From: nv91-asa@eufrat.nada.kth.se (Anders Sandberg)
Newsgroups: alt.magick
Subject: Mathematical Magick
Date: 14 Dec 1994 10:58:00 GMT
Some Random Thoughts about the Occult Correspondences of the Platonic
Solids and Their Symmetries
By Anders Sandberg
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
polyhedra available to visualise 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
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
publicised their results, which is the reason they bear his name. Here
is a table with their properties (based on [C]):
Faces Edges Vertices Schäfli 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 Water
[ The Schäfli 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 polyhedra are duals if the vertices of one correspond one-to-one
to the centres 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 Icosahedron 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, Saggittaurius 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 centre).
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):
Moon Water-Fire
Mercury Air-Fire
Venus Water-Earth
Mars Fire-Earth
Jupiter Air-Water
Saturn Earth-Air
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 centre 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
The Cube naturally corresponds to Earth. It is stable, the basis of
western architecture and salt crystallises 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 centre. 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 stabilise it
completely (spirit stabilises 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 quasicrystals belong to the icosahedral
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 doedecahedron, belonging to 5.3.2 have links to the five elements
and the trinities and dualities. Thus they correspond closer to the
whole system than the more material elements, which deal with just the
four elements.
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
corresponences (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 tetrahedons 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 betwen 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
magickal interest.
Other Polyhedra
One such set is the Kepler-Poinsot polyhedra, also known as star
polyhedra. They are a generalisation of the platonic solids, where
faces no longer have to be normal polygons but can be star-polygons
(like pentagrams) instead, and they may intersect. The four star
polyhedrons are called the small stellated dodechahedron, the great
stellated dodechahedron, the great dodecahedron and the great
icosahedron. They all belong to the same symetry group as the
dodecahedron and icosahedron. These fascinating polyhedrons can be
seen to correspond to the four elements. The small stellated
dodecahedron has pentagrams as faces. The great dodecahedron, its
dual, has intersecting pentagonal faces. The great stellated
dodecahedron has also pentagram faces and its dual, the great
icosahedron has triangular faces.
I would say the great icosahedron corresponds to Fire (its faces are
triangles, and it opens almost like an erupting flower). Its dual, the
great stellated dodecahedron correspond to Water. The small stellated
dodecahedron, with its pentagonal pyramids rising from the
pentagrammal faces correspond to Air, and the almost asteroidlike (it
looks a bit like a sphere with dents) great dodecahedron as Earth.
Like I mentioned above, the {5.3.2} group seems to exist on a higher
level (Perhaps in Briah if the {4.3.2} group exists in Yetzira),
containing the symbolism and patterns of the other group in abstract
form.
Beside these polyhedrons, there are the Archimedian polyhedrons. These
are polyhedrons where the faces can be different regular polygons (no
intersections or star-faces allowed). There are 14 of these, with no
known occult connotations. This is an area where much further
discovery is possible.
The Magick of Groups
Groups are among the most useful mathematical concepts, and can be
readily applied to magick (what cannot be applied?). A Group is a set
with an associated binary operation * on the set, with the following
three axioms:
1 There is an element e in the set such that e*x=x*e=x for all x.
2 The operation * is associative (a*b)*c=a*(b*c).
3 There exists an element x' for every x in the group, such that
x'*x=x*x'=e.
Note that these three axioms fit quite well with the three supernal
sepiroth. However, the * operation seldom seems to have any obvious
occult interpretation.
The simplest group (except the unit group with just a unit element) is
the cyclic 2-group Z2. It has two elements which correspond to the two
sides of a duality. The next simplest is of course the cyclic 3-group
with three elements, corresponding to a trinity.
The most important group in algebraic magick is of course the group of
the four elements. However, there are two groups with four elements,
the cyclic group Z4 and the Klein 4-group. The respective
multiplication tables are:
Z4 Klein
e a b c e a b c
e e a b c e a b c
a a b c e a e c b
b b c e a b c e a
c c e a b c b a e
I use e= Earth, a=Air, b=Water c=Fire.
To which group does the four element belong? This depends a bit on
your perspective on the elements. If one sees them as a cyclic
organisation, where each element is succeeded by the next, Z4 is a
natural choice. However, if one carefully studies the Klein group, one
sees that it consists of two parts. The material elements form a sub-
group, where the interactions between Air and Earth form only
themselves. When the astral elements are added, the interactions
between themselves also form the material elements (the descent from
the astral to the material plane). However, when they interact with
matter, it can be elevated to the astral level. The group is also the
cartesian product of two 2-groups, which fits in well with the
division between actives/passives and astrals/materials. This is the
reason I think the group fits best for the elements.
If the elements correspond to the Klein 4-group, what does the cyclic
group correspond to? If one studies the quadruplicities in the
litterature, one quickly find that practically all sets of four
symbols correspond to the four elements. However, there seems to be
one important quadruplicity which fits the cyclic group, the INRI
formula. It denotes a linear progression, but at the same time the
first and last step are the same. In terms of group theory and the
Golden Dawn system, one could say that the first Yud represents the
stable, unenlightened state. Nun, death and destruction, forces a
change which leads to Resh, rebirth and light which becomes the second
Yud, representing the relatively enlightened state where the process
can begin anew ("Every day is an initiation").
Multiplication table:
I1 N R I2
I1 I1 N R I2
N N R I2 I1
R R I2 I1 N
I2 I2 I1 N R
[I1 and I2 represents the first and second yud, respectively]
Its worth noting that the orbit of Nun (the elements x, x^2,x^3...)
generates the whole group, it goes forward all the time. The orbit of
the first Yud is simply itself, the unenlightened state cannot change
without any external stimuli. Resh, the sun, has an orbit spanning
itself and the first yud. Light in itself can only become no light or
more light, not something else. However, the second yud has an orbit
which moves backwards along the sequence, ending up at the first yud
and then continuing around to itself. The enlightened are able to move
as they want, and have no fear of light, death or being unenlightened.
If we look at the orbits of the Klein group in the same way, we find
that each element has an orbit consisting of itself and the unity
element earth; a pure element can become material, but not create
anything else. A mixture of element is necessary to create the whole
universe.However, it is necessary only to start with two elements to
create it; fire and water can produce earth and air (but not vice
versa).
There are just one group of order five, the cyclic 5-group Z5 since
the only groups of prime orders are cyclic (this follows from the very
useful theorem of Lagrange which states that the orders of subgroups
must divide the order of the group). It is interesting to see that the
elements in themselves might form a non-cyclic group, but when Spirit
is added, the result is cyclic. Fire emanates from spirit, and is then
in turn transformed through Water, Air and Earth until it returns to
its source.
The orbits of the Z5 group naturally lead to the theory of lineal
figures (pentagrams, hexagrams etc.). A cyclic group can be generated
by an element in different ways, depending on its size. Z5 can be
generated in essentially two ways: by going through each of its
elements in turn (1234512345123...) or by going through every two
elements (135241352413...). This corresponds to the pentagon and
pentagram respectively. It can be shown that for a cyclic group of
order n, every choice of a "step length" k which is relatively prime
to n creates a lineal figure (other choices doesn't generate the whole
group). For example, the cyclic group of order 7 can be generated by
steps of length 1 (the heptagon), 2 (the "even heptagram)" and 3 (the
"spiky heptagram"). The other choices will just create these three
figures. For the cyclic group of order 8 there are just the octagon
and octagram (stepsize 3 or 5), the steps of size 2,4 and 6 produces
just two nested squares. For further correspondences of these figures,
see the paper on lineal figures in [IR].
The most important group of order six is the permutation group of
three elements (also known as the symmetric group of three letters),
S3. It consists of the six possible permutations of three letters.
These permutations are mentioned in the Sepher Yetzirah (which in fact
is a precursor to the study of permutation groups), where the 6
permutations of the three different letters Yud, Heh and Vau of the
Name were used to create the directions (this is also used in the
Yetziratic Sealing Rite). These permutations fit with the corners of
an octahedron or the faces of a cube.
Such symmetric groups are very important, since it can be shown
(Cayley's Theorem) that all groups are isomorphic to a group of
permutations (which are subgroups of the symmetric groups). This means
that if we have a group (say the group of elements), we can interpret
each elements as an operation on a word. For example, if we look at
the permutations of a four-letter word, the sub-group of permutations
generated by the interchange of the first two or the last two letters
is isomorphic to the klein group corresponding to the elements (the
elements correspond to the identity,(12),(34) and (12)(34) in cycle-
notation). From an occult standpoint this is of course a natural
interpretation of temura, which essentially deals with operations on
different entities through the mediation of letter-permuations, not
unlike how physicists study physical objects by studying the equations
describing them.
This introduces an interesting question in the Cabala, what other
permutation-subgroups of the tetragrammaton have magickal
correspondences? There are 24 elements in the group. This makes it
possible to have subgrups of order 2,3,4,6,8 and 12. The subgrups of
order 2,3 and 4 have already been discussed.The sub-group of order 12
is known as the alternating group, and consists of all the even
permutations of the tetragrammaton. Its interesting to note that 12
permutations of the tetragrammaton are used linked to the zodiac and
12 tribes of Israel [1], but they are unfortunately not purely the
even permutations.
The zodiac quite obviously corresponds to the cartesian product of a
trinity (the alchemical elements) and the four elements (the klein
group, which is Z2 * Z2). The result is Z3 * V = Z12, a quite neat
cyclic group wich fits well with the cyclical nature of the zodiac.
Various elements generate the whole group or subgroups (like the
quadruplicities and trinities).
However, this essay would become far too long if I elaborated on the
interesting details of this (and more complex groups). Instead I
encourage the reader to study and contemplate the meanings of the
various mathematical concepts I have described, and find new
structures. Mathematics is filled with fascinating objects, and their
subtle interrelations and symbolism is an source of endless beauty.
~References:
[CL] Colin Low, Some Notes on the Quabalah
[C] H.S.M. Coxeter, Regular Polytopes
[DK] Donald Kraig, Modern Magic
[F] John B. Fraleigh, Abstract Algebra
[IR] Israel Regardie, The Complete Golden Dawn System of Magic
[RW] Robert Williams, The Geometrical Foundation of Natural Structure
--
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Anders Sandberg Towards Ascension!
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