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For thousands of years, sacred geometry has been used as a means to illustrate states of universal order. The archetypal patterns that occur through the relationship of numbers have appeared the world over in the form of medicine wheels, mandalas, mosaics, weaving, pottery, architecture, stained glass windows etc.
The circle is the master shape in sacred geometry and is a two dimensional representation of the sphere. It symbolizes a state of undifferentiated potential from which all primary shapes emerge. The three primary shapes that develop from uniform divisions of the circle are the equilateral triangle, the square and the regular pentagon. The generic term for them is regular polygons.
They can then be combined to form other polygons:
On the three dimensional plane, it is through the uniform divisions of the sphere that five primary forms are created, known as the five regular polyhedra, or the five Platonic solids (named after Plato who used these forms as a metaphor of universal order in his dialogue Timeaus).
The primary shapes and forms are the fundamental archetypes of creation, describing the self-organizing forces that shape and define the world. Each one has the ability to generate the proportional mathematical systems that are found in nature.
The shapes (polygons) indicate how energy is compressed and organized. Energies are controlled, specified and modified through the effects of angulation. The diagonal measurements that exist between the vertices (the points of the polygons) are called the root powers or root functions.
The ratios (relationships) that are generated by the three primary shapes are the same proportional systems found throughout nature.
(N.B. Because a triangle does not have an opposing point the diagonal extends from point to base).
The mathematical root can, in certain regards, be likened to the vegetal root. In both cases the root represents the principle of growth and transformation. The vegetal root is embedded in the earth. It secretes acids to break down (divide) the fixed, dense mineral compounds of the soil into miniscule constituents that the plant can assimilate and transform into its own tissue, transforming mineral to vegetal.
The mathematical root, on the other hand, is embedded in the polygon. It divides (breaks down) whole numbers into unending ratios—numbers that stretch off to infinity and cannot be defined numerically (except as an approximation). In doing so, the finite (the whole number measuring the shape) is seeded with the infinite (the root power with its unending ratio).
In theosophical geometry the root power symbolizes the interplay of spirit and form; force and matter; the finite and the infinite; the known and the unknowable. For thousands of years sacred geometry has been used as a path for the journey home—the integration of mind-body-spirit. The intrinsic root powers of the primary shapes and forms are unique in their ability to consistently replicate through endless divisions in space. The scale changes continuously – evolving or devolving depending on whether it is being multiplied or divided – but the ratio remains constant in the relationship of one level of subdivision to the next.
Through certain applications of division and multiplication, the process of infinite contraction and infinite expansion can be seen. In the example shown below, the circle is uniformly divided at three points (i), producing an equilateral triangle (ii). An inverted triangle is then included, producing a six pointed star – a hexagram (iii).
The six smaller triangles that comprise the star are then surrounded by six smaller circles (iv) Six more triangles are included to complement the ones existing in the small circles (v). The smaller triangles that compose the smaller hexagrams are then surrounded by smaller circles (vi), and this process can be continued indefinitely.
| The initial boundary of the original circle will never be exceeded, while its inner space is divisible to infinity. Alternately, the process can be applied the other way, so that the initial circle is enclosed by a larger triangle and inverted triangle, making a hexagram star which is then surrounded by a circle which is then surrounded by another hexagram star, ad infinitum. |
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Similarly, the square can be divided and multiplied into processes of infinite contraction or expansion.
| Of the three primary shapes, however, it is the pentagon that has the most transformational qualities. Not only does the pentagon contain an infinite bed of golden ratios; it is also a golden key that opens to a geometric world of profound order and sublime beauty. |
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When the pentagon is dissected by its internal radiants (the root powers), two types of golden proportioned triangles are produced.
These in turn can be reduced to smaller, similar triangles as well as miniature pentagons (below). Every triangle and pentagon created through these divisions relates to all the others through the harmonic of phi.
Combined in a different way, the triangles are able to generate a golden proportioned spiral. As in the previous example, the areas of the triangles are all in phi proportion with each other.
By following harmonic pathways, intricate and exquisite patterns unfold and a profound order is revealed. Diverse shapes join together in math-magical compliance, revealing continuous, timeless, universal actions that are otherwise invisible and can only be known to us through geometry.
Many universal principles are portrayed, such as balance, coalescence, coherence, equilibrium, harmony, integration, interconnectivity, optimalization, stability, sustainability, symbiosis, symmetry, synergy, unity etc.
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There is truth and purity in natural things and our contact with them nourishes the soul and illumines the mind. Plato said that “Geometry purifies the eye of the soul, since it is by it alone that we contemplate the truth”. |
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