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Pardon the mess, we're still
under construction.
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Sacred
Geometry or Platonic Solids and Their Symmetries
Some
Random Thoughts about the Occult Correspondences of the Platonic Solids and
Their Symmetries
The 5 Platonic solids are ideal, primal
models of crystal patterns that occur naturally throughout the world of
minerals, in countless variations. These are the only five regular polyhedra,
that is, the only five solids made from the same equilateral, equiangular
polygons. Pure quartz crystal has been cut into these primal shapes for use
in meditation, healing and manifestation.
Very
little of modern mathematics has been used in the Cabala, which relies mainly on
simple arithmetic operations and some basic combinatorics (an area which it in
fact partially founded). I think this is regrettable, since there is a plethora
of interesting mathematical results which could be applied to occultism. In the
following I will discuss a few interesting areas of solid geometry and abstract
algebra. The discussion will be rather non-mathematical, and I will not attempt
stringency, which anyway is a bit hard to apply when discussing occult matters.
It is helpful to have models or good renditions of the various polyhedral
available to visualize the various properties I will discuss below (like [RW] or
[C]), since I cannot include pictures of the often quite complex structures.
The
Five Platonic Solids
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The
Cube
(Hexahedron)
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The
Tetrahedron |
The
Dodecahedron
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The
Platonic solids, also known as the regular polyhedrons, are the
three-dimensional bodies whose surfaces consist of identical, regular polygons
which meet in equal angles at the corners. There are five such polyhedrons, the
Tetrahedron, the Octahedron, the Cube, the Icosahedron and the Dodecahedron. The
first three have apparently been known since ancient times. The others was
definitely known by the Pythagoreans, since one of them, Timaeus of Locri,
invented the "Platonic" correspondence between them and the elements.
Plato later publicized their results, which is the reason they bear his name.
Here is a table with their properties (based on [C]):
Faces
Edges Vertices Schfli Dual Plato symbol Tetrahedron 4 6 4 {3,3} Tetrahedron Fire
Octahedron 8 12 6 {3,4} Cube Air Cube 6 12 8 {4,3} Octahedron Earth Icosahedron
20 30 12 {3,5} Dodecahedron Spirit Dodecahedron 12 30 20 {5,3} Icosahedron
[
The Schfli symbol represents the type of polygons making up the faces and the
number which meet at each vertex. A cube consists of squares (4) and three
squares meet at each corner (3), thus its symbol is {4,3} ]
[
Two polyhedral are duals if the vertices of one correspond one-to-one to the centers
of the faces of the other. ]
The
Platonic Correspondences are Tetrahedron: Fire,
Icosahedron: Water,
Octahedron: Air, Cube: Earth
and Dodecahedron: The Quinta Essentia. While
this is pleasing from a traditional and aesthetic standpoint, I have not found
it workable from a more mathemagickal standpoint.
These
solids naturally fall into three groups, based on their symmetries and duals.
The Octahedron and Cube, which are duals of each other, form one group, while
the Dodecahedron and Icosahedrons form another. The Tetrahedron form a third
group with only itself as a member since it is its own dual. Note that the five
elements are similarly divided: the spiritual elements are duals to the material
elements (and a similar duality holds for actives and passives), and the fifth
is left out or its own opposite (one is reminded of the concept of positive and
negative aethyr in [CL]). Thus, from my mathemagickal standpoint, Quintessence
belongs more naturally to the Tetrahedron, the Cube and Octahedron corresponds
as normal to Earth and Air while Fire and Water correspond to the Dodecahedron
and Icosahedron respectively. I will now discuss the properties of the various
polyhedrons from different perspectives.
The
Tetrahedron
The
Tetrahedron classically represents Fire, and each face is also the alchemical
triangle of fire. The Golden Dawn called it the Pyramid of Fire, and used it as
the admission badge for the path of Shin. The three upper triangles represents
Solar Fire, Volcanic Fire and Astral Fire, while the bottom triangle, often
hidden from view is the latent heat. The upper triangles are also linked to the
three fire-signs Aries, Sagittarius and Leo.
Note
that each face and each vertex can be put into a one-to-one correspondence with
an element. Each element touches the others, showing that the superficial
divisions of Fire and Water, Air and Earth are really unities. No element is
superior to any other, and they all balance each other into a very stable
structure (Buckminster Fuller designed his entire mathematics and architecture
on this simple fact). This represents is in my view the state before the
divisions between the elements, and thus resonant with the Quinta Essentia, from
which the element were formed.
Its
worth noting that the tetrahedron is its own dual. At the same time it belongs
to the 4.3.2 symmetry group, the same as the octahedron and the cube belongs to.
In a way this reflects "Keter is in Malkuth, and Malkuth is in Keter",
the material world subtly reflects the spiritual world and vice versa.
The
four elements are linked with six edges, which may correspond to the hexagrams
and the planets (the Sun is as usual in the center). Seeing things this way,
each planet can be seen as a path between two elements. Some possible
correspondences (this probably requires more thought, and I would be happy to
hear other possibilities):
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Moon
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Water-Fire
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Mercury
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Air-Fire
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Venus
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Water-Earth
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Mars
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Fire-Earth
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Jupiter
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Air-Water
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Saturn
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Earth-Air
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We
will see that this planetary/double-element correspondence is common in the
other structures too, making it very interesting.
The
Octahedron
The
Octahedron corresponds classically to Air. It has 8 faces (corresponding to
Hod and mental activity?), 6 vertices and 12 edges. The edges naturally
correspond to the zodiac. They can be arranged in such a manner that the four
triplicities border a triangular face each without overlap. These faces cover
half the surface, leaving 4 incomplete faces with signs from three elements
along each edge (this may signify an absence of the left-out element. The
octahedron thus consists of both the abundance of each element and its
absence). At each corner two elements meet (creating the same planetary
correspondences as in the tetrahedron, with the sun at the center as usual).
In this arrangement, each square "equator" corresponds to one
quadruplicity.
Another
common use of the octahedral symmetry is used in banishing rituals (mainly the
LBRP and the Rose-Cross Rite). The sphere encircled by three orthogonal
circles is the natural projection of the octahedron onto the surface of a
sphere. In most rituals the horizontal equator corresponds to the cherubic
signs. This also corresponds to the six directions of the Yetziratic Sealing
Rite [DK], see below for the discussion of the symmetric group.
The
octahedron fits air very well, since the various symmetries and
correspondences are so clear and easily viewed. As we will see in the case of
the cube, many of these symmetries are hidden or hard to discern in the case
of Earth, perhaps signifying that the intellect allows us to see the structure
of the world more easily than our physical senses, which are parts of the
system we try to study.
The
Cube (Hexahedron)
The
Cube (hexahedron) naturally corresponds to Earth. It is stable, the basis of
western architecture and salt crystallizes into cubes. It has six faces,
making some groups attribute it to Tiphareth. The six faces naturally fit the
sephira, and can of course be linked to the planets except for the sun, which
is placed in the center. Another natural link is the folded out cube, which
forms a cross. The eight corners of the cube neatly corresponds to three
complementary dualities. When two dualities interact, the four elements are
created. Now the four elements are dualized again, and we get eight corners
representing the relative absence and abundance the each element. This is
naturally dual to the faces of the octahedron. In the same way the six faces
correspond to the six vertices of the octahedron (i.e. meetings between two
quadruplicities). It is however not possible to arrange the three
quadruplicities along the edges to enclose whole faces without overlaps. Does
this signify the imperfections and limitations of the material world?
Its
an interesting fact that the cube isn't stable. If a model is made using
toothpicks and peas, it can easily be shown that it tends to distort or
collapse. It is however possible to inscribe a tetrahedron inside a cube so
that its vertices meet four corners of the cube and its edges lie in the faces
of the cube. This will stabilize it completely (spirit stabilizes and orders
matter). If two tetrahedrons are inscribed using different sets of vertices,
they intersect and form a geometric body known as the "Stella
Octangula" (which is an octahedron with pyramids added on its faces).
This is a very neat representation of the complementarity between positive and
negative forces, which seems to underlie much of the structure of the cube.
It
is worth noting that the duality of the cube and octahedron fits the duality
between Air and Earth. Both belong to the same symmetry family (called 4.3.2),
to which all normal minerals and crystals belong (only the so-called quasi-crystals
belong to the icosahedra symmetry family). It is also an interesting fact that
of the platonic polyhedrons, only the cube can fill space completely, without
interstices or overlaps. Thus we see that despite that the only way to create
a completely consistent universe out of one element is to use matter. The
other elements are not able to bind together in the right way to form a stable
world, but will either move around or form imperfect patterns.
The
Icosahedron
This
polyhedron traditionally corresponds to water, possibly because it rolls quite
easily. Its 20 faces could correspond to the sephiroth and qlippoth, but I
have so far not found any significant arrangement. While the octahedron and
cube, belonging to 4.3.2 have many symmetries involving the four elements,
trinities and dualities, the icosahedron and dodecahedron
In
nature these symmetries are rare, and are usually found in viruses and
radiolaria. One reason for the rarity of these symmetries may be that they
don't interconnect as well as the 4.3.2 group. In crystals, molecules and
viruses with 5.3.2 symmetries organize according to the 4.3.2 group instead,
subjugating their own symmetries. The higher elements decay into the lower in
order to form the world.
These
symmetries are harder to discern, since traditionally we humans have a
tendency to avoid high-order groups, especially odd symmetries (its worth
noting that the number five is sacred to the Discordians since it is the
smallest number of factors the human mind is unable to handle at once).
The
12 vertices can of course be viewed as the zodiac. In this case each sign is
linked to five other signs along the edges which corresponds to the five
elements, a quite interesting set of correspondences (this is of course
reflected in the faces and edges of the dodecahedron in a similar way). This
seems to imply a network between the signs, where each sign is transformed
into five others by the actions of the five elements. I have so far not seen
any uses for this system, but it is potentially interesting.
One
obvious way of arranging the elements in such a pattern is the following:
choose two edges opposite to each other and assign them to an element. Then
there are four edges along the "equator" if the two edges are
regarded as the poles which can be assigned the same element. These edges are
orthogonal to the first, and each pair of opposite edges are orthogonal to all
others. In fact, if the opposite edges are joined with lines through the
interior, a very neat structure of interlocking rectangles result, where each
rectangle locks the other rectangles without touching them. Each pair of
rectangles doesn't interlock, but together they form a synergetic whole. In
this way each element can be assigned to its own edges in a proper way. It is
interesting to note that the pattern inside each element belongs to the 4.3.2
symmetry group.
The
icosahedron can be inscribed in the octahedron if its vertices are placed on
the octahedron-edges in the golden ratio. In this case eight faces of the
icosahedron lie in the plane of the faces of the octahedron, and the rest lie
in the interior. As a general rule, the golden ratio is intimately linked to
the 5.3.2 family of solids. This construction is symbolic of how the
creativity and feeling of Water is needed to form the rational thought of Air.
In
my own system the icosahedron corresponds to water. It seems to tie together
things in complex, apparently random ways and encompass them without
necessarily elucidate their interrelationships. As one can see, the complexity
of the icosahedron and dodecahedron "liquiefies" the various
correspondences. The number of possible arrangement is much larger than for
the relatively simple cubes and octahedrons.
The
Dodecahedron
This
solid is classically attributed to spirit, probably because it was the last
discovered and because of the pentagonal faces. Its twelve faces has naturally
been attributed to the zodiac, and there have even been dodecahedral calendars.
The symmetries discussed above exist in a dual form here too.
The
dodecahedron can be seen as the union of five intersecting cubes, whose corners
touch the vertices of the dodecahedron (this is a rather complex structure and
hard to visualize). At each vertex three different cubes meet. Along each side
of the dodecahedron an edge from a cube runs, creating a rather neat system of
correspondences between the five elements and the edges like the system
mentioned above for the icosahedron.
Another
way of placing polyhedrons in the dodecahedron is to use five intersecting
tetrahedrons, whose corners touch the vertices. This is a most elegant
configuration where the tetrahedrons seem to twist around each other. It exists
in two different forms, essentially corresponding to clockwise and
counterclockwise rotation. The space occupied by all five tetrahedrons is a
smaller icosahedron, another nice example of the power of duals. It could
perhaps be seen as a "construction drawing" of Fire, where the Quinta
Essentia takes on its various elemental properties, and combines them in an
eternally rotating and twisting form.
The
evolution of the Quinta Essentia into the four elements may thus be described as
follows: The original form of the Tetrahedron is created out of the primordial
chaos by being the simplest and most stable form. It combines in various ways
with itself, either by moving and mixing, forming the Dodecahedron and Fire, or
by linking together and building the Icosahedron and primordial Water. However,
while both polyhedrons are close to being perfect spheres, they don't fit
together. These imperfect interactions between the growing numbers of
polyhedrons force them to order themselves according to cubical symmetries, and
Earth and Air are formed. As we will see, this fits with some results within
group theory.
Before
we shift our focus to the abstract properties of groups, its worth mentioning
that there exist other polyhedrons of potential magical interest.
Click
on pictures to enlarge!
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