A STUDY OF THE PLATONIC SOLIDS IN KEPLER’S MYSTERIUM COSMOGRAPHICUM; INCLUDING AN EXPOSITION BASED ON A NEW HELIOCENTRIC QABALAH
By Jaime Paul Lamb
ABSTRACT
In the following, we will introduce the five Platonic solids, addressing their history, development and elemental attributions. We will also address Johannes Kepler’s Mysterium Cosmographicum of 1596 CE, in which he utilized the newly (re)established heliocentric model to apply the geometry of the Platonic solids in the metrics and relative distances of the planetary orbits. We will then rectify the hitherto geocentric qabalistic diagram, known as the Etz Chaim (“Tree of Life”), by transposing the Sephiroth, Malkuth and Tiphareth – thereby, rendering a new heliocentric qabalah. Finally, we will apply this rectified cosmological model in an interpretation of the Platonic solids, using the established methods and correspondences of the Western Esoteric Tradition, particularly that of Hermetic Qabalah.
THE PLATONIC SOLIDS
The Platonic solids are five regular, convex polyhedra, with regular polygonal faces meeting in the same number at each vertex, in three-dimensional space. There are precisely five solids which satisfy the above criteria, they are: the tetrahedron, the hexahedron, the octahedron, the dodecahedron and the icosahedron. The Greek Neoplatonist, Proclus (412-485 CE), credits their discovery to the Pythagoreans sometime before 450 BCE, although there is evidence that the Egyptians identified at least three of the five regular, convex polyhedra.1 Although Theaetetus of Athens (417-369 BCE), a contemporary of both Socrates and Plato and, like Plato, a student of mathematician, Theodorus of Cyrene, has been credited with the first formal proof of the five regular, convex polyhedral2, Plato most popularly described the Platonic solids in intuitive and philosophical terms, theorizing that these were the literal building blocks of phenomenal reality. In this supposition, Plato’s conceptualization of the solids may be said to be somewhat built upon notions in circulation among pre-Socratic philosophers such as Empedocles (494-434 BCE), who posited that the arche (the primordial element or principle underlying and permeating the universe) was not a single element but a combination of earth, water, air and fire, and Atomist philosophers such as Democritus (460-370 BCE) and Leucippus (5th century), who posited that the universe was composed of indivisible, microscopic units they called “atoms”. Similarly, in his Timaeus (360 BCE), Plato attributed each of the five solids to one of the classical elements (earth, air, water and fire, plus æther); the tetrahedron being associated with fire; the hexahedron (cube) with earth; the octahedron with air; the dodecahedron with æther;and the icosahedron with water, thus implying their status as the primal units of matter.3 4 In what may be seen as a nod to Pythagoras’ meta-arithmetical numerology applied to astrology, Plato’s association of the dodecahedron with æther is apparently based on the twelve faces of the dodecahedron corresponding to the twelve signs of the zodiac, as the philosopher somewhat cryptically wrote that “[…] god used [it] for arranging the constellations on the whole heaven.”5 At the dawn of the 3rd century BCE, Euclid devoted the entirety of the XIIIth and last book of his Elements to the mathematical description and construction of the Platonic solids, arguing in Proposition 18 the impossibility of a sixth convex, regular polyhedron.6
KEPLER’S MYSTERIUM COSMOGRAPHICUM
In his first major astronomical work, the Mysterium Cosmographicum7 (Tübingen, 1596 CE), 16th century German astronomer, astrologer and mathematician, Johannes Kepler (1571-1630 CE), argued that the distances between the planets Mercury, Venus, Earth, Mars, Jupiter and Saturn could be described using the Platonic solids. In this cosmological schema – which was based on the Copernican, heliocentric model – the convex, regular polyhedra were concentrically enclosed within the sphere of Saturn and were explained as being indicative of the handiwork of God, who Kepler imagined to be a Geometrician. The Mysterium Cosmographicum marked the first attempt since Copernicus to prove the theory of heliocentrism.8
Anyone with the least knowledge of geometry […] would immediately recall the five regular solids with the ratio of circumscribed orbs to the inscribed ones, and there would immediately appear before his eyes the Euclidean […] proposition […] where it is proved that there cannot be, or be imagined, more regular solids than five […] As a memorial of the event [Kepler’s discovery] I present you with this idea, just as it occurred, in the words conceived at that very moment: The Earth is the circle which is the measure of all. About it circumscribe a dodecahedron. The circle enclosing this will be Mars. About Mars circumscribe a tetrahedron. The circle enclosing this will be Jupiter. About Jupiter circumscribe a cube. The circle enclosing this will be Saturn. Now in the Earth inscribe an icosahedron. The circle inscribed in this will be Venus. In Venus inscribe an octahedron. The circle inscribed in this will be Mercury. Here you have the reason for the number of the planets.9
A HELIOCENTRIC QABALISTIC ANALYSIS OF THE PLATONIC SOLIDS
Using Kepler’s cosmological model of the solar system as an interpretive key, we find that the Platonic solids may be attributed to the several paths which are distributed between the Sephiroth (“Emanations”) on the Etz Chaim (“Tree of Life”), thereby providing us with further hermetic and qabalistic insight due to their correspondences – elemental, planetary, zodiacal, mythological, tarotic and so forth. Planetarily, the sephirothic distribution of the Etz Chaim is but a linear reworking of the concentric planetary spheres devised in the first few centuries of the Common Era by the ancient Hermeticists and Neoplatonists. In their geocentric cosmological model – and, consequently, that of the Qabalists – the planets are sequenced in their typical Ptolemaic, or Chaldean order, viz; Luna, Mercury, Venus, Sol, Mars, Jupiter, Saturn and, finally, the firmament, or sphere of the fixed stars and the constellations of the zodiac; each of these spheres being nested within the other with the Earth at the center, enveloped by the lunar sphere, which is itself enveloped by the mercurial sphere, etc. However, in Kepler’s Mysterium Cosmographicum, we are dealing with the newly (re)discovered, Copernican, heliocentric model – not the geocentric, Ptolemaic or Chaldean models, upon which the Etz Chaim was constructed – and, thereby, the Earth and Sun are transposed, leaving the Earth inhabiting the position of the sixth Sephirah, Tiphareth, and the Sun situated in Malkuth, the tenth Sephirah. With this transposition, we find the planets in their true, heliocentric order on the Etz Chaim.10
Beginning with the octahedron, we find this solid representing the twenty-seventh path, which runs between the Sephiroth, Hod and Netzach, just as it runs between Mercury and Venus – the planets respectively corresponding to Hod and Netzach – in Kepler’s model. In the standard correspondences established in the cipher manuscripts which lead to the foundation of the Hermetic Order of the Golden Dawn11, the Tarot card assigned to this path is XVI The Tower. This Tarot card represents transformation and may be interpreted to imply the necessity of a mediating, or rather, synthesizing force between two poles. Interestingly, the twenty-seventh path connects the sphere of Hermes (Hod) with that of Aphrodite (Netzach), whose synthesis is the hermaphrodite or Rebis, a synthesis and union of opposites, as depicted in alchemical codices wherein it represents the end product of the magnum opus.12 The octahedron, representing air in the Platonic correspondence, may be seen to represent the action of the intellect in this operation, as the element air is associated with this process. The Hebrew letter Peh, meaning “mouth”, is assigned to this path as well; which may be interpreted as representing the Logos (Greek: λόγος; from λέγω, meaning ‘I say’), simultaneously extrapolated to mean both “word” and “reason”, thereby strengthening the aforementioned association of the intellect in the process of transformation. Furthermore, the gematriac value of Peh is eighty, which reduces to eight by Theosophical addition (8 + 0 = 8), thus illustrating its connection with the octahedron, which has eight sides.
The next Platonic solid we encounter is the icosahedron, which Kepler inscribed between the spheres of Venus and the Earth. The icosahedron is qabalistically represented by the twenty-fourth path, which connects the Sephiroth of Netzach (representing Venus) with that of Tiphareth (which houses the Earth, in the heliocentric model). The Tarot assigned to this path is XIII Death, to which the zodiacal sign of Scorpio – a water sign – is attributed in the established Hermetic correspondences. In his Timaeus, Plato attributed the element of water to the icosahedron, thus reinforcing the watery nature of this path. The Hebrew letter associated with the twenty-fourth path is Nun, which means “fish”, further elucidating the concept of life, or consciousness, within the waters. As we had previously discussed, words sharing a gematriac value are believed to have an occult resonance, from the qabalistic perspective. The gematriac value of the word Netzach is one-hundred-twenty-eight; this value is also shared by the Hebrew words הגמלים, meaning “camel”; מנחל, meaning “river”; and מבועי, meaning “spring”. The words for river and spring are obviously aquatic; but the camel also affords us a unique insight, in that it may be seen to represent the inverse of the fish – as the fish lives within the water, the store of water lives within the camel. Thus, we have constructed several qabalistic “proofs” verifying Plato’s intuition as to the watery nature of the icosahedron.
The dodecahedron, which Kepler inscribed between the Earth and Mars, is associated with the twenty-seventh path on the Etz Chaim, spanning from Tiphareth to Geburah. The Tarot card attributed to this path is XI Justice, which represents balance and equilibrium. Just as Mercury and Venus may be said to be a matched pair of opposing forces brought into a unity by the path which spans their Sephiroth, so may the Earth and Mars be viewed. For the Earth represents the fertile mother and receptive femininity, while her antithesis, Mars, represents the active, fiery violence associated with the god of war. So may these forces – like those of Mercury and Venus – be brought into a state of equilibrium and balance, as suggested by Justice’s scales as well as by the symbol of Libra, the zodiacal sign associated with this path.
Between the spheres of Mars and Jupiter, Kepler inscribed the tetrahedron. The reader will recall that Plato associated the tetrahedron with the classical element of fire, which figures quite prominently in our hermeneutic of this particular solid. From the perspective of classical Greek mythology, we find the archetypal presence of the Olympians, Ares (Latin: Mars) and Zeus (Latin: Jupiter) in these planets; Ares, being the son of Zeus, was subordinate to him. Being the god of war and of military valor, Ares’ qualities were considered necessary – yet, he was treated ambivalently by the votary and populace, and was despised by his father, Zeus, who recognized his tendency to pervert his archetypal characteristics into arbitrary violence and cruelty. For, in the Vth book of the Iliad, Zeus addresses Ares thus…
Most hateful to me art thou of all gods that hold Olympus, for ever is strife dear to thee and wars and fightings. Thou hast the unbearable, unyielding spirit of thy mother, even of Hera; her can I scarce control by my words. Wherefore it is by her promptings, meseems, that thou sufferest thus. Howbeit I will no longer endure that thou shouldest be in pain, for thou art mine offspring, and it was to me that thy mother bare thee; but wert thou born of any other god, thus pestilent as thou art, then long ere this hadst thou been lower than the sons of heaven.13
Qabalistically, Mars and Jupiter correspond to the Sephiroth, Geburah and Chesed, respectively; and the path on the Etz Chaim between these two Sephiroth is nineteen, which is associated with the VIII Strength Tarot card. The zodiacal sign of Leo, which is a fire sign, is also attributed to this path, as is the Hebrew letter Teth, meaning “serpent”. Bearing these standard correspondences in the highly syncretized system of Hermetic Qabalah in mind, we find that this path, and the sephirothic conjunction it represents, communicates the sublimation of the fiery, warlike agency of Geburah into the kingly righteousness of Chesed – a process perhaps best symbolized by the serpent; as the serpent may be interpreted to represent the wisdom necessary to transmute the raw and potentially destructive martial power of Geburah into a dynamic force, wielded with the jupiterian agency of Chesed. Thus, the serpentine wisdom feeds the regal lion, whose dominion is described by the circumference of his will.
Lastly in the Keplerian planetary distribution of the five Platonic solids, we encounter the hexahedron, or cube, which is inscribed between the spheres of Jupiter and Saturn. At this planetary junction, we may note that, as the sphere of Saturn concentrically encapsulates the inner planets circumscribed by its orbit, it was Saturn (Greek: Κρόνος; Cronus), the titanic father of the gods in classical mythology, who ostensibly swallowed Jupiter and the rest of his offspring whole.14 The cube also affords us valuable symbolism for our contemplation, in that it represents materiality and, when unfolded into a two-dimensional, planar diagram, it becomes a cross – itself a symbol of the material world, or the Earth. This may be part of the reason why Plato attributed the hexahedron to the classical element of earth in his Timaeus. The cube is also a representation of the space created by the aforementioned Cartesian coordinates: x (side-to-side), y (front-to-back) & z (up-and-down) – rendering the cube a microcosm of the phenomenal world; and when the fourth dimension of time is added to this spatial schema, we are able to locate ourselves, and other points, in the continuum. In the Sepher Yetzirah (Hebrew: ספר יצירה; meaning “Book of Formation”), the oldest known qabalistic text, dating to sometime between the 2nd century BCE and the 2nd century CE15, the six directions are said to have been “sealed” by Yah, the Lord of Hosts, thus describing a cubic spatial model at least a millennium before Descartes…
He selected three letters from the simple ones, and sealed them as forming his great Name, IHV and he sealed the universe in six directions. He looked above, and sealed the height, with IHV. He looked below, and sealed the deep, with IVH. He looked forward, and sealed the East, with HIV. He looked backward, and sealed the West, with VHI. He looked to the right, and sealed the South, with VIH. He looked to the left, and sealed the North, with HVI.16
Qabalistically, Jupiter and Saturn correspond to the Sephiroth, Chesed and Binah, respectively, and only an invisible path runs between them; this invisible path, however, is the only leg of the “Lightning Flash” not following an established sephirothic path. This path has no number or tarotic attribution – however, there are two qabalistic touchstones corresponding to this gulf-like region of the Etz Chaim; the first is the “hidden” Sephirah, Da’ath (Hebrew: דעת; meaning “Knowledge”), and the second is referred to as the “Abyss” – both denoting the only passage by which one may access Atziluth (Hebrew: אֲצִילוּת), the qabbalistic “World of Emanation”. Da’ath may be seen as the guardian, or gatekeeper, of the Divine Light of the three supernal Sephiroth: Binah, Chokmah and Kether. The Abyss is an imaginary line which separates the three supernals from the lower Sephiroth. It is a principal aim of the Practical Qabalah to attain this atziluthic state – in the gnosis of an unmediated apprehension of the supernal emanations – but it does not come without experiencing the perils of crossing the Abyss.
The Paths pursued by the Emanations in these successive overflowings is represented upon the Tree of Life by a Lightning Flash, or in some diagrams by a Flaming Sword. It will be observed by reference to diagram I that the Lightning Flash must proceed from Kether outwards and downwards to the right to reach Chokmah, and then turns on a level course to the left and proceeds an equal distance beyond Kether upon that side, and there establishes Binah. The result is a triangular figure upon the glyph, and it is called the Triangle of the Three Supernals, or the First Trinity and is separated from the rest of the Sephiroth by the Abyss, which normal human consciousness cannot cross.17
CITATIONS
1. “The Platonic Solids”, University of Illinois, retrieved at: uiuc.edu
2. Allman, Greek Geometry from Thales to Euclid, Hodges, Figgis & Co., 1889, p. 206
3. Plato, Timaeus, 360 BCE, 55c4-6
4. Zeyl & Sattler, Stanford Encyclopedia of Philosophy, Entry: “Plato’s Timaeus”, 2017, retrieved at: plato.standford.edu
5. Plato, Timaeus, 360 BCE, 55
6. Euclid, Elements, Book XIII, Propositions 13-18, 300 BCE
7. The actual title of which is the unwieldy Prodromus dissertationum cosmographicarum, continens mysterium cosmographicum, de admirabili proportione orbium coelestium, de que causis coelorum numeri, magnitudinis, motuumque periodicorum genuinis & proprijs, demonstratum per quinque regularia corpora geometrica (English: Forerunner of the Cosmological Essays, Which Contains the Secret of the Universe; on the Marvelous Proportion of the Celestial Spheres, and on the True and Particular Causes of the Number, Magnitude, and Periodic Motions of the Heavens; Established by Means of the Five Regular Geometric Solids).
8. Voekel, Classics of Astronomy by Johannes Kepler, 2010, retrieved at: chapin.williams.edu
9. Kepler, Mysterium Cosmographicum, 1596
10. This transposition constitutes a step towards a rectified heliocentric qabalah, for those who are wishing to investigate this rewarding alternate magical paradigm. Of course, there is the problem of Yesod and maintaining twenty-two attributable paths for the Hebrew alphabet and other correspondences, but these are relatively easy to solve – their solution being outside of the scope of the present work. Modifications such as this, along with sidereally accurate astrological adjustments to the largely cardinal and tropical system currently in use by the vast majority of Western occultists, allow the practitioner to work within the schema of astronomically accurate cosmic sympathies, while taking stellar displacement due to axial precession into account.
11. Küntz (editor), The Complete Golden Dawn Cipher Manuscript, Holmes Publishing, 1996
12. e.g.,Valentine, Azoth of the Philosophers, 1613, and Nollius, Theoria Philosophae Hermeticae, 1617
13. Homer, Iliad, Book V, 890-897
14. Hesiod, Theogony, 8th – 7th century BCE, 453-467
15. Reitzenstein, Poimandres, Leipzig, 1904
16. Sepher Yetzirah, Westcott (trans.), Ch. I, v. 11, 1887
17. Fortune, The Mystical Qabalah, London: Society of Inner Light, 1935, p. 27
Jaime Paul Lamb is a consulting astrologer and taroist living in Phoenix, AZ. He is a member of the American Federation of Astrologers (AFA) and is certified in Hellenistic Astrology through Chris Brennan’s premier course on the subject. Lamb is the author of Myth, Magick & Masonry: Occult Perspectives in Freemasonry (The Laudable Pursuit, 2018), Approaching the Middle Chamber: The Seven Liberal Arts in Freemasonry and the Western Esoteric Tradition (The Laudable Pursuit, 2020) and The Archetypal Temple (Tria Prima, 2021). For more information, visit: www.jaimepaullamb.com.
Lamb is the Instructor for the Institute for Hermetic Studies course, Classical Astrology and the Western Esoteric Traditions. For more information click on the link below.
https://institute-for-hermetic-studies.teachable.com/courses/1788803
