Botticelli & the Spiral

Cestello Annunciation, 1489–90, Sandro Botticelli, Geometry: 2021, DS.

 

Above is a Christmas greeting card of sorts, composed of Sandro Botticelli's enigmatic Annunciation* and my own minor embellishment (meant to dazzle you, of course). And, while I'm not saying Botticelli actually used the pentagonal Golden Spiral to design his masterpiece, it does somewhat explain Mary's unusual posture.  She seems to be holding the geometrical apparatus up with both hands!

It doesn't explain the most important relationship in the image, however... that is, the odd relationship between Gabriel's and Mary's hands... but, perhaps an additional spiral might tell us something.

 

Put the two together and we have this:

 

Note the diagonal the two triangles have in common; when extended, it pierces Mary's hand. It's as if Gabriel was throwing her a curve-ball and she caught it!

Hmmm... but is that the end of the analysis....?

Alas, no...

The Restoration of Symmetry: The Philosopher's Stone

The illustration for Michael Maier's 21st alchemical Emblem
from Atalanta Fugiens, by Swiss engraver, Matthäus Merian, 1617.
(All images in this post can be clicked for their original, larger size.)

The Philosopher's Stone

"Make of the man and woman a Circle, of that a Quadrangle, of this a Triangle, of the same a Circle and you will have the Stone of the Philosophers.

...In like manner the Philosophers would have the Quadrangle reduced into a Triangle, that is, into a Body, Spirit and Soul, which three appear in the three previous colours before Rednesse: that is, the Body or earth in the Blacknesse of Saturn, the Spirit in the Lunar whitenesse as water, and the Soul or air in the Solar Citrinity. Then the Triangle will be perfect, but this again must be changed into a Circle; that is, into an invariable rednesse, by which operation the woman is converted into the man and made one with him, and six the first of the perfect numbers is absolved by one, two having returned again to an unity in which there is Rest and eternall peace."

- From Emblem 21 of Michael Maier's alchemical test, Atalanta fugiens, 1617.

"The theoretical roots outlining the stone’s creation can be traced to Greek philosophy. Alchemists later used the classical elements, the concept of anima mundi, and Creation stories presented in texts like Plato's Timaeus as analogies for their process. According to Plato, the four elements are derived from a common source or prima materia (first matter), associated with chaos. Prima materia is also the name alchemists assign to the starting ingredient for the creation of the philosopher's stone. The importance of this philosophical first matter persisted throughout the history of alchemy. In the seventeenth century, Thomas Vaughan writes, "the first matter of the stone is the very same with the first matter of all things".

- From the Wiki entry for Philosopher's Stone.

"Dr. Sigismund Bacstrom believed that if a physician could establish harmony among the elements of earth, fire, air, and water, and unite them into a stone (the Philosopher's Stone) symbolized by the six-pointed star or two interlaced triangles, he would possess the means of healing all disease. Dr. Bacstrom further stated that there was no doubt in his mind that the universal, omnipresent fire (spirit) of Nature: "does all and is all in all." By attraction, repulsion, motion, heat, sublimation, evaporation, exsiccation, inspissation, coagulation, and fixation, the Universal Fire (Spirit) manipulates matter, and manifests throughout creation. Any individual who can understand these principles and adapt them to the three departments of Nature becomes a true philosopher."

- From the The Secret Teachings of all Ages by Manly P. Hall, 1929.

"Associated with spontaneous symmetry breaking is the phenomenon of symmetry restoration. If one heats a system that possesses a broken symmetry it tends to be restored at high temperature. ... Above the critical temperature the system exhibits rotational symmetry. Such a transition from a state of broken symmetry to one where the symmetry is restored is a phase transition. We believe that the same phenomenon occurs in the case of the symmetries of the fundamental forces of nature. Many of these are broken at low temperatures. Very early in the history of the universe, when the temperature was very high, all of these symmetries of nature were presumably restored. The resulting phase transitions, as the universe expanded and cooled, from symmetric states to those of broken symmetry have important cosmological implications."

- An excerpt from David J. Gross's The role of symmetry in fundamental physics, 1996.

***

I suppose the "Restoration of Symmetry" seems like a rather coldly analytical approach to Love, but, for a "geometer moth"  - that is, those of us for whom connecting-the-dots, so to speak, is an integral part of our nature - finding the hidden codes which help describe the world in which we live is not so much what we do, but what we are. And, like the geometrid, we do not tenaciously hide this information from view - as if it were knowledge we, alone, had access to - but, instead, wear the information on our metaphorical wings... or the skin of our backs. That is to say, we display information; we are unable to secret it away.

A contemporary glyph for Maier's diagram shown in his 21st emblem (above).*

Symmetry is a word that has tremendous importance in the world of science - in physics, chemistry, mathematics, biology, and, yes, even philosophy - in which its definition varies somewhat, but, ultimately, refers to the similar phenomenon one finds in art and geometrical figures. Basically, it refers to physical parts, properties, or processes which are equivalent in two or more directions. The circle is a figure which, for instance, is geometrically equivalent in all directions, and is thereby described as having rotational symmetry. The Philosopher Stone glyph shown above - a modern interpretation of German alchemist (and counsellor to Emperor Rudolf II) Michael Maier's emblem (circa 1617) (artist unknown) - has bilateral symmetry, in that if a line is down its center, each side is exactly equivalent to the other, although seen in reverse. This can also be referred to as reflective symmetry as one side effectively mirrors the other.

A Mojo Magic Square (with update)

Mojo Magic Square - digital - © 1981, 2014, DS

(click to enlarge)

"Mojo is magic, magical ability, and the power to get things done. "Mojo" first appeared in the 1920s in the southern United States from the Gullah word "moco" (magic), Gullah being Creole spoken by some groups of African-Americans.  The ultimate root of "mojo" was the word "moco'o," which means "shaman or medicine man" in the African language Fulani.  "Mojo" spread first into mainstream Black English and then general usage primarily through the popularity of jazz and blues music.  Muddy Waters got his 'Mojo working' and Jim Morrison of The Doors called for the Mojo Risin'."
- Quote found here.

"Think joie de vivre, that sense of being alive, joyful, and fully present in the moment. The path to finding your mojo differs for everyone. For some, mojo is a sense of purpose and meaning in your life. For others, it’s reclaiming optimal health or sparking their creativity. Some seekers find mojo by getting in touch with the divine, while others get their mojo by getting in touch with their sexual prowess. For many, it’s all of the above."
- Quote found here.

"So what is a mojo? It is, in short, the staple amulet of African-American hoodoo practice, a flannel bag containing one or more magical items. The word is thought by some to be a corruption of the English word "magic." Others state that it is related to the West African word "mojuba," meaning a prayer of praise and homage, as it is a "prayer in a bag" - a spell you can carry. A third possible derivation is from the Bantu/Kongo word "mooyoo," the magically-charged ashes and ground up bones of an ancestor that are encased in the front of a nkisi ndoki - a fetish-statue made in the form of a human being or animal. This connection with the bones of the dead is interesting, because historically, many mojo hands have indeed contained small bones, either of animals or of humans.

Some root workers top off their mojo bags with parchments upon which are printed medieval European grimoire seals and sigils of talismanic import, particularly the Jewish-derived seals from the Greater Key of Solomon and The 6th and 7th Books of Moses, both of which are sold as sets of seals printed on parchment paper, and are used without reference to the rituals given in the texts of the original grimoire books.

These last items surprise many Caucasians, who are unaware that a strong vein of Germanic and Ashenazi Jewish folklore runs through traditional African-American hoodoo. Still, however strange it may seem to cultural anthropologists in search of "African survivals" in hoodoo practice, it is a fact that John George Hohman's "Pow-Wows or the Long Lost Friend" - first published in America in 1820 and translated into English in 1856 - has long been a staple source of inspiration for conjure-workers in both the African-American and European-American Appalachian traditions, and many a black hoodoo practitioner can cite chapter and verse of "Albertus Magnus," "The Black Pullet," "Secrets of the Psalms," "The 6th and 7th Books of Moses," "8th, 9th, and 10th Books of Moses," and other occult books of European origin."
- From an article found on Lucky Mojo.

A Sator Magic Square in Oppède, France.

"The earliest dateable Sator Square was found in the ruins of Pompeii, which was buried in the ash of Mount Vesuvius in 79 AD. Examples may be found carved on stone tablets or pressed into clay before being fired. Its translation has been the subject of speculation with no clear consensus.

It seems that, in all early examples, it is a Rotas Square that reads "ROTAS OPERA TENET AREPO SATOR" from top to bottom, and from left to right.

The Sator Square is a four-times palindrome, and some people have attributed magical properties to it, considering it one of the broadest magical formulas in the Occident. An article on the square from The Saint Louis Medical and Surgical Journal, vol. 76, reports that palindromes were viewed as being immune to tampering by the devil, who would become confused by the repetition of the letters, and hence their popularity in magical use....

The square has reportedly been used in folk magic for various purposes, including putting out fires (the spell is "TO EXTINGUISH FIRE WITHOUT WATER" in John George Hohman's Long Lost Friend), removing jinxes and fevers, to protect cattle from witchcraft, and against fatigue when traveling. It is sometimes claimed it must be written upon a certain material, or else with a certain type of ink to achieve its magical effect."
- From the Wiki Sator Square entry.

***

Well, it's officially a new year in the western world, but, in the Far East, the lunar new year doesn't occur till February 2nd. (More about that later...) So, it's a relative thing, and if your "mojo" isn't quite working the way you'd like it to, you've got a whole month to get yourself in gear. Now, while "gear" isn't necessarily the operative term when defining the word mojo, dynamo, or dynamic kind of is; so, in the last analysis, you know your mojo is working when your inner world is moving seamlessly in synch with the outer world. My guess is, if one can get that down, then, regardless of all the circumstances beyond one's control, he or she will be A-OK... and as much of a contender in this game of life as anyone else is.

As it happens, mojo is a word that's been used - in various corrupted forms - quite a lot in recent years. Originally it was Hoodoo charm (although not to be confused with this hoodoo) bringing good luck and psychic protection to its bearer. Now, it's a catchword most commonly utilized by motivational speakers, executive coaches, "life" coaches, and self-help gurus when describing formulas for increasing ones productivity*, particularly in the workforce.

But, just exactly what do I mean by the word Mojo?

Mojo is animism... the dynamic energy fluctuation that makes an organism tick. If ones wiring is in top form, ones inner force pulls back and then moves forward in sync with ones conscious desires and motivations. In other words, ones Mojo is "working". On the other hand, if ones wiring has gone awry - for instance, in the case of a series of set-backs, intense trauma or injury of any kind, or just plain chronic disappointment - then ones Mojo is thrown off its game. It isn't as if you've lost it... it's that it's blocked from true expression; in other words, it's broken.

Case in point: the magic square (first image above) is something I've had on my desktop for the past 2 years. Why is this? Well, that's a good question! Before this period, I had just gone through a series of unfortunate events, and was holed up in an unfamiliar town with absolutely no clue - and no hope - for a "brighter future." Without going into details, my mojo was, in fact, at the lowest point of its existence...

The Language of Form - Part 3 - Infinity

Mathematics, by itself, is a cold, dry vista; but when it is accurate then it's the closest thing we have to an utterly objective truth. Art, however, is its other face; colorful and flamboyant in comparison, it reflects subjective truths, and truths of the intelligent and emotional organism. The form language then, is a bridge between the two, giving art its structure and mathematics its vitality and meaning.

So, I think the main point I am trying to make in my discussion, is that the form language is a code that connects the more obtuse, abstract mathematical zone to the more voluptuous, organic geometry of living structures... and, beyond that, to sentience, animism, and the webs of awareness that weave these seemingly disparate expressions into one all-encompasing whole. In this way, it represents a language that can be understood on a universal level and, perhaps, one that can be better understood by intelligences more evolved and/or advanced than ourselves. That the form language is also a unconscious code, the code of the psyche, so to speak, may mean that it is the code-breaker, enabling us to communicate with species that are not human; those of this earth, and possibly any that might exist beyond our present corporeal reach.

In a sense, this post is an intimation of a sort of Alpha/Omega point in what I've tried to describe in my two previous posts (Language of Form: Part 1, Part 2). Which is, more or less, how we've arrived at that strange image above. This image was the "Infinity" card in that enigmatic series of images I initiated over 20 years ago in my search for the form language (mentioned somewhere at the beginning of this essay). This was the "Pelaneiron" version (of my original Metastructures image), when, with my new tool, the computer, I decided to superimpose all the many geometric images onto a series of stones. Why I attempted to do this doesn't really concern us here but, for whatever reason, I "ran out of steam" very early on in the process, so, it's all rather moot.

Of course, within those 20 years, numerous scholars, scientists, theorists, etc. put forth literature that inspired me, informed me and validated my own - albeit weird - work, (although, oddly enough, never changed it). There were all those wonderful books... ones describing Synchronicity by F. David Peat, and Marie Louise Von Franz; Rupert Sheldrake and his Morphalogical fields; Heinz Pagel and Ian Stewart on Symmetry; David Bohm and the Implicate order; Paul Davis and James Gleick on Chaos theory; Mandelbrot and his Fractals; Michael Talbot and the Holographic Universe; Michio Kaku on dark matter and parallel worlds; and there were the philosphies of Aliester Crowely and P.D. Ouspensky; Timothy Leary and Aurthur Koestler; Einstien, Schroedinger, Jung. There was Buckminster Fuller and Synergetics; Dean Radin and Quantum Entanglement; Rudy Rucker and Infinity; Jill Purce and Nigel Pennick on Sacred Art & Geometry; Fritjof Capra and Gary Zukov on the "Tao of Physcis"... And the list goes on and on (and on). (links to come... maybe!)

On a personal, intellectual level I am indebted to them all, but, chances are, you won't find them quoted on this blog. This blog belongs to art - at least, that is my intention! But art would be poorer without ideas, as it would be poorer without music and literature. It is part of a dynamism that includes them all, as the form language is the dynamic code from which they arise.


Previous: The Language of Form - Part 1, Part 2

The Language of Form - Part 2

Fractal, Form and Field - the Inevitable Symmetries

The interesting thing about fractals (example shown above) is that they illustrate more than just pretty patterns. They, too, employ the same language of form we're discussing, as well as mathematical equations and/or sequences. A new science - the Science of Chaos - is gradually emerging. and fractal theories have been absorbed within that paradigm. But don't be deceived. "Chaos" in this sense, is really the intimations of a higher order, specifically the orders of symmetry, and the fractal nature of the physical world at large. Fractals, in other words, bridge the micro to the macro, and do so by the marriage of mathematics to the language of form. Symmetry, on the other hand, has become the catch-word across the board; in all areas of science, as well as art, music, history and societal analysis. Symmetry, in other words, is a very transdimensional term. It describes both the singular entity and the encompassing field. It can take us from a simple geometric figure to a parallel universe. We can follow it from the atomic to the galactic, from the child's "cat's cradle" to the Super-string universe and beyond.

Artists, of course, generally take both the fractals, the strange geometry, and the form language for granted in their work - for them it is an instinct - and this has been the case since pre-history. Whether it was a lattice or spiral drawn on a cave wall or the crystalline structures tiled onto the walls of the great mosques, artists, intuitively, have generally "been there first" on an unconscious level.

For another intimation of the modern "fractal" for instance, we need only look at some of the illustrations of German naturalist, Ernst Haeckel, (1834-1919). Consciously, he was "drawing from life"... but, observing his work (an example is shown above), I can't help but feel that, once again, on a more unconscious and metaphysical level he was trying to intuitively connect the dots, so to speak, to a larger, more profound picture... the organization of the organism, if you will, its inherent symmetry, and its relation to all phenomenal form.

When it comes to attempting to define a form language, however, it is a mistake to take any form too literally and so superficially that one overlooks the primary source - and/or the larger picture. I'm afraid that those in scientific fields and mathematical fields often have a habit of doing just this. For example, take the Sierpinski "gasket" (my version) shown above. It needs no explanation for the mathematician, but for an artist, myself in particular, the complexity in which it's presented and the way in which it's developed and described is rather daunting and off-putting. In actuality, this triangle describes a portion of a very simple grid, the very same grid from which the Platonic solids emerge. This grid, in turn, is created from a very simple field of inter-penetrating circles, the very same field from which my geometrical figures - the "Platonic" cyclohedra - arose and/or emerged (in 1984). (A graphic of these figures is shown below. Sorry, I no longer have my documentation of these figures online, as there was a decided lack of interest previously shown; in the U.S.A., that is. Interestingly, some of my related diagrams were published in China in 2007, with my permission.) 

The Cyclohedra

The point I'm trying to make, using simple geometrical observations, is that the language of form is inherently transdimensional. Both figuratively and demonstratively it is an emergent - and yes, I am referring to the theories of David Bohm - and, as such, describe form as both a singular entity and the component of a field. In other words, there is symmetry, and then again there is field symmetry. There is the stratum of one dimension as the component of higher dimensions, and what we "see" and cannot "see" is relative to our placement in that stratum... a sort of "Flatland" fable.


Fractals, and geometric observations aside, the interesting thing about more transfigurative forms in artwork - Dali and H. R. Giger come to mind - alchemical symbols and drawings, even the illustrative work (like Haeckel's) often have a "dirtiness" about them. This type of "dirtiness", I think, is the type of "dirtiness" Matta refers to when he speaks of "hallucinations" - that is, inspired images which are inseparable from the unconscious repertoire from which they arise. That symbols from unconscious and imaginal origins ultimately obey the same "laws", the same unwritten code that is used to represent the actual, phenomenal world, is our "cause for pause", so to speak. In my opinion, it intimates one way in which we might someday come to understand the Transdimensional quality of the form language, whereby shapes, patterns, fractals, and "diagrams of forces" become the keys to unlock perceptional doors to a more profound concept of space - and its hidden symmetries - that is, as a tissue, a fabric, a synchronistic and living connective.


Credits:


Fractal form: found here.
Mosque photo: found here.
Ernst Haeckel illustration: found here.


Also: another Fractal link.


"Language of Form" post links: Part I, Part 3

The Language of Form - Part 1

"The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty."

- D.W. Thompson, On Growth and Form, 1917

Whether you, as an artist - or a creative, so to speak - work consciously or unconsciously, you will be dealing with the same forms, the same fundamental shapes and patterns over and over again. And these shapes are part of a language, a language that all humans - and, most likely, organisms - use to perceive and create (and, possibly, transcend) the physical world. Oddly enough, this same language is also used to construct the actual symbols we use. This includes sacred symbols, arcane symbols, and even corporate logos. Regardless of how complex a form we're dealing with, it can always be broken down into several rudimentary, or "root shapes" - a handful of basic forms - that may change in color, arrangement of mass, etc., but never meaningfully diverge from the fundamental code, the language of form this article addresses.

D. W. Thompson, 1860-1948, biologist, mathematician, and philosopher, considered organic form to be a "diagram of forces."  Many somehow convoluted his basic tenet as being anti-Darwinian and/or creationist, and therefore, his immense body of work, "On Growth and Form" is generally overlooked by contemporary theorists. My own opinion is that Thompson was grappling with the very thing that I'm attempting to discuss here and, that is, a form language. From his perspective, it is a language shared by all organisms in the expression/ construction (even) of their own anatomy. He went on to illustrate dozens of examples of how, by mapping each organism with Cartesian Coordinates, one could prove that all members of any given species were simply a deformation of one basic morphology that inadvertently applied to all of them. Another tenet that he held, was that all things physical conform to a specific order or forms because "they must". And by this, he meant that all things physical must obey this fundamental order because it is the only true order which exists... and this order is, for the most part, a self-generating, geometric order, the connective between both the macro and micro "worlds". Today, Thompson's ideas might be seen as a precursor to some of those which fall under the heading of Biophysics.

The 4 Suits of the Pelaneiron and/or Metastructures - 2007, DS

In my younger days - as a student of Sacred Geometry, that discipline which examines the metaphysical meaning of number, patterns and form - I was obsessed with this code and my obsession took many strange turns. The four symbols (above) were an example of this, initially inspired by my interest in that dense body of esoteric work referred to as the Tarot, specifically the Minor Arcana sans the "court cards". The Tarot deck of cards, whose origins are rather mysterious to begin with, have inspired countless artists. (For those interested, here is one roster here.) The point I'm trying to make - and was trying to make with my own 4 symbols - is that everything that we see and experience can, ultimately, be broken down into a code. I wrote in 1981: "After years of frustrating myself with abstract intellectualizations, and confusing myself with complex philosophies, I suddenly discovered what my unconscious mind had probably been establishing for years... that is, a theory of structure." 

This "structure", however, did not merely apply to physical construction, but forms and patterns that might transcend the 3-dimensional as we know it. Over a period of several years during the 1980's I drew around 100 geometrical images based on the 4 root shapes; specifically numerical aggregates and radiant figures that resulted in 4 interpenetrating fields. I referred to this work under the title "Temp L" (1976), then "Metastructures" (1982), and, finally, a word that "came to me" out of nowhere: The Pelaneiron (2007). (and, no, I do not consciously know what the word means, and it is not a word that can be found in any dictionary that I know of...)

There are, of course, sheerly mathematical codes, and certainly these were (explicitly) a large part of Thompson's, and (implicitly) my own system. But, generally speaking, breaking things down into actual numerical codes is a "language" that many people (myself included) find too dense and complex. It is, however, this exact type of "language" one uses in certain forms of digital art - specifically that of the "fractal", which I'll address in Part 2 of this article: Fractal, Form and Field - Inevitable Symmetries. 

Upper Photograph credit: Diatoms by Andrew Syred