The basic shapes are crucial to mysticism and to magick. Shapes are perhaps even more fundamental in our understanding of Universe than numbers. They describe spatial relationships, usually between things at particular points in space. While a set of equations or numerical expressions can be used to describe a shape, they do not really describe it; for example, the equation
describes a circle, but does not really hint at the fundamental roundness of the circle. This roundness is a visual and tactile experience rather than an abstract one, but is no less an important part of the circle than its diameter. Since it engages the senses in a direct way, shapes affect the brain in different ways from numbers. (The sense of hearing has, as well, its own pattern methodologies that humans have held sacred -- certain harmonies and musical relationships, rhythms, speech sounds, mantras, and so on-- and some of these will be explored on the Math and Magick page as well.)
The circle is an almost universal symbol of harmony and wholeness. While fractal geometers such as Benoit Mandelbrot can point out that "mountains are not cones" and "rivers are not straight lines," some of the basic shapes have been discovered and used as sacred symbols by cultures the world over. But why the circle for harmony and wholeness? The fact that we find this symbol not only in European designs but in pre-Columbian American designs as well, often with a cross in the center, indicates that it creates some sort of resonance in the mind. It is aesthetically pleasing and relatively simple to construct.
It is relatively easy to envision the circle as a symbol of harmony. All points on the circle are the same distance from the center; there are no gross distortions, nothing gains at the cost of anything else. Mystics who understand the principle of balance can appreciate the utter balance of the circle. The sun and moon (when full) appear as circular discs as well. Since currency was invented, coins were made into circles which appear roughly the size of the sun-disc (and moon-disc) when beheld at a natural 2/3-arms-length. The motion of sun across the sky, and the shape of the rainbow, are suggestive of the circle as well. The notion of circle as a symbol of wholeness might have derived from the observation that one can turn around in a circle and see roughly the same distance in every direction. Many cultures in human history, particularly those with a keen sense of astronomy, have been aware that the Earth is spherical; many of those cultures who were (or are) unaware of Earth's spherical shape postulate a disc shape.
This disc of earth now manifests as the magickal weapon of the pantacle, and its power is the same -- the power of harmony, of stability, of wholeness. The magician might wield this weapon and draw from its strength when she senses that a fundamental imbalance or lack of harmony has occured.
The circle with a cross at the center, dividing it fourfold, is a universal figure as well, and can be found in designs the world over. This is the shape of the mandala; images drawn in this way, from the formal mandalas of India, to the spontaneous images of the kaleidoscope, to the "atavistic" paintings of Jungian enthusiasts wishing to understand the subconscious on a more basic level.
The circle of sky is important too -- this appears in the guise of the star chart; it is interesting to note that even cultures which utilize only rudimentary technology will draw round charts depicting the sky (Hadingham, p. 109, has the photo of an elliptical Pawnee star chart drawn a hundred years ago). Many believe that the stone circles of England, as well as many of the temples of Meso- and South America, were built on astronomical principles -- in particular the "SoLunar" cycle, which describes solar and lunar eclipses. The motions of the planets can be described as orbits around the Sun, or as orbits around Earth with smaller, circular "epicycles" which account for their occasional retrograde motion. The latter perception is depicted in the modern astrological chart.
The dome, or half-sphere, is a primal symbol. The sky appears to be a dome; the dome of sky is treated as a deity in many mythologies. The bowls with which one prepares food are half-spherical. Most huts and other dwellings made of whatever materials are on hand will be dome-shaped or roughly dome-shaped. For these and other reasons the dome is a symbol of shelter, and of being contained by the sacred.
For the later temple architects of Greece, the rounded ends of the early wooden temples were seen as unelegant signs of simpler, less civilized worshippers. These architects strove for an angular, starkly rectangular look to distance their designs from earlier ones. Thus we find something of a dichotomy between rounded structures, which might in a sense represent closer ties to the Earth; and rectangular ones, which might represent the distinct handiwork of humanity seeking to distinguish its works from those of the gods.
The square, because of its fourfold symmetry, and its six-sided three-dimensional expansion the cube, is a universal representation of created order, or multiplicity of form.1 As Bucky Fuller points out numerous times in Synergetics, the square is not a "stable" form when seen as a shape made of vectors. If one makes a square by taping four sticks together, it can be pushed out of shape. It is thus a contrived order. This does not make it less important, for the works of humankind are more contrivance than they are mirror of nature. Thus we often see things divided into fourfold or sixfold symmetry in reminiscence of the square. For example, there are four elements and four directions (though there could just as easily have been six directions or five elements). Since huts and tents are usually round, those seeking to build more formal-appearing structures will opt for a square or rectangular design. (The history of temple building in Greece, for example, exemplifies this.) A square or rectangular structure needs supports to hold it up, hence the origin of the pillar or column as a symbol of preservation and continuance.
The Greek pursuit for the proceedure to "square the circle" is analogous
to the pursuit of Medieval alchemists for the Philosopher's Stone. To create
a circle one needs three two bits of information: where the center is going
to be, what the diameter will be, and the ratio of diameter to circumference,
which is a constant (which we now call
"
").
, to the horror of
the Greek geometers, turned out to be irrational. If one could draw a square
such that its perimeter is equal to the circumference of a given circle,
or whose area is equal to the area of a given circle, the notion of
's irrationality might
be disproven. No way to do this exactly has been found, though there is a
way to accomplish this approximately, using the Golden Ratio
.
x 6/5 = 3.1416404,
which is very nearly equal to
. The proceedure is
sadly too complex to outline here, but it involves the use of pentagon or
pentagram, since these shapes exemplify the Golden Ratio, and also the Vesica
Piscis, which resembles in shape the eye, mouth, and yoni. One who
is familiar with the language of alchemy might recognize the similarity,
at least in feel, this procedure has to alchemical ones. See Lawlor,
pages 74-79.
Robert Lawlor, in Sacred Geometry, shows a Japanese drawing of the process of manifestation from principle. I have approximated this image:
On the right is a circle, representing unity and principle. In the center, moving left, is an equilateral triangle, and on the left, the finished product, a square. Not only does this image have an eerie resemblance to the English spelling of the word DAO, or TAO, this picture does an excellent job of illustrating the relationship, mystically, between the circle, triangle, and square. The circle and square are somehow "static," one representing underlying or abstract reality while the other represents manifest form in its final state. The triangle though represents the process, the verb, the way one thing becomes another. This notion of the triangle as dynamic seems to be just as primal.
Bucky Fuller attempted to see Universe through a child's eyes, and what he came away with is the notion of the triangle, rather than the square, as the fundamental shape of nature. He based this on a personal form of vector geometry, where a line is not a static line but representative of energy. Thus the stability of the triangle, and the shapes based on it (tetrahedron, octahedron, icosahedron, so on) is a stability of dynamic forces pushing against one another, restricting each other's movement, cooperating to make more.
Claude Levi-Strauss used the triangle as a symbol of process as well, by positing a dialectic duality represented by two opposites (like, say, raw and baked) and a third position which represents something in between, a composite (e.g., boiled).
The triangle's angles are "sharp" enough to indicate a lack of inertia; it must always "point" in some direction. In magick and some mystical traditions, triangles pointing up represent fire, as this is suggestive of fire's upward motion; triangles pointing down represent water, as this is suggestive of the cup or vessel. Fire and water are easily vexed, and are in this way fundamentally different from air and earth, which don't, for the most part, appear to be frequently or easily vexed. The triangles of fire and water overlap and interlace to become the hexagram, a symbol (among other things) of human desire to Know the divine.
In Goetic evocation the magician stands in a circle, a place of harmony, while the demon is evoked into a triangle pointing at the magician. The triangle here, being not square, and being only a partial hexagram, represents the disharmony of the demon, it's "sharp edges," and also defines its relationship to its conjuror.
There is a way in which the triangle can know fourness: this is in the diagram of the tetractys:
*
* *
* * *
* * * *
The "triangle" of 4 is 10, and 10 was the number of most importance in the system of Pythagoras; so this relationship, and the diagram of it, was an important one. Tyson, in his edition of Agrippa's Three Books of Occult Philosophy, included a diagram by Robert Fludd of the tetractys as the process of the first four days of creation (p. 256):
Lux
Tenebrae Lux
Tenebrae Aqua Lux
Terra Aqua Aer Ignis
Plato used a triangle to similarly describe numerical processes (Tyson, ed., p. 252):
1 |
Unity | Point |
2 3 |
Prime | Line |
4 9 |
Square | Surface |
8 27 |
Cube | Solid |
1. One might argue that many ancient sources believed, as Agrippa pointed out, that four is the basis of all things whether their origin is divine, human, or natural. However this notion of an inherent four-fold symmetry to all things seems to occur hand-in-hand with the idea that the Universe is arranged around some sort of detailed divine plan. Cultures that do not see a detailed plan behind the workings of Universe seem less dependent on four-fold symmetries and tend to see five-fold, six-fold, and eight-fold patterns just as easily. Thus the idea that four-fold symmetries abound may be itself a fabrication, an imposition of tidiness on Universe. This may be, at least in part, what Crowley was refering to when he wrote: "[Four is] the terrible number of Tetragrammaton, the great enemy. .... The Dyad made Law" (p. 43). We must be careful here -- is Crowley calling four the great enemy, or Tetragrammaton, or both? His wording is unclear.
Bibilography and Works Cited.
Agrippa, H. C. Three Books of Occult Philosophy. Edited and annotated by Donald Tyson. St. Paul, MN: Llewellyn, 1998.
Bamford, Christopher, ed. Rediscovering Sacred Science. Edinburgh, UK: Floris Books, 1994.
Crowley, Aleister. 777 and other Qabalistic Writings. Edited by Israel Regardie. York Beach, ME: Samuel Weiser, 1977.
Fuller, R. Buckminster. Synergetics: Explorations in the Geometry of Thinking. E. J. Applewhite, collaborator. New York, NY: Macmillan Publishing, 1975.
Govinda, Lama Anagarika. The Inner Structure of the I Ching. San Francisco, CA: Wheelwright Press, 1981.
Hadingham, Evan. Early Man and the Cosmos. New York, NY: Walker and Company, 1984.
Lawlor, Robert. Sacred Geometry: Philosophy and Practice. London, UK: Thames and Hudson, 1982.
Terry, Leon. The Mathmen. New York, NY: McGraw-Hill, 1964.
Copyright notice.
This is an original work by Callisto Radiant (T. Roberti) that has been placed on the Web for public use. Callisto Radiant may be reached at Sabrin1315@aol.com. You may share it, copy it, print it, etc., so long as this copyright notice is shared, copied, printed, etc., along with it.