 Couple
Participation+ 4
Participation- 7 Mystery
Participation+ 4 Harmony
Objectivation- 6 Knowledge
Objectivation+ 1 Communication
Objectivation- 6 Knowledge
The Dominion, PG 1999

The diurnal houses are situated above the horizon, the nocturnal houses below. The Angles (Houses 3, 5, 7, 1) as well as the intermediate houses (8, 2, 4, 6) follow the sequence Individuation, Alligation, Participation, Objectivation, which define the degree of openness of perception (narrower in Individuation and Alligation, wider in Participation and Objectivation). The Houses as well as the zodiac signs and the planets, are opposed in pairs, the diurnal to the nocturnal, individuation to objectivation, allegation to participation, and more precisely the Houses 1and 8, 2 and 7, 3 and 6, 4 and 5.

A planet which is dignified in a given house (e.g. the Sun in House 2) is debilitated in its opposite house (the Sun in House 7). The Northern part of the Dominion brings together the diurnal planets (linked to excitation) , the Southern part brings together the nocturnal planets (linked to inhibition). The Western part of the Dominion brings together the idealist planets (the Sun, Mercury, Uranus, Pluto), associated to fixed signs, the Eastern part brings together the realistic planets, associated to the other signs (called cardinal and mutable signs). The scheme also shows the planetary hours of the day (real solar hour) : the mercurial hour from 4:30 till 7:30, The solar hour from 7:30 till 10:30, the jupiterian hour from 10:30 till noon, the Martian hour from noon till 13:30, the neptunian hour from 13:30 till 15:00, the chironian hour from 15:00 till 16:30, the moon hour from 16;:30 till 18:00, the Venusian hour from 18:00 till19:30, the ceresian hour from 19:30 till 21:00, the saturnine hour from 21:00 till 22:30, the plutonian hour from 22:30 till 01:30, and the uranian hour from 01:30 till 04:30.
Houses Rulers, Planetary Pairs and Dominion, CURA 2015

The twelve planets are arranged in a star-shaped figure according to the order of the Houses: Houses 1 and 2 Northwest, 3 to the North, 4 Northeast, 5 Southeast, 6 to the South, 7 and 8 Southwest (see left figure below). The figure is symmetrical along the horizontal axis: house 1 and Mercury above the horizon are opposed to House 8 and Uranus below the horizon. The planets are also opposed in pairs along a central symmetry : reported in the zodiac according to their domicile, the pairs of planets are arranged symmetrically as shown in the figure on the right.

Planetary Star (Order of the Houses), PG, CURA 2015
Planetary Star Couples, zodiacal projection, PG, CURA 2015

The organisation of the Cyclade, or the structuration of the Aspects and Figures, makes a similar figure possible. Let us propose the following equivalence (keeping only the main and median aspects, which support the harmonic figures):

House 1, Ascendant, conjunction 0°
House 2, Semi-sextile and sextile, 30° and 60°
House 3, Midheaven, square 90°
House 4, Trine and quincunx, 120° and 150°
House 5, Descendant, opposition 180°
House 6, Bi-quintile et sesqui-square, 144 and 135°
House 7, Imum Coeli, tri-decile and quintile, 108° and 72°
House 8, Semi-square and decile, 45° and 36°

or alternatively, when grouping the minor aspects, the Aspects Compass:

DominionGreek Houses
Objectivation+ 1 Communication The Horoscope conjunction 0°
Alligation+ 2 Friendship The Good Daimon sextile 60°
Individuation+ 3 Situation Midheaven square 90°
Participation+ 4 Harmony
The God trine 120°
Alligation- 5 Couple The Setting opposition 180°
Objectivation- 6 KnowledgeThe Bad Fortune 150°, 144° et 135°
Participation- 7 MysteryThe Underground 108° et 72°
Individuation- 8 FameThe Gate of Hades 45°, 36° et 30°

The Compass of Aspects. Dominion and Cyclade (houses and aspects)


"In a given field, the first "phase" of structuration consists in recognising four poles divided into two antagonist pairs, which engage the dialectic movement by two rotations in opposite direction."
(Raymond Abellio, La structure absolue, 1965, p.21-22)

The Wheel of the Rulers, shaped in four circles of Correspondences, brings together Signs, Planets and Houses according to a specific arrangement which brings forward the dual character of the Zodiac, the Planetary and the Dominion. The figure is globally symmetrical along the horizontal axis. The Northern hemicycle brings together, from the centre, the diurnal houses in a sequence according to their degree of openness (from Individuation to Objectivation), the diurnal signs in a sequence according to their zodiacal order, then their domiciles and the exaltations in the outer semicircle. The Southern hemisphere brings together the nocturnal houses according to the same sequence, the nocturnal signs in the zodiacal order, their domicile and exaltations. The horizontal axis is the axis of symmetry opposing complementary houses, signs, and planets.
Wheel of Rulers. Correspondences Houses Signs Planetary Domiciles and Exaltations, CURA 2015

Other planetary pairs result from the planetary organisation and its zodiacal distribution, in such way that each so-called active or positive planet is opposed to each of the so-called passive or negative planets, and reciprocally.

Pairs 1
Pairs 2
Pairs 3
Pairs 4

These planetary pairs, redistributed according to the Domiciles, are arranged in the zodiac according to the following figures, all four symmetrical along the equinoctial axis :

Planetary Pairs and Zodiac

The four groups of planetary pairs, positioned according to their Domicile in the Dominion, admit the following symmetrical distributions :

Planetary Pairs and Dominion

A denary model of the Planetary is found in the Sepher Yetsira. The Sephiroth could designate the ten planets including Pluto excepting Ceres and Chiron : Keter, Hhokmah and Bînah being the three "invisible" planets (the trans-Saturnian), unknown at the time of when the text was written, and the other Sephiroth being the planets of the Septenary (although the modern hermetic tendency is to associate the nine planets excluding Pluto to the nine superior Sephiroth of the Sephiroth tree, and the Earth to Malkhouth, the Kingdom). Carlos Suarès attributes to all the Sephiroth the energetic qualities of the PLanetary: "As to the Sephiroth, we have compared them to energy transformers, which define the "principles" ruling the ten hierarchic degrees of structuration of this energy." (in Le Sefer Yetzirah : Le livre de la Structuration, Genève, Mont-Blanc, 1968, p.50).

In the last chapter of the kabbalistic treatise, specific directions are attributed to the zodiac signs: "East-North, East-South, East-Low, North-High, North-Low, West-South, West-North, West-High, West-Low, South-High, South-Low" (Sefer Yetzirah, éd. Suarès, p.112). In this way the zodiac can be deployed "in space". I reconstituted this tridimensional model in the scheme where the zodiac signs are grouped according to three planes: above, the plane of spring excitation (Aries/Gemini) and autumn excitation (Libra/Sagittarius), in the middle and in grey the fixed signs characterised by positive induction (Leo/Aquarius) and negative induction (Taurus/Scorpio), and below, the plane of summer inhibition (Cancer/Virgo) and winter inhibition (Capricorn/Pisces). Each sign is linked to the following one/ Leo and Aquarius appear at the apex of ascending triangles, Taurus and Scorpio at the apex of descending triangles.

ZODIACAL CUBE. Sefer Yetzirah ZODIACAL CUBE. Nested quarters

Other cubic arrangements are possible, where the Houses are located at the eight corners and the zodiac signs or planets at the twelve edges, as shown in the second cube above (zodiac quarts are hatched). Planetary-sectorial arrangements can also be envisaged on the eight corners and the twelve edges of the tetrahedral star. One can also imagine a projection of the twelve planets on two parallel circles of a sphere : arranged according to their zodiacal domicile, they are opposed in pairs along a vertical plane dividing the sphere in two halves and aligning the planets on three planes: above, the equinoctial planets ( linked to equinoctial signs), in the centre, the planets of the fixed signs, and below, the solsticial planets (see figure below). On the icosahedron, the planetary pairs are opposed through the centre, and the Northern and Southern hemispheres regroup the positive or negative planets around a double centre (Sun/Pluto, extremities of the Solar system (see figure below).

Planetary Sphere, zodiacal order, PG CURA 2015 Planetary icosahedron, PG CURA 2015

The icosahedron delimits a positive zone to the North and a negative zone to the South, and otherwise five zones of 2, 3, 4, 5 and 6 triangles: The planets are arranged on a ball covered by three orthogonal circles delimiting six intersections and eight triangles. Each of the 12 planetary operators is placed on a quarter circle.
On the black circle, going to the right: VEN, CER, NEP, MAR ; on the blue circle, going up: MOON, SAT, CHI, JUP ; on the red circle, going up: the induction planets MER, PLU, URA, SUN. The positive planets are on a kind of bidental fork with curved handle Uranus Pluto, and the negative on a fork interlocked with the latter, with handle Mercury Sun.

Each planet is opposed to its complementary planet. Planetary triplets are located at the center of the eight triangles inscribed within the ball, as follows:
- the spring triplet : URA, MAR, JUP
- the winter triplet: MER, CER, SAT
- the triplet of the fast planets : MOON, MER, VEN (at the center of the figure)
- the triplet of the slow planets: CHI, URA, NEP
- the triplet of the "beneficial" planets : JUP, VEN, SUN
- the triplet of the "maleficent" planets: SAT, NEP, PLU
- the "solar" triplet : SUN, CER, CHI
- the "lunar" triplet : PLU, MAR, MOON

Planetary Bowl, PG CURA

The Matrix organisation and thus the distribution of the psychic potential can be comprehended along various planes : the Tetrade, the Ogdoade (Yi King), the Decade (Sephiroth tree), or the Dodecade - all perspectives of the astral Matrix which through its planetary, sectorial and zodiacal infrastructures, is nothing else than the field of attribution of the psychic dynamic. The relationships (Rulers) between these infrastructures allow the coordination of their astrological and "eso-astrological" models, whose function is to order and to symbolise the psychic operators. In short, there is no "absolute structure" but a system of correspondences fecundating the structural process in its modelling possibilities.

The Matrix is one, is global - or she is not. The theory of Rulers forbids the random manipulation of unrelated symbols, with empirical and approximate content, or linked to arbitrary mythological equivalences. It founds the coherence of the astrological discourse and the logical unification of its infrastructures (Zodiac, Panetary, Dominion, Cyclade) by a series of internal equivalences. A cyclic dynamic is inherent to the various operators : one is Venusian at the age of 4 months, Jupiterian at the age of 7 years, Uranian at the age of 35 (according to the planetary Ages linked to the sidereal revolution of the planets), but Taurus in May, Capricorn end of December, and in "diurnal individuation" at noon and in "nocturnal participation" at midnight. These cycles are universal.

Zodiacal forms, planetary forces, sectorial locations, and cyclical moments are related to a single Matrix, according to the differentiation modes specific to each of the four conditional areas (Structure, Energy, Space, Time). Each psychic-astral operator can be related to its associated operators. The Exaltation of a planet in a zodiac sign is a modulation of the main attribution (the Domicile) and makes this sign a vector of crystallisation of the planetary pair. In this way the "meanings" of the Signs, Planets, and Houses are comprehended through their correlations, in their mutual interdependence.

These astral infrastructures are organised according to the logic of the first four natural numbers. Through the Rulers, the relationships between the Triad and the Tetrade are assigned: the planets are organised in ternaries within the zodiacal and sectorial quarts. In particular, every zodiacal quarts is associated to three planetary functions. The Tetrad qualifies the Zodiac, the Triad the Planetary (and its three intrinsic levels of perception), the Dyad the Dominion (with the terrestrial horizon as limit of the planetary diurnal and nocturnal arches, the Monad the Cylade (the temporal moment being a zodiacal, planetary and sectorial snapshot).

The set of astrological factors (with 8 houses, 10 planets, and 12 signs) could have been perceived by the gnostic Valentin at the onset of the 2nd century AD, at a time when the Greek model of the Eight Houses had not yet been completely forgotten or concealed: "At the beginning, two eons have produced two more, then the second produced another pair, and so forth until the total of eight. The fifth and the sixth then produced ten eons, the seventh and the eighth produced twelve. The total : 8 + 10 + 12 = 30" (Robert Grant, The gnosis and the Christian origins, transl. Jeanne-Henri Marrou, Paris, Le Seuil, 1964, p.49-50). The first eight eons designate the astral houses: the Participation houses would be related to the initial Monad, being the dynamics of the primordial unity et the possibilities of merging the being to the Being, the Alligation houses to the intrinsic polarity of each phenomenon with its double positive and negative aspect, the Individuation houses to the Triad intrinsic to the Planetary, the Objectivation houses to the final Tetrad, foundation of the Zodiac.

Signs, planets, houses and cycles, unequally distributed in the birth chart, show for each individual, for each organism, its own psychic-astral perspective, which reflects on the relationships it maintains with its environment. The zodiac signs translate the reactivity modes of the organism, characterised by the mobility or the slowness of the nervous processes in their specific phases. The planets translate the perception and decomposition modes of the external world according to the energetic variations of excitability. Astral houses translate the integration modes of the organism into its environment, thus its existential rooting or releasing situations. Finally the cyclical operators translate the temporal progress, historicity and evolution of these operators.

Reference of the page:
Astral Matrix III : Matrix Geometry
Translation of 'Géométries matricielles'
French version :
All rights reserved © 2017 Patrice Guinard

Centre Universitaire de Recherche en Astrologie
Web site Designer & Editor: Patrice Guinard
© 1999-2017 Dr. Patrice Guinard