top of page
  • Writer's pictureDr. Peter Duchemin

Occult Interlanguage – MatheMagick Squares and Emergent Esoteric Synthesis

Updated: Jun 21

As I write this article, the film Tenet, by Christopher Nolan is showing in theatres. Like its namesake, it is a fascinating palindromic enigma. I predict that, as with several films that introduced philosophical and esoteric principals into the mainstream occulture (Inception, The Matrix, Fight Club) etc, we will see the famous Sator square become something of a household name…. at least for a time.

(The Article Continues Below the Image)

“Sator Skyline” by a6rax1s

It is interesting to consider magic squares as an example of the magic of pure pattern. By definition, they assemble individual elements into a whole that is more than the sum of its parts. Their presence in religious, mystical, and esoteric practice extends into the far distant past – they seem to have exercised a fascination on the human mind from the very dawn of literacy itself. My goal in this article is to demonstrate how a philosophical occultist or artist might use magic squares to affect their frames of consciousness in order to put diverse magickal systems into contact with one another as overlapping and interteracting “fields” of meaning, contemplation, and energy.

Starting at the beginning, then – the most anciently known of all mathemagic figures is the order 3 Lo Shu Square, known to the West as the Kamea of Saturn. It has been a talisman of wholeness and power for at least 2600 years of recorded history. Though it can be oriented in several directions, or reflected, there is really only one way of arranging the consecutive numbers 1-9 on a 3×3 array to produce a consistent magical sum (of 15), across all the horizontals, verticals and diagnols.

(The Article Continues Below the Image)

“Lo Shu” by a6rax1s

This square was so fundamentally striking to the human mind that the binary Chinese oracle, the I Ching, is hosted upon its matrix, & not only that, but the I Ching’s (potentially older) ternary cousin the Tai Hsuan Ching can be mapped onto the Lo Shu point for point (for more on this intruiging system see Bill Whitcomb’s The Magician’s Companion). Legend has it that the Lo Shu was first discovered – by none other than the “Yellow Emperor” – marked on the back of a Giant Turtle.

(The Article Continues Below the Image)

This myth puts the Lo Shu symbolically at the Foundation of the world itself. In a way, it very much is this: as a synthesis of deep associational chambers, it is unparalelled in its strength and integration. Several traditions drawon it for its sheer stability and cohesion. It is the World Tortoise, it is Saturn’s Seat. It is the radiant lucency of the Chasmal (See David Chaim Smith’s The 32 Keys). Hermeticists might do well to identify it with the Ennead, and visualize it in the Qabalistic sphere of Yesod.

Inquiring more broadly, however, we can ask: what happens generally when a given system of mystical signs gets structured by the organizational frame of a mathemagical square or another similar figure? For that matter: what is a mystical system? What is a mathemagic figure?

  1. A mystical system is a set of symbols that becomes loaded with emotional, philosophical, and aesthetic meaning such that they can be accessed individually but link together into a cohesive whole. Such systems lead the mind to a sense of one-ness and interrelation and provide our awareness with quick access to a complete suite of important attentional considerations (i.e. “energies”). The symbol systems act as ‘loci’ or places of memory-association that organize human attention around a web of interconnected cultural meanings. They are meant as experiential maps for mystical and magickal experience.

  2. A mathemagic figure is a pattern of numbers or letters whose individual parts come together in such a way that an emergent property which none of the parts possesses alone is revealed. Commonly that is a magical sum, as in the Lo Shu, or a multidirectional palindromic phrase such as in the SATOR square.

Epiphanies are the result of this combination of structure and psyche. To appreciate this, first consider what happens when a person becomes intimate with a system such as say, the Hermetic Qabalah. The Hebrew letters, which may once have been simple signs begin to evoke a rich series of memories: some real, some fantasy – some attractive, some repellant. It is as if through meditation, those signs have become vibrantly synaesthetic: cathexed (a Freudian term meaning “psychicly charged”) with a vivid emotional and imaginal content that is activated whenever the reader sees a Hebrew letter. This associational “charge” extends into numbers as well as other signs, and creates for the magician an emersive and potent world of connections- invisible to the vulgar eye.

(The Article Continues Below the Ad)

Now consider what might subsequently happen to such a consciousness when the letters of the charged magickal language are put into relationships and arrangements wherein emergent mathemagical patterns appear and serve to create an aesthetic unification of the system – the appearance of a “more than the sum of it’s parts” pattern is shared between the bare mathematical matrix and the “inhabited” code of the mystical system. The beginner’s psyche will be supported by this cohesion of elements. The adept’s psyche will be triggered into epiphany.

This approach is not unlike tuning an instrument so that it may be used to evoke resonant frequencies. In this case the magical arrangement evokes a total structural harmony of the whole and the parts. A kind of synaesthetic gnosis blossoms directly out of the mathemagic matrix and scales far beyond it. Cliff Pickover calls such an experience Arithmetic Satori:

“Arithmetic satori is the psychological result and aim of the practice of magic square meditation… this practice induces an awareness, and experience of joy emanating from a mind that has transcended its earthly existence. Experience is no longer mediated through concepts… the satori experience has a paradoxical quality, such as a feeling of oneness that is inexpressible in a language posited on a subject-object dichotomy. The existence of a separate self is viewed as a fiction through the satori experience. Awareness seems to take place directly, unmediated by conscious thought, and without consciousness of the process.” (Pickover, p xv)

He has, in the same work, rigorously compared these squares to “frozen music”, noting that the pattern of magickal sums very often resembles the Chladni patterns produced through bowing a steel plate of sand with a violin bow.

(The Article Continues Below the Image)

Chladni Plate Image from William Henry Stone (1879) Elementary Lessons on Sound, Macmillan and Co., London, p. 26, fig. 12 . Now in Public Domain.

The fact that mathemagic patterns can import their ‘musical’ character into a symbol system and thus raise it to the next level has led to the following method of meta-magickal investigation:

  1. This methodology allies itself to the “field theory” of mind and magick that may be drawn from diverse sources, including in substantial part from Dr. Rupert Sheldrake’s work on “morphogenic fields”. For Sheldrake, there is a suggestion that the mind is more than an epiphenomenon of the biological brain, and may extend beyond the body as a form of influential field. Adapted into the language of magick, what we hypothesize is that our occult praxes, such as circle castings, astral projections, and sigil firings are a) effective, and b) effective because they are operating as an extended field of influence that exploits a permeable boundary between mind and body, matter and spirit. In short, there is something dream-like about reality and magick influences this (albeit dense) dream by generating fields of attentional focus. Systematic magicks constitute elaborate assemblies of diverse attentional foci, organized into a harmonized synthesis of awareness: a magickal awareness field.

According to this approach, the mathemagickal triangle, square, star or other figure becomes the guarantee, or proof, that such a field is being generated: if it exhibits the properties of frozen music, then it is potenitally productive of a potent magickal awareness field.

Lets look at some examples of how this approach can radically affect how we perceive and use specific magickal systems:

(The Article Continues Below the Image)

Magen Hive Schematics” by a6rax1s

In the above image are two versions of the same magickal hexagon arrangement. This arrangement comes from Cliff Pickover’s book (p 335). He credits Harvey Heinz as the source.In the first image, you can see the numbers 1-30 arranged in an order-3 magick hexagon with seven cells knocked out. This creates nine hexagonal “rings” (seven small ones, a middle one, and a large one) – each of six cells. Every ring has the same magic sum: 93. In the second version I have substituted the names of the 30 magickal aethers derived from the work of John Dee, and explored in depth by Aleister Crowley in Liber 418 – the Vision and the Voice. In this case, however, they are unusually arranged, and the question is: does putting the aethers into this special arrangement also then allow for the magical resonance of 93 to emerge from them as a whole, once they are “charged”? What would such a resonance mean for the magickian’s experience in exploring these aethers? What would the nine special combinations of aetheric energy, unique to this array – derived from the nine rings – signify in terms of the larger body of Enochiana?

To explore these questions I performed a series of scrying experiments in which I called the name (and governors) of the aether in meditation, and then recorded the images that came to mind. I drew these images in the 30 cells, and the result was the following map:

(The Article Continues Below the Image)

The Magen Hive – Scryer’s Map” by a6rax1s

It has taken me a little while to glean an inkling of what this channeled composition might actually mean. Clearly it is a map of the Aethers, arranged so as to generate a field of awareness that is resonant to the sum of 93 – in this, it connects to the philosophy of Thelema – a pre-modern folk tradition first recorded by Rabelais in his satire Gargantua, and fitted to the framework of Golden Dawn-style Victorian occultism by Aleister Crowley. It is well known that 93 is the isopsephic sum of both agape (universal love) and thelema (will). It is also the gematria equivalent of magen, as in “magen star” – an earlier term for what is now called the Star of David. Isopsepy and gematria are Greek and Hebrew numerology, respectively. In any event, this transformation of Enochian redirects its energies to a different kind of field – one similar to a ninefold hive that thrumbs with volitional and agapic energies – suggestive of the work of the respected Ma’atian Thelemite, Nema, founder of the Horus-Ma’at Lodge. An integrated love-will hive. I thus present this set of field maps to advanced workers as a Ma’atian synthesis of the Enochian and Thelemic material.

Another interesting example relates not (directly) to Enochian, but rather to the Solomonic arts of the Shem Ha Mephorash (ie, Angelology and Demonology).

(The Article Continues Below the Image)

Magickal Torus with Shem Ha Memphorash”

This diagram, which looks like a letter “Z” is in fact a map of the surface of a hexagonal torus, or ring, with a total of 72 triangles composing its surface. These fall quite neatly into 12 hexagons, and each of these hexagons, when “stellated,” forms into a star of David style Hexagram – a “Magen Star!” Both the stars, AND the hexagons total the magick sum 219, (which is 73×3). This image is also presented in Pickover’s magnificent book. It won’t take long for an adept to see where I am going with this: this “Ring of Solomon”, can in fact be used in conjunction with the 72 names of the Shem Ha Mephorash and its attendent angels to arrange the Goetic spirits of the grimoire Lemegetton in a kind of oroboroic vessel that teases a harmonic overtone out of their collective assembly! This discovery has a lot of nuances that are beyond the scope of this article, but it should be clear that there is something very potent about this methodological approach – it suggests an advanced psychic science of shadow integration, among other things.

(The Article Continues Below the Image)

The Ring of Virtues Assembles the Goetia” by a6rax1s

Consider what is happening— in all these cases we are taking a whole system, and arranging it so that it creates an emergent mathemagic overtone insofar as its elements have been arranged into a particular shape, and exhibit a consistent magickal sum. Hypothesis: This mathemagick figure contributes to and emits a field of influence that conditions time by means of virtually framing events. In other words, the shape now expresses the wholeness of the system as it has been so magickally arranged, like an energy transformer that assembles the system into a radiance of awareness-energy of a particular character and flavor. In the Solomon example, the 72 malevolent spirits are equalized within the magickal torus, and become part of a much larger, and more integrated picture. It is as if the noise of the Goetic “Howling” were transformed, in the toroidal vessel, into a chorus – the “lead” of dissonance, into a “gold” of resonance.

Such a mathemagickal field array does something else as well: it allows multi-layer integrations to be achieved between the systems that have been so shaped. Take the two above examples. Since we have created a torus out of the Solomonic energies – we can now also “fit” the magen-hive hexagons overtop the 12 ‘plates’ of said torus, like floating islands, or like scales, that now clothe it in a second layer of compressed “mathemagicized” energy compositions. One can imagine an energy exchange occuring between the two frozen musics – the magen hive and the solomonic oroboros, developing ultimately into a higher layer synthesis not unlike the symbiosis between mitochondria and cell – two independent organisms achieving an arrangement of benefit to both.

So to recap: We have brought the Enochian and Solomonic systems into strikingly elegant synthesis around this mathemagickal ‘ring’. A ring of virtues. A Torus-Hive. This in itself is a rather significant achievment, but it does not stop at this example. In fact, there is no real limit to how diverse systems might be combined structurally on the level of their fields using this method, and in effect, we are presenting a novel mode of performing syncretism between magickal systems that preserves each system’s intrinsic wholeness.

We are building a metamagickal inter-language.

I have personally applied this approach to several systems. So far it has proven fruitful for the Runes, the Oghams, the I Ching, the Qabalah, etc… Any set of symbols that can be counted serially and enumerated exhaustively may be brought into an elegant topological/geometric synthesis with any other through these techniques. In brief, a detailed exploration of magickal figures in as many dimensions as possible and in as many shapes as concievable will lead, I believe, to a renaissance in terms of how magickal and occult systems are approached, what they mean, and how they can be put into communication with one another. I invite you to think of other applications of this meta-magick, and to consider how we might find various suitable ways of achieving a mathemagickal “speaking-together” of as many arcana as possible.

Works Cited:

  1. Pickover, Clifford A. The Zen of Magic Squares, Circles, and Stars: An exhibition of Surprising Structures across Dimensions; Princeton U.P. ; Princeton 2002.

  2. Stone, William Henry Elementary Lessons on Sound; Macmillan and Co., London, 1879.

#esoteric #sator #magick #square #magic #occult

4 views0 comments

Recent Posts

See All


bottom of page