It is said that above the threshold of Plato’s Academy at Athens, the words, “Let no one ignorant of geometry enter here,”1 were inscribed. As the “first and noblest of sciences,” it is by geometry that, according to one Scottish luminary, “we may curiously trace nature through her various windings to her most concealed recesses […] and view with delight the proportions which connect this vast machine.”2 Plato taught that, behind the everyday facade of this “vast machine,” is the true or ultimate reality, a veritable transcendent Baumian “Wizard,” which issues and dictates all that takes place here, on our dim side of the veil. One’s experience of this world, Plato insists, with all of its complexities, connections, and convolutions, is but a complicated caricature; it is an imitative, imperfect image or reflection of the simple, ultimate reality, normally seen by man only as “through a glass darkly,”3 to quote the apostle.

The mysterious, intelligible, divine essences which dictate the events, entities, objects, and all else that appears to exist in our sensible world here below, Plato referred to as “Forms.” Following the ancient Ionian Greek philosopher, Pythagoras, Platonists have consistently interpreted these archetypal Forms, which “inform” reality, in terms of mathematics, especially geometry. Four of the five so-called “Platonic solids,” for example, i.e., the cube, the octahedron, the icosahedron, and the tetrahedron, to which the Forms are often linked, were related by Plato in his Timaeus to the four classical elements, earth, air, water, and fire, respectively. Plato’s famous Peripatetic pupil and founder of the Lycaeum, Aristotle, on the other hand, would later identify the dodecahedron with the “fifth element,” aethyr.

The direct contemplation of the Forms, Plato discussed in terms of epopteia, that is, in terms of whatever it was that the initiates “saw” during the ritual celebration of the Greater Mysteries performed in the telesterion at Eleusis in ancient Greece. Plato knew to make this correlation firsthand, having been initiated into the Mysteries himself, as was Pythagoras before him. The Platonist philosopher and priest at Delphi, Plutarch, too had been passed through the telesterion at Eleusis, as well as the later Neoplatonist, Proclus. Plutarch said of the Mysteries, for example,

“[Upon dying] the soul suffers an experience similar to those who celebrate the great initiations… Wanderings astray in the beginning, tiresome walking in circles, some frightening paths in darkness that lead nowhere; then immediately before the end all the terrible things, panic and shivering and sweat, and amazement. And then some wonderful light comes to meet you, pure regions and meadows are there to greet you, with sounds and dances and solemn, sacred words and holy views; and there the initiate, perfect by now, set free and loose from all bondage, walks about, crowned with a wreath, celebrating the festival together with the other sacred and pure people, and he looks down on the uninitiated, unpurified crowd in this world in mud and fog beneath his feet.”4

Proclus wrote similarly,

“They cause the sympathy of the souls with the ritual in a way that is incomprehensible to us, and divine, so that some of the initiands are stricken with panic, being filled with divine awe; others assimilate themselves to the holy symbols, leave their own identity, become at home with the gods, and experience divine possession.”5

That hallucinogenic drugs likely played a role in the Greater Mystery celebrated at Eleusis has been proposed by a number of scholars, most notably by the father of ethnomycology, R. Gordon Wasson, the Swiss chemist who discovered the psychedelic effects of LSD-25, Albert Hofmann, and Boston University Classical scholar, Carl A.P. Ruck, in their 1978 classic, The Road to Eleusis: Unveiling the Secret of the Mysteries. In the work, the authors entertain the possibility that ergotized rye, a precursor to LSD, could have provided the entheogenic means to intoxicate and illuminate hundreds, perhaps thousands of candidates at a time, year after year, for nearly two millennia. Their argument was later supported by the discovery of ergot fragments inside a vase and within the dental calculus of a young man, both found at a Mas Castellar temple dedicated to the two Eleusinian goddesses, Demeter and Persephone, in Girona, Spain. Hungarian classical philologist, Karl Kerenyi, on the other hand, has argued convincingly for Demeter’s (and thus Eleusis’) association with Papaver somniferum, the infamous opium poppy whose use the Greeks inherited from the ancient Cretes, which the maternal, cereal goddess is frequently shown holding.

Remarkably, one of the most common effects of hallucinogenic drugs is the induction of complex geometrical patterns in the visual field. In 1926, German and American psychologist, Heinrich Kluver, studied the subjective effects of mescaline, the psychedelic component of the Lophophora williamsii cactus, better known as peyote, on himself in his laboratory. In charting the phenomenology of mescaline intoxication, in addition to the bright, “highly saturated” colors and vivid imagery that characterize the psychedelic experience, Kluver identified what he termed “form constants,” a series of recurring geometrical motifs that prevail among those who have taken mescaline and other psychedelic drugs. Like Plato’s solids to which he attributed elemental correspondences, the categories of Kluver’s form constants are four in number: spirals, cobwebs, tunnels or funnels, and lattices (including hexagons, triangles, and checkerboards). Significantly, as we shall see, all four of these motifs are consistently found in sacred sites the world over, such as in cave paintings, petroglyphs, and even churches and mosques.

Newgrange entrance stone

Spirals, for example, adorn the ancient Celtic Newgrange entrance stone and passageway, located in County Meath, Ireland, and dating back some 5,000 years. Spiral petroglyphs may also be found in the Saguaro National Monument in Arizona, left behind by the Hohokam people between 300 and 1500 CE. Perhaps the most famous example of spirals in ancient art is found with the Maori of New Zealand, who, for thousands of years, have frequently tattooed their faces with a double spiral motif, one moving clockwise on one cheek and the other, counterclockwise, on the opposite. Similarly, at altitudes lower than 2,600 meters above sea level, spiral petroglyphs moving in both directions cover a number of stones throughout Columbia. The famous petroglyph panel of McKee Springs in Jensen, UT, on the other hand, actually shows a shaman with one hand resting on a spiral portal and having held aloft in his other hand the fruiting body of a mushroom. The portal motif will be addressed further below. In regard to the double spiral motif, the Swiss Psychiatrist, Carl Jung, once noted that a number of North American Native cultures associate counterclockwise spirals with the notion of ascension, while the clockwise spiral was associated with descent.

Notably, the spiral has famously been associated with the number phi, known as the golden mean, which has been referred to as the “divine ratio” on account of its appearance in the dimensions of a number of natural phenomena – including meteorological patterns, the genealogy of cows, rabbits, and honeybees, and the growth patterns of various minerals and flora. Phi even dictates the complex ratios and dimensions that define the very human face and body, especially the skull.6

The motif of the cobweb, on the other hand, another of Kluver’s form constants, is pronounced among Chumash rock art in Santa Barbara County in southern California, where a number of cobweb designs in ochre decorate the stones. Cobweb motifs also adorn the floor and allegedly the ceiling of Vieille Major, a 12th century romanesque cathedral located in Marseille, France and dedicated to Saint Magdalene. Unfortunately, the old Marseille Cathedral is no longer open to the public, so we have not been able to corroborate the latter claim. Although, the design is also prominent in the art of Chapelle Saint-Blaise des Simples located in Milly-la-Foret, France, which French poet, Jean Cocteau, the alleged last Grand Master of the fictional Priori of Sion of The DaVinci Code fame, decorated as his final resting place.

Further, tunnels and funnels, in the form of encircled asterisks or concentric circles, such as those found in Morocco or among the petroglyphs of the Dinetah, ancestors of the Navajo, are some of the widest depictions in cave art known. It is suspected that these represented to the prehistoric mind a sort of portal whereby other worlds or dimensions might have been accessed during tribal initiations and shamanic flights. In fact, it has been proposed by Froese, Woodward, and Takashi that, based on a comprehensive study of over 40,000 years-worth of cave paintings and petroglyphs, the inspiration behind the geometrical designs repeatedly discovered in prehistoric cave dwellings and sacred sites was very likely hallucinogenic drugs.7 Indeed, as we’ll learn below, in more than a few instances, these drawings have been found in a vicinity where powerful psychotropic plants are known to flourish.

Lastly, lattices, whether they be composed of honeycombs, triangles, or checkerboards, regularly adorn ancient caves and sacred spaces. While honeycomb designs are less common, they are frequently found decorating the ceilings of Islamic mosques, such as the Fatima Masumeh Shrine in Qom, Iran, and the Sheikh Luft Allah Mosque in Isfahan, Iran. However, triangles and checkerboards are far more common.

Over a stretch of the Chihuahuan Desert from Carlsbad to Las Cruces, for example, dozens of triangle lattices painted in red, yellow, and black were discovered among the petroglyphs of southern New Mexico in 2014. Significantly, at every single one of these sites, hallucinogenic plants were found growing directly beneath the triangle designs, including a particularly potent species of wild tobacco and the powerful dissociative psychedelic nightshade, datura. While tobacco is not normally considered hallucinogenic, some varieties are indeed known to share a number of psychedelic alkaloids with so-called deadly nightshades, such as datura, mandrake, henbane, and belladonna.8

Finally, checkerboard-style latticework, the most common of the lattice motifs, appears in the Neanderthal ‘artwork’ of a Gibraltar cave dating back some 40,000 years.9 Another checkerboard lattice motif, colored in the same red, yellow, and black tones as the triangle lattices of New Mexico, also adorns Lascaux Cave in the Dordogne region of southwestern France, inhabited roughly 17,000 years ago. And, strangely, identical etched checkerboard-style lattice grids have been documented in caves as far apart as Koonalda Cave in Australia and Seneca Cavern in Ohio. Similar Paleolithic carvings, also thought to be associated with the use of psychedelic drugs, have also been found on Daufuskie Island, South Carolina and in Blombos Cave in South Africa.

In addition to the effects of hallucinogenic drugs, form constants have also been observed during traditional trance states, with and without the use of entheogens. Six decades after Kluver’s experiments, South African archaeologist, James David Lewis-Williams, incorporated the German-American’s form constants into his ‘three stages of trance’ model, with the geometric shapes comprising the visuals observed in the initial stage.

After studying extensively the rock art of the southern African San people in relation to altered states of consciousness (ASC) and the various levels of trance reported in shamanism, Lewis-Williams determined that the first symptoms to occur in the traditional trance state are characterized by the same geometric motifs that constitute Kluver’s quartet of form constants.

In Lewis-Williams’ second stage, Kluver’s form constants proceed to evolve from their purely geometric manifestation and develop into any number of seemingly random and possibly infinite apparitions, usually culturally familiar, whose shape and appearance is at least remotely suggested by that of the form constant itself. Examples of this phenomenon might include hexagonal lattices, for instance, developing into honey-filled combs, or, if one is Irish, into the famous basalt columns of the Giant’s Causeway in Northern Ireland. A professional chemist, on the other hand, might see, among other similar shapes, something like indole rings. The basic notion is that, in stage two of Lewis-Williams’ model, the “seed” forms from stage one “blossom” into one or more or perhaps even all of the possibilities inherent in their own unique structures, especially if those possibilities are familiar culturally.

In stage three of Lewis-Williams’ model, the seemingly infinite array of possible envelopments suggested by the form constant(s) in question then “settles” into a definite, subjective, imaginal scenario, wherein the observer, now in an advanced stage of hallucination, both visual and somatic, has transcended the invisible fourth wall and entered as a participant the theater of his own psychological projection. In the case of the aforementioned hexagonal lattices, for example, this condensation might manifest as the experience of being enclosed within a beehive, or of the observer himself transforming into a bee, or interacting with bees, etc. That is, in stage three, to borrow language from the domain of quantum electrodynamic physics, the waves of potential manifestations developed in stage two have finally collapsed into particles of actual subjective scenarios — again, usually ones that are culturally familiar.

Lewis-Williams’ model has been criticized, most notably by Helvenston and Bahn, in their paper, Archaeology or mythology? The model of the “three stages of trance” and rock art of South Africa. The authors’ main point of disagreement is that the variety of hallucinations constituted by form constants are most common to serotonergic drugs that bind to the 5-HT2A receptor, that is, the so-called “true hallucinogens,” e.g., mescaline, LSD, psilocybin, DMT, etc., none of which, the authors claim, are known to grow in Africa.10 There are at least three points that need to be addressed here.

Firstly, as the Italian ethnobotanist, Giorgio Samorini, discovered in a mural found in Algeria, North Africa, the local use of the psilocybin-containing mushroom, Psilocybe mairei, may stretch back some 10,000 years.11 Also, according to Gary Stafford et al, the roots of the African Acacia karroo, suspected to contain DMT, is used in Zimbabwe as a traditional aphrodisiac and pain reliever.12 Moreover, another species of acacia, Acacia nilotica (also native to Africa), according to Bep Oliver-Bever, is possessed of both DMT and harmane derivatives.13 Ergo, serotonergic drugs that commonly precipitate form constants certainly are known to have existed in Africa.

Secondly, recall that stretch of the Chihuahuan Desert, from Carlsbad to Las Cruces, mentioned above, where dozens of triangle lattices painted in red, yellow, and black were discovered among the petroglyphs of southern New Mexico in 2014. In every case, hallucinogenic plants were found growing beneath the triangle designs, including a wild tobacco and datura. The effects of tobacco and datura are not commonly associated with form constants. And yet, here we find Kluver’s triangle lattices apparently being used as markers for hallucinogenic plants which do not qualify as serotonergic.

Lastly, a recent discovery by David W. Robinson et al at Pinwheel Cave in California, which is decorated with datura flower-like pinwheel spirals, unambiguously confirmed the ingestion of hallucinogenic plants by the Natives — namely, Datura wrightii, the chewed quids of which were found tucked behind rocks inside the cave.14 Pinwheel Cave was inhabited by none other than the above mentioned Chumash people, who also decorated their quarters prominently with another of Kluver’s form constants, the cobweb. Thus, we have reason to think that Lewis-Williams’ model should not be so easily dismissed.

It may be surprising to learn that Lewis-Williams’ ‘three stages of trance’ model is possessed of another correlation that will bring us back full circle to Plato in ancient Greece, and, especially to his successors, the Neoplatonists.

Following the founder’s death, the so-called “Middle Platonists” began to move the Academy away from Platonic philosophy, and in the direction of Stoicism and Skepticism. To remedy this divergence, a Hellenistic philosopher from Roman Egypt, named Plotinus, founded what is now known as the Neoplatonic school of philosophy. To Plato’s dualistic framework of Intelligible and sensible worlds, the former being the realm of the Forms and the latter the realm of everyday human experience, Plotinus elaborated a third, mediating principle, called the Anima Mundi or ‘World Soul.’ It should be noted, however, that Plotinus did not see himself as making an innovation in Plato’s philosophy. Rather, Plotinus saw himself as emphasising an obscure point at which Plato, in his dialogue, Timaeus, had merely hinted.

The World Soul may be seen as a sort of buffer between the Intelligible and sensible realms. As the container of all that has, may, and will take place in the sensible realm, the World Soul is envisaged as the amalgam of all souls in existence. Further, it is the domain wherein all of the potentiality inherent within each of Plato’s archetypal seed Forms blossom and bloom into all of their multifaceted, multivalent manifestations. And, it isn’t just quantities into which the Forms develop. The Forms engender qualities too. If the number four were taken to be a Form, for example, in the realm of the World Soul, it would not simply expand into quantities of objects and events associated with quartets, squares, right angles, or cubes, etc., but also into related qualities, such as ‘honesty,’ ‘stability,’ and ‘foursquareness.’ The number two, on the other hand, might produce the notion of ‘dualism’ or ‘opposition,’ and the number three, ‘harmony’ or ‘dialectic.’ Just as we saw with stage two of Lewis-Williams model, in the World Soul, all of the theosophic “signatures” and esoteric “correspondences” enveloped within the Forms have developed into all of their possible, potential morphous manifestations. To quote the late, Lithuanian Platonist, Dr. Algis Uzdavinys, whose description of the soul is reminiscent of the culturally-driven phenomenology described in stage two of Lewis-Williams’ model,

“Though the [Forms] are without any visible shape or figure, they may be viewed as possessing figures in the psychic realm of the imagination […], since the soul is the pleroma [or ‘fullness’] of reality. So within the soul, as on a magic screen, all things are contained inwardly in a psychic mode. […] [The] soul is capable of seeing and knowing all things, including figures of the [Forms that are] essentially without any shape or figure, in this […] mode of entering into itself and awakening the inner powers which reveal the images […] and symbols of the universal reality. Neither the outward nor the inner psychic seer is capable of seeing without images. Thus the nature of things seen corresponds in each case to the nature […] of the seer himself, that is, to the particular archetypal measures or configurations […] and to the actual contents of his existential and culturally shaped consciousness.”15

By the time these energeia reach the sensible realm, they coalesce, condense, and crystallize into the experiential world that manifests to our everyday senses, with all of its subjective narratives and complex scenarios.

Adopting a neuropsychological framework, Lewis-Williams proposed that Kluver’s form constants arose as a result of entoptic phenomena; that is, visual effects whose source is within the eye itself. Notably, nearly two decades prior to Lewis-Williams’ findings, a similar theory had been offered to explain what is actually taking place when our brains trick us into seeing geometric patterns. According to mathematicians, Cowan and Ermentrout, and later, Paul Bressloff, professor of Mathematical Computational Neuroscience at the newly established Oxford Center for Collaborative Applied Mathematics, fMRI scans suggest that Kluver’s form constants may originate in the portion of the brain known as V1, the visual cortex.

“Hallucinations comprising spirals, circles, and rays that emanate from the center correspond to stripes of neural activity in V1 that are inclined to given angles. Lattices like honeycombs or chequer-boards correspond to hexagonal activity patterns in V1.”16

This is significant as stripes and hexagons are precisely what the famous mathematician, Alan Turing, observed in the emergence of stripes and spots in animal coats, skins, and scales. For, as their connection to Platon’s Forms indicates, these particular patterns which Kluver observed while tripping out on peyote in his lab are by no means unique to humans alone.

“Turing hypothesized that these animal patterns are a result of a reaction-diffusion process. […] For example, if [an] inhibitor diffuses faster than [an] activator, then it quickly spreads around the point or perturbation and decreases the concentration of activator there. So you end up with a region of high activator concentration – in other words, you have a spot of activator on a background of inhibitor. Depending on the rates at which the two chemicals diffuse, it is possible that such a spotty pattern arises all over the skin of the embryo, and eventually stabalises. If the activator also promotes the generation of a pigment in the skin of the animal, then this pattern can be made visible.”17

And thus, we come back to Cowan and Ermentrout’s theories regarding the brain, in particular, the visual cortex. Although neural activity in the brain is not a reaction-diffusion process, according to Bressloff,

“There are neurons which are excitatory – they make other neurons more likely to become active – and there are inhibitory neurons, which make other neurons less likely to become active. The competition between the two classes of neurons is the analogue of the activator-inhibitor mechanism in Turing’s model.”18

Ergo, Kluver’s kaleidoscopic chimeras appear to occupy a mediating position between Plato’s Ideas or Forms in the macroscopic, noetic realm, and their convoluted manifestations on the microscopic, material plane — including in the patterns of the animal kingdom, such as the coats, scales, and skins of various fauna.

While it is clear that there are undeniable (and unexpected) correlations between Lewis-Williams’ ‘three stages of trance’ model and the model of cosmology as developed by Plato and his successors, we are reluctant to draw any definite conclusions. To begin with, we do not know if Lewis-Williams’ research was influenced by the linguistic concepts or cosmological models proposed by Plato or the Neoplatonists. We also cannot rule out cryptomnesia on the part of Lewis-Williams. Nor are we in a position to draw the more extreme conclusion — that Lewis-Williams’ ‘three stages of trance’ were observed by the ancient Greeks directly, with or without the aid of hallucinogenic drugs, and it was that direct observation that contributed to the development of the cosmological model among Plato and his successors. For the present, we must be content to simply note the correlation.