THE HALLS OF WISDOM

       This painting is part of a much larger work called The Upper Aeons - the first painting in my Apocryphon Chapel project. It is, in fact, the background of the lower half, and much of it is now covered over by allegorical figures. But, I decided to photograph it and present it as a work on its own.
      It was painted in Vienna (2018 - 2020) while writing my book Sacred Codes and serving as director of The Academy of Visionary Art. During that time, I had a lot of insights into the relationship of Sacred Geometry to Harmony and Composition, which went into both the book and this painting. 
     For those who are interested, they may consult the related chapter of Sacred Codes online at https://www.academia.edu. Below, I'll endeavour to break down the images on the columns, explaining the relationship of Sacred Geometry to Harmony and Composition.

MIRRORED CUBES

       After meditating for some time on the idea of the Cosmos as a Mirrored Sphere, I realized that, in the beginning, the Divine Eye created a limited number of perfect polyhedra within that sphere - mainly, the Five Platonic Solids. These appear in the sacred geometry surrounding the Divine Eye above. 
     I painted each of the Platonic Solids as if it were made of pure light with an intense spectral glow. Each shape is shining from within and surrounded by a circle. After meditating on these images, I came to see the bounding circles as mirrored spheres and that each Platonic Solid, when seen from within, is also a mirrored shape. In other words, the first shapes to appear in the cosmos, after the Mirrored Sphere, would be the Mirrored Tetrahedron, the Mirrored Octahedron, the Mirrored Dodecahedron, etc. And - strangest of all - the Mirrored Cube... In The Halls of Wisdom, each of the images on the columns is, in fact, engraved on a mirrored cube...

INTRODUCTION:
SEEING IN THE MIRRORED SPHERE

     Before Vienna, I lived in Bourgogne France for three years. In 2012, I began writing Sacred Codes while also working on a large drawing called Vishnu-Christ Avatar (right). As I was drawing, I kept noticing a recurring geometrical shape behind its ever-evolving imagery which - towards the end - I drew quite consciously at the top and bottom of the work. 
     At the bottom, I drew a squared circle with a triangle set into the base (see first detail right). Since that underlying shape had become the armature for so many of its images, I called it my Trimorphic Matrix Subocularis.
     At the top of the drawing, I created a design based on that shape, which I called The Archetype of the Divine Eye (see second detail right). In my meditations, it became the Divine Eye which creates the cosmos through vision. I became very interested in the idea that, for the Visionary artist or creator, vision is active rather than passive - that light is projected from the eye. So, in the case of Visionary Seeing - the eye actively creates what it sees.

     During the seven years that I lived in Vienna (2013 - 2020), I developed this idea further, realizing that the Cosmos is a Mirrored Sphere in which the Divine Eye sees and creates everything around it (see the graphic representation, bottom right).
     In my reading of a Gnostic text called The Apocryphon of John, I received confirmation of this idea. In their visions, the first century Gnostics saw Divinity as One and so the entire creation is a unified whole. This Divine Source creates the Cosmos through vision... 
     In The Apocryphon of John, the creation begins when Divinity thinks upon itself, and thus sees an image of itself (the Father), which then expands into two (the Mother and Father), followed by the three (the Trinity) and so on (see the detail from my Upper Aeons painting below. Progress shots of this painting may be found in The Chapel section of this website).
     Hence, Divinity created the All by beholding countless images of itself in the Mirrored Sphere of the Cosmos (...an idea I continue to explore in my autobiographical novel Reflections in the Mirrored Sphere).
     

THE MEASURES OF 2D SPACE

      The images on the columns portray the first perfectly-harmonic shapes that the Divine Eye envisions and creates. In Sacred Geometry, all operations are made with a compass and straight rule. Proverbs 8:27 describes the creation as a process wherein the Divine Craftsman  'traced a circle over the face of the abyss.'  Hence, the All is created Ad Circulum - 'from the circle.' For the next step, the Divine Architect would have used either a compass to create another circle, or a straight rule to cut the first circle with a line.

     If the first circle was cut by another circle, then six more circles would naturally follow to create the Seed of Life. This shape becomes the matrix for the Triangle, and hence, the Ad Triangulum measure of space.
     If the first circle was cut by a line, then four more circles would naturally follow to create the Quatrefoil. This shape becomes the matrix for the Square, and hence, the Ad Quadratum measure of space.

FROM THE TRIANGLE

       When the Divine Eye first projected its luminous vision into the cosmos, it also created the measures of space. In the Seed of Life, the vision projecting outward from the Divine Eye measures 2D space in triangles - an infinite grid of triangles that we call Ad Triangulum space ('from the triangle').

FROM THE SQUARE

       But, in the Quatrefoil, the vision projecting from the Divine Eye is measured in squares - an infinite grid of squares that we call Ad Quadratum space ('from the square') 

THE OUTERMOST COLUMNS

       In The Halls of Wisdom, these fundamental ideas appear in the outermost columns. On the column to the left, space is measured out Ad Quadratum while, on the column to the right, it is measured out Ad Triangulum.
     However, these columns are not just ordinary columns. I painted them as collections of mirrored cubes. Each cube is an idea in the Divine Mind. It has a divine spark within, projecting its light onto the surface of the cube, where its idea appears in image-form.

     Personally, I see these mirrored cubes as partly-transparent, partly-opaque. They are intended to be viewed from both within and without...

THE PYTHAGOREAN-PLATONIC SYSTEM OF HARMONY

      The inner columns manifest the Divine Mind's first ideas on Harmony.

       Having established two basic measures of space, the Divine Architect establishes Harmony within the Cosmos. Plato describes this in The Timaeus, where the Demiurge creates the Cosmos step-by-step in a harmonic and orderly progression. Plato's system was based on Pythagorean ideas, and their combined system is now generally called the Pythagorean-Platonic System of Harmony. 

      Basically, Pythagoras established that the open string of a monochord, when it is plucked, establishes the fundamental note. When it is pressed half-way, then the octave is heard - the most harmonious interval between two notes, called the 1:2 Diapason. When the monochord string is pressed two-thirds of the way, then the second-most harmonious interval is heard, called the 2:3 Diapente. Finally, when the monochord string is pressed three-quarters of the way, then the third-most harmonious interval is heard, called the 3:4 Diatessaron.

     For painters, this same system of harmony can be seen in space - particularly the space of their compositions. Raphael was aware of this and depicted Pythagoras in The School of Athens - kneeling and taking notes from a tablet which an angelic child is holding before him. The Pythagoras Tablet is a graphic representation of a four-stringed lyre (or tetrachord) showing the relationship between the 1:2 Diapason, the 2:3 Diapente and the 3:4 Diatessaron  (explained below).

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     All Academically-trained artists, such as Gustave Moreau, knew this system of harmony. Through a series of diagonals (which I explain in Chapter 13 of Sacred Codes), the harmonic points of the 1:2 Diapason, 2:3 Diapente and 3:4 Diatessaron can be established on any canvas. The painter can then treat his canvas like a piece of music, melodically moving the viewer's eye around the composition by placing figures on harmonic points. We can see this in Moreau's compositional sketch below, where he intuitively places Pasiphae's bowed head and turned knee on the 3:4 Diatessaron points.  

HARMONY IN THE RIGHT INNER COLUMNS

     The inner columns consist of another series of mirrored cubes where ideas are expressed in images. The columns on the right express Harmony through Ad Triangulum measures while those on the left express the same Harmony through Ad Quadratum measures. 

     Beginning with the inner column on the right, we see three representations of the same harmonic idea. At the top (first image below) there is a mirrored cube with the Tetractys, a design created by Pythagoras to expresses his fundamental insight that the All proceeds from the One in a series of harmonious ratios, moving from the 1:2 to the 2:3 to the 3:4. Or, more simply, as 1:2:3:4.

     In the middle of the column (second image below), there is another mirrored cube which shows the geometry underlying the Tetractys. The ten small circles of the Tetractys are, in fact, constructed in Ad Triangulum space with the aid of two opposing triangles in the hexagram.
     At the bottom of the column is the final mirrored cube (third image below), which depicts Plato's Lambda, an idea he develops in The Timaeus to translate Pythagoras' Tetractys numerically into ratios and progressions. It show how the first androgynous number (1) divides itself into the first female number (2) and the first male number (3). On the left side of the numerical triangle, the 1:2 ratio then progresses through a series of even ratios, as 2:4, then 4:8 and so on. On the right side of the numerical triangle, the 1:3 ratio makes a similar progression through odd ratios, as 3:9, then 9:27 and so on.
     However, the numbers in the middle of the triangle (6, 12 and 18) become 'the arithmetic and harmonic means' through which the male and female series of ratios may be re-united. Now, the male & female 2:3 Diapente appears (horizontally) and it progresses as 4:6, 8:12 and so on. The male & female 3:4 Diatessaron also appears (diagonally) and it progresses as 6:8, 9:12 and so on (more on this below).

HARMONY IN THE LEFT INNER COLUMNS

     The inner columns on the left express Harmony through Ad Quadratum measures.
     Beginning at the bottom of the left inner column (the first image below), I have painted a mirrored cube with a familiar design - the squared circle with a triangle set into its base. As I mentioned above, this three-fold shape became the underlying armature for many of my images. Indeed, it appeared so often that I called it my Trimorphic Matrix Subocularis. After performing a series of meditations on this three-fold shape, it revealed to me how the Divine Architect was able to create Harmony in the cosmos through its design - which is fundamentally Ad Quadratum because the triangle is set into the base of the square, and not set into the circle, which is the case in Ad Triangulum space.
     In the middle of the left inner column (the second image below), we see another mirrored cube with the same design, but now inscribed with a series of interpenetrating circles.
     Finally, in the top mirrored cube (the third image below), I have put in the numerical notations that reveal how the middle image is, in fact, a more-expanded representation of Pythagorean-Platonic System of Harmony (explained below). 

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THE GEOMETRY OF HARMONY

       The first image below reproduces, once again, the design on Pythagoras' Tablet from Raphael's School of Athens. Now we can begin to understand that the ten "I" shapes at the bottom of this design depict the Pythagorean Tetractys using I-shapes instead of small circles. Nevertheless, the same relationship holds - of Divinity creating the Cosmos in a succession of harmonic ratios, as 1:2:3:4. 
     Pythagoras' Tablet has a U-shaped design which shows the proportional unity of the 1:2 Diapason to the 2:3 Diapente and to the 3:4 Diatessaron. Strictly speaking, the relationship of one ratio to another is called a proportion. In the U-shaped design, the proportions between all three ratios (1:2, 2:3 and 3:4) are worked out mathematically. We see this with the numbers at the top of each line: 6, 8, 9, 12. 

     A ratio may be written as a fraction, so that 1:2, for example becomes 1/2. To relate the three ratios (or fractions) to each other, we need to expand their bottom denominators to 12. So, the 1:2 Diapason becomes 6:12; the 2:3 Diapente becomes 8:12; and the 3:4 Diatessaron becomes 9:12.  The numbers at the top should therefore be read as fractions of 12 - the 6/12 Diapason, the 8/12 Diapente, and the 9/12 Diatessaron as related to the 12/12 (the whole number 1, the fundamental note or complete string length of the monochord).

     Let us not forget that the Pythagorean Tablet depicts the four-stringed lyre or tetrachord. The string on the right (12) is the fundamental note or complete string length, while the other three strings (6, 8, 9) are the ones that ring harmoniously with it and (as we shall see) with each other...

     In the second diagram, I have reproduced Pythagoras' Tablet, except now the vertical lines representing the tetrachord strings have been stretched out to their true lengths (on a lyre or tetrachord, they are wound to different tensions, but roughly the same length). This allows for a visual comparison among the various string-lengths, and how those string lengths apply to a painter's composition, such as Moreau's Pasiphae above. 

     Returning to Pythagoras' Tablet on the left, there is a U-shape near the bottom marked 1/2 Diapason. It shows the Diapason interval between the 6-string and the 12-string. That relationship of 1:2 is represented more clearly in my middle diagram, where we can clearly see that the 6-string is half the length of the 12-string .

     In the middle of Pythagoras' Tablet, there are two U-shaped designs marked 2/3 Diapente. These two U-shapes show that the 2:3 Diapente occurs twice in the tetrachord: first in the interval of the 6-string to the 9-string, and again in the interval of the 8-string to the 12-string. In my diagram, we can see how the 6-string is two-thirds the length of the 9-string; and how the 8-string is two-thirds the length of the 12-string.

     Finally, near the top of Pythagoras' Tablet, we find two more narrow U-shapes marked 3/4 Diatessaron. These two U-shapes show that the 3:4 Diatessaron also occurs twice in the tetrachord: first in the interval of the 6-string to the 8-string, and again in the interval of the 9-string to the 12-string. In my diagram, we can see how the 6-string is three-quarters the length of the 8-string; and how the 9-string is three-quarters the length of the 12-string. 

     In this way, the various strings are not only harmonious with the fundamental note (the 12-string) but with each other. And the same is true with harmonic ratios in painting - each figure placed on one of those points will establish a harmonic relationship with all the other points.

       Finally we come to the diagram on the top right, which is my own invention. It shows the geometry underlying all these numerical relationships. Basically, I have translated the design on Pythagoras' Tablet into circles, then inscribed them in my basic armature, the Trimorphic Matrix Subocularis (directly below it) to show their fundamental relationship. 

     Hence, in my Geometric Model of Harmony, the largest circle is like the largest U-shape in Pythagoras' Tablet. At the top, it is marked Diapason = 1:2 and at the bottom 6:12. It represents the Diapason harmony, but now as one circle expanding outward from the centre point. The ratio of the radius to the diameter is the 1:2 Diapason or 6:12. We could also say that the centre of the circle is harmonically located in the middle, which is 1/2 the distance of the circle as a whole.
     Next are the two large intersecting circles marked Diapente = 2:3 at the top, with the expanded ratios at the bottom as 6:9 = 2:3 and 8:12 = 2:3. In my Geometric Model of Harmony, the left and right circles each constitute two-thirds of the greater circle - the Diapente harmonic. From the centre, the left circle moves harmonically two-thirds to the left, while the right circle moves harmonically two-thirds to the right. 

     Finally, three small intersecting circles are marked Diatessaron = 3:4 at the top, with its expanded ratios at the bottom: 6:8 = 3:4 and 9:12 = 3:4. In this case, the centres of the left and right smaller circles mark three-quarters of the greater circle - the Diatessaron harmonic. From the centre, the centre-points of the left and right circles move harmonically three-quarters to the left and right.

     When all these circles are set into my armature (my Trimorphic Matrix Subocularis), a new geometric relationship arises. The apex of the triangle meets at the point where the two Diapente circles intersect. I have marked this as TONUS 8:9 because that is the point from which Pythagorean Harmony expands beyond the first three harmonic ratios.

     The mathematics is a bit complex, but basically, the 1st century Neo-Pythagorean philosopher named Nichomachus of Gerasa used Plato's Lambda to decrypt Pythagoras' original calculations. The 1:2 Diapason minus the 2:3 Diapente leaves behind the 3:4 Diatessaron (2/1 - 3/2 = 4/3). Continuing in this direction, the 2:3 Diapente minus the 3:4 Diatessaron leaves behind the 8:9 Tonus or whole note (3/2 - 4/3 = 9/8).
     Once the whole note is established, Pythagoras was able to calculate the next harmonic notes on the Pythagorean Scale, using the same method (4/3 - 9/8 = 32/27 etc). However, the intervals between the notes become less and less harmonic, the further they move away from the octave (causing the Pythagorean Scale to be modified in the 1600s to Just Tuning and Equal Temperament). Nevertheless, the 8:9 ratio at the apex of the triangle establishes the whole note for Pythagorean Tuning.  

(For further research into Pythagorean-Platonic Harmony, I recommend Ernest F. McCain's Pythagorean Plato: Prelude to the Song Itself - 1984; and Joscelyn Godwin's The Harmony of the Spheres - 1992)

THE CAPITALS AND FLOOR

       Last of all, I would like to explain why the tops of the columns are reverberating in space.
     On the capitals, we can see that I've inscribed the Roman numerals for 6, 8, 9 and 12. Using the top cube as the base measure up to 6, the capital ascends in measures of 8, 9 and 12 - the Pythagorean-Platonic ratios. These same measures then echo into the heavens, reverberating through space. 

     On the floor of the Halls of Wisdom, I have engraved a tessellated dodecagon because it divides the perfect circle into twelve facets. The number twelve, which brings the 1:2 Diapason, 2:3 Diapente and 3:4 Diatessaron into one accord, is used throughout the painting. Indeed, it is like a secret cypher that I use in all my works...