The Sacred Geometry of the Platonic Solids
Stephen M. Phillips
Flat 4, Oakwood House, 117-119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
“Geometry is the knowledge of the eternally existent.”
What is ‘sacred geometry’? Whether it refers to Stonehenge, the Egyptian pyramids, Gothic cathedrals or Tibetan mandalas, this notion is problematic in such contexts because whether one should regard the geometry of ancient monuments or religious artefacts as sacred depends upon one's own religious beliefs (or lack of them). Indeed, whether the word ‘sacred’ should be attributed to anything at all is contingent upon whether one has such beliefs. What is sacred to Christians or ancient Egyptians is usually not so to Jews or Hindus. However, true sacredness transcends cultural relativism and religious differences; it must exist in its own right instead of being merely an attribute projected onto some object by a particular religious frame of mind. Geometry that is sacred only in the eyes of the believer cannot be truly such. So what makes a geometrical design sacred? Indeed, does sacred geometry actually exist in this universal sense? Supposing that God exists (otherwise the word ‘sacred’ is meaningless and such geometry becomes mere religious doodles), sacred geometry would be that which embodies in a geometrical representation not just symbolic but quantitative aspects of cosmic reality (spiritual as well as physical) designed by the mind of God. It must be neither man-made symbolism, such as a cross or a six-pointed star, nor just a metaphor of universal order expressed in the doctrinal ideas of a particular religion or spiritual philosophy, which followers of another religion may not accept. Instead, sacred geometry must be a geometrical pattern or form depicting some quantitative aspect of the divine design of reality transcending all religious metaphors. It would be sacred because it encapsulates what actually exists — whether this is known or unknown — and not merely what fallible human beings speculate about in their mythologies and theologies and incorporate into the architecture of their churches. True sacred geometry does not embody human ideas about the nature of reality or God; it necessarily transcends them. But how then would we discern the divine design of reality, which this sacred geometry is supposed to express? Indeed, how would we recognise its sacredness? For those mathematicians or theoretical physicists who are Platonists, one of the signatures of mathematical truth is its beauty. However, sacred geometry has to possess more than just beautiful properties that no human mind could have artfully fabricated. Beauty may be a necessary attribute but it is not sufficient. Although what is sacred truth must be beautiful, what is beautiful may not even be true, let alone of divine origin. Sacred geometry must have such extraordinary properties (and so many of them as to discount the possibility of their being due to chance) that they can only indicate the existence of transcendental intelligence as well as artistry behind the sacred object. There must be no rational, conventional explanation for its possession of so many mathematically miraculous properties. Given this stringent requirement, how many examples of sacred geometry discussed in so-called ‘New Age’ books would meet such a criterion?!
This article will prove that the five Platonic solids (called in mathematics the five ‘regular polyhedra’ because they are the only 3-dimensional shapes with equal sides) have far deeper significance than what hitherto has been known to mathematicians. It will show that their geometry is sacred in the above sense — that is, to say, not because God constructed the world out of these forms (as the ancient Greeks believed) but because they collectively embody the Divine blueprint underlying all levels of existence, including the physical universe.
2. The Tree of Life
The Tree of Life (Fig. 1) has claim to be sacred geometry par excellence because, according to the Jewish mystical tradition called ‘Kabbalah,’ it is God’s blueprint for Creation. It portrays the ten divine qualities, or Sephiroth (sing: Sephirah), as spheres arranged on three Pillars and connected by 22 Paths. The uppermost Supernal Triad of Kether, Chokmah and Binah signify the triple Godhead outside Creation. The seven Sephiroth of Construction, Chesed, Geburah, Netzach, Hod, Yesod and Malkuth, represent aspects of God’s immanence in Creation. The ‘Gulf’ or ‘Abyss’ of Daath separates them from the Supernal Triad, which is not a Sephirah. Malkuth, the lowest Sephirah of Construction, signifies the material manifestation of the Tree of Life, whether a subatomic particle, the human body or the whole universe. The Sephiroth manifest in four Worlds that are stages in the descent of the Divine Life into matter. They correspond to the traditional Christian divisions: Spirit, soul, psyche and body. The Godname of a Sephirah is its essence or expression in the highest World of Atziluth (the Archetypal World). Through their many elaborations by the human mind, it is mostly the Godnames that became anthropomorphized into the gods and goddesses of ancient mythologies. (Judaism and Christianity focussed on YAHWEH, the Godname of Chokmah, although other Godnames appear in the Old Testament).
In the ancient practice of gematria, numbers are assigned to the 22 letters of the Hebrew alphabet so that hidden meanings in the texts of scriptures may be extracted from phrases and sentences. Hebrew words acquire number values that are the sum of their letter numbers. Table 1 shows the number values of the Godnames of the ten Sephiroth:
(All Godname numbers appearing in the text are written in boldface type)
In keeping with their primary nature, the Godnames prescribe — geometrically as well as arithmetically — the Tree of Life topography of all possible levels of reality, including the 4-dimensional space-time domain of physical (brain) consciousness. Godname numbers also define the dynamics and structure of the basic building blocks of matter, which is what some particle physicists call a ‘superstring’ (see Article 2). The reason for this is that the superstring is the microscopic manifestation of the Tree of Life blueprint, whose geometry is defined by Godname numbers. The Godname numbers associated with the Sephiroth are potent ‘master numbers’ quantifying the Tree of Life nature of reality, this prescription becoming ever more concrete, further down the Tree its associated Sephirah is located.
The presence of all ten Godname numbers defining properties of a geometrical pattern or set of geometrical objects such as the Platonic solids is a necessary condition for it to constitute ‘sacred geometry.’ If this criterion is upheld for a pattern or object, it will embody one or more numbers of universal significance and therefore of scientific importance because they pertain to systems that are holistic, even though science does not recognise them as such. In the case of the 3-dimensional Tree of Life, it can be shown that its projection on a plane is minimally generated from 21 points arranged in rows of 1, 2, 3, 4, 5 & 6 points, where 21 is the number value of EHYEH, the Godname of Kether. The Tree of Life consists of 10 corners of 16 triangles assembled by the joining in pairs of 34 of their 48 sides to create 22 sides or ‘Paths,’ of which 12 are the edges of two tetrahedra. 26 sides disappear in their joining, and this is the number value of YAHWEH, whilst 50 (the number of corners, edges, triangles and tetrahedra comprising the Tree, i.e., its geometrical elements) is the number value of ELOHIM, the Godname of Binah expressing the most abstract archetypes about divine form. How the remaining Godnames prescribe the geometry of the Tree requires consideration of the tetractys, which is discussed next.
3. The tetractys
At the heart of the Pythagorean philosophy based upon the power of numbers was
the tetractys, a triangular array of ten dots, or ’yods’ (1), arranged in four rows (Fig. 2). With no insight into what Pythagoras taught other than what is
provided by the few remnants of his teachings distorted by later generations of commentators,
scholars of ancient Greek mathematics assume that the tetractys represents the numbers 1, 2, 3
& 4 summing to 10, which, as their sacred Decad, the Pythagoreans regarded as the perfect
number. But it meant to
them far more than this. According to H.P. Blavatsky: “The ten Points inscribed within that ‘Pythagorean triangle’ are worth all the theogonies and angelologies ever emanated from the theological brain” (2). Partly the reason for this is that the tetractys signifies the same as the Tree of Life, namely, the ten-fold nature of God. (Fig. 3 shows this equivalence in detail). But — more important for the present discussion — the main reason for the Pythagorean reverence for the tetractys is that numbers expressing information about the nature of reality (space-time and beyond) manifest in objects possessing sacred geometry when they are re-assembled from tetractyses. In other words, the tetractys is the key that unlocks information about reality encoded in sacred geometry. This is the real reason why the followers of Pythagoras esteemed the symbol at the heart of their master’s teachings, swearing by the oath:
The numbers generated by triangulating objects with sacred geometry and transforming the resulting triangles into tetractyses are of three types:
The yod populations of sacred geometrical objects can also be divided into two other classes: the numbers of yods on their boundaries that delineate their shape and the numbers of yods inside them.
4. The inner form of the Tree of Life
As pointed out in Article 1, the outer Tree of Life has an inner form made up of two identical sets of seven enfolded, regular polygons (triangle, square, pentagon, hexagon, octagon, decagon and dodecagon), which share what I call the ‘root edge’ (Fig. 4).
This extends between Daath and Tiphareth, the centre of the Tree of Life in both a physical and a metaphysical sense. Remarkably, these 14 enfolded, regular polygons have 70 corners — the same number as the number of yods in the outer form of the Tree when assembled from tetractyses. This is not coincidental but the manifestation of a hidden regularity or pattern of order and design. Just as a DNA molecule encodes the development of a living organism from a single cell, so this inner Tree is found to encode in a geometrical way the self-replication of its outer form to map all possible levels or states of consciousness attainable by evolution. The complete map is called the ‘Cosmic Tree of Life’ (CTOL) (4). It consists of 91 overlapping Trees of Life, where
91 = 12 + 22 + 32 + 42 + 52 + 62.
They correspond to the 91 subplanes of the seven cosmic planes described in Theosophical literature (the cosmic physical plane comprises seven planes with 49 subplanes (49 is the number value of the Godname EL ChAI assigned to Yesod) and the six cosmic superphysical planes comprise 42 subplanes. Calling the Sephirah of each Tree a ‘Sephirothic Level’ (SL), CTOL is found to consist of 550 SLs, where
This is the first indication of the central role played by the Pythagorean tetractys in mathematically representing the properties of CTOL.
As these equations indicate, this map of all levels of being is a 3-dimensional, geometrical object the beauty of whose mathematical proportions reflects its divine design. As another example of its beautiful properties, this ‘Jacob’s Ladder’ consists of 3108 triangles and their corners and sides, where
3108 = 14 + 34 + 54 + 74.
There are many other examples. With their triangles turned into tetractyses, the lowest 49 Trees of CTOL mapping the cosmic physical plane have 2480 yods, which is the number of yods in 248 tetractyses. Compare this with the prediction made by superstring theory that the unified force between superstrings in ordinary matter is transmitted by 248 particles (so-called ‘gauge bosons’). Each is described by a mathematical function called a ‘gauge field’ having ten independent components because superstring theory predicts that space-time has ten dimensions (the components are measured along the directions of these dimensions). This demonstrates that the number characterising the kind of perfect, unbroken, mathematical symmetry of superstring forces described by the so-called ‘E8 group’ is encoded in the map of the cosmic physical plane, the ten yods of the equivalent 248 tetractyses denoting the ten components of each of the 248 gauge fields. Another 248 particles are predicted to mediate interactions between superstrings making up what theoretical physicists call ‘shadow matter.’ This is an as yet undetected, invisible kind of matter that may comprise some of the ‘dark matter’ believed by astronomers to make up about 90% of the mass of the universe. Its invisibility is due to the prediction that only the force of gravity acts between superstrings of ordinary matter and shadow matter. Most of these particles play no part in the physics of the cooled-down universe today because they are too massive to be created by the typical energies with which subatomic particles interact. The lighter particles of shadow matter are predicted to form a kind of parallel universe that co-exists with the one visible to human sight but which is ever beyond the ability of the five human senses to detect.
The superstring parameters 248 and (248+248=496) are also encoded in the inner form of the Tree of Life whose 49-fold replication maps the cosmic physical plane. (It is remarkable and no coincidence that 496 is the number value of Malkuth, the tenth Sephirah, signifying the physical universe). The seven separate, regular polygons of the inner Tree have 48 corners (Fig. 5). Together with the two endpoints of the root edge considered separately (which formally correspond to corners) they constitute 50 corners. This is how the Godname ELOHIM with number value 50 prescribes this pattern of sacred geometry. With their 48 triangular sectors turned into tetractyses, the seven polygons have 55 corners of tetractyses and 240 hexagonal yods, so that the two sets of polygons contain 480 hexagonal yods. But the numbers
240 and 480 are part of the group-theoretical description of the mathematical symmetry exhibited by the unified superstring force, for 240 is the number of non-zero roots of the gauge symmetry group E8 and 480 is the number of non-zero roots of the superstring symmetry group E8×E8! So we see that a parameter characterising the group-mathematics of the unified superstring force is encoded in the inner form of the Tree of Life. This is just one example among many to be discussed in later articles of how its sacred geometry incorporates the mathematical structure of superstring theory and embodies numbers of
significance to particle physics.
Inspection of the list of Godname numbers given above shows that 240 is the sum:
240 = 21 + 26 + 50 + 31 + 36 + 76
of the Godname numbers of the first six Sephiroth of the Tree of Life, which are located at the corners of a hexagon or, equivalently, a six-pointed star. The encoding of the number 240 in the inner form of the Tree of Life has its counterpart in the 49 trees representing the cosmic physical plane. As we have seen, they contain 2480 yods. The lowest tree in CTOL contains 80 yods. There are therefore 2400 yods above the lowest tree in this section of CTOL, i.e., the yods in 240 tetractyses. The 248 roots of E8 consist of eight zero roots and 240 non-zero roots. This is paralleled in, respectively, the eight and 240 tetractyses whose yod populations are the number of yods in the lowest tree and the 48 trees above it mapping the cosmic physical plane.
It is also remarkable that there are 67 yods below Binah of the lowest tree and 73 yods up to Chokmah (Fig. 6), for the former is the number value of the Hebrew word ‘Binah’ and the latter is the number value of the Hebrew word ‘Chokmah.’ This obviously cannot be coincidence but demonstrates a profound connection between the names of the Sephiroth (as well as their Godnames) and numbers generated in the Tree of Life when its triangles are changed into tetractyses. Correspondence between the inner and outer forms of the tree is also demonstrated by:
These and other remarkable correspondences are not coincidental. Instead, they indicate design.
5. The Platonic solids
Plato propounded in his Timaeus the Pythagorean doctrine that particles of the four Elements Fire, Air, Earth and Water had the shapes of, respectively, the tetrahedron, octahedron, cube and icosahedron (Fig. 9). The Pythagoreans associated the fifth regular polyhedron, the dodecahedron, with the cosmic sphere,
which came later to be identified with the fifth Element, Aether. But the five regular polyhedra have a more profound significance than this — one that truly justifies their geometry being called ‘sacred.’ In order to uncover it, one must regard the tetractys as the universal template out of which all sacred geometry is built, whether it be the Tree of Life, the Sri Yantra or the Platonic solids. We found in the last section how, when CTOL was considered as assembled from tetractyses, a number of significance to superstring theory appeared in a region of CTOL having ‘physical meaning’ (the word ‘physical’ is used in its widest possible sense, pertaining not only to space-time, which Article 2 indicates is mapped by the lowest seven trees of CTOL, but to its cosmic counterpart — the lowest 49 trees). Finding such a scientific parameter of cosmic relevance is evidence of the sacred geometry of the Tree of Life. But it requires reconstructing the Tree of Life from tetractyses so as to make manifest the information encoded in its geometry. Likewise, to uncover information encoded in the five Platonic solids, we must imagine their faces constructed from tetractyses. A triangular face transforms into three tetractyses, a square face is made up of four tetractyses and a pentagonal face is composed of five tetractyses (Fig. 10). Notice that 5, 4 & 3 are the sizes of the well-known, right-angled triangle illustrating Pythagoras’ Theorem. Notice also that the centre of a solid and any two of its adjacent corners form a triangle in its interior. Each polyhedron is built also from such triangles in its interior. A Platonic solid will be called ‘Type A’ if its internal triangles are turned into single tetractyses and ‘Type B’ if they are divided into three tetractyses (internal triangles divided into three tetractyses will also be called ‘Type B’ in order to distinguish them from the internal triangles of Type A solids).
Table 1 displays the numbers of corners (C) edges (E) and faces (F) of the five Platonic solids and the yod populations of their faces and interiors. The former are related by Euler’s formula for a convex polyhedron:
C – E + F = 2.
Inspection of the Godname numbers listed earlier tells us that the most obvious sign that the Platonic solids constitute sacred geometry is that they have 50 corners and 50 faces, i.e., their shapes are collectively prescribed by the number value 50 of ELOHIM, the Godname of Binah, which is the third member of the Supernal Triad in which the possibility of form — the abstract notion of spatial relationship — first arises. Indeed, there are many examples (to be discussed in later articles) of how geometrical forms or patterns of numbers expressing parameters of superstring theory or the Tree of Life are always defined arithmetically by the ten Godname numbers. Below are listed examples of how Godname numbers define the geometrical composition and yod population of each solid, together with other properties illustrating their correspondence with the outer and inner forms of the Tree of Life:
Transformation into the Type A tetrahedron generates 67 yods (number of Binah). Transformation into Type B generates 73 yods (number of Chokmah) other than the 48 hexagonal yods in faces (also, 73 internal yods and corners of solid). 120 new yods surround centre, where 120 = 22 + 42 + 62 + 82. 95 yods are on edges of 30 tetractyses (Type B), of which 91 are generated by transformation.
FIRST FOUR PLATONIC SOLIDS
128 vertices, edges and faces, where 128 = 21 + 26 + 50 + 31. This is the sum of the Godname numbers of the first four Sephiroth;
Number of yods on edges of four solids = 150 = 15×10;
Average number of yods (including centres) in first four solids (Type A) = 168;
Average number of yods inside first four solids (Type A) = 31;
Average number of yods in faces of first 4 solids = 137 (1370 yods in (7+7) enfolded polygons with Type B triangular sectors). The number 137 is the nearest integer approximation to the reciprocal of the fine-structure constant α = e2/ħc ≈ 1/137;
150 (15×10) yods on polyhedral edges, of which 30 are corners and 120 are hexagonal, where 30 = 12 + 22+ 32 + 42, and 120 = 22 + 42 + 62 + 82;
Numbers 1, 70 and 272 denote, respectively, the yod at the centre, the number of internal yods surrounding centre and number of yods in faces;
Number of internal hexagonal yods (Type B) = 310 = 31×10;
Number of hexagonal yods (Type A) = 310 = 31×10;
Number of hexagonal yods (Type B) = 550;
Number of yods without faces divided into tetractyses = 91;