Geometric Symbolism and Sacred Forms

However, in other contexts, especially in pre-Christian times, the same geometric figure could have been placed in relation to other fundamental numbers. Similarly, the solid form of the pyramid is given by the combination of several distinct numerical elements such as the point, triangle, and square. However, this discourse is not exclusive to solid figures, but also applies to plane figures, especially irregular ones. For example, the sacred triangle with sides 3, 4, and 5 reconciles within itself all the first ten numbers even though it primarily manifests the number 3. Similarly, a single line refers to the number 2 when thought of as connecting two distinct points, but to the number 1 when thought of as a side in the formation of polygons. However, restricting ourselves to the realm of two dimensions and regular forms, generally, the characteristic number of a geometric form is given by the number of distinct vertices. Number 4, for example, can correspond to a square as well as to a cross, and number 5 to a pentagon as well as to a pentagram. Therefore, both the pentagon and the pentagram will express the virtues of number 5, although the pentagram (the stellar polygon) in a more active, dynamic, and exteriorized aspect, while the pentagon (the convex polygon) in a more receptive, stable, and interiorized aspect. The city is square-shaped, its length equals its width. The angel measured the city with the rod: twelve thousand stadia; the length, width, and height are equal. (Revelation 21:16) The cube is the geometric figure that more than any other expresses the idea of stability and strength, suitable therefore to symbolize the Holy City, in which the presence of God dwells in strength and without any diminution. The cube in fact corresponds symbolically to the full identification of the celestial principle with the terrestrial principle insofar as a sphere of unit radius contains inscribed within itself a cube of unit side. The two figures, the cube and the sphere, are thus perfectly harmonized with each other, in perfect resonance to form a single living reality. Geometric forms, by their nature, are more linked to a dynamic aspect of number than to a qualitative aspect of number. A regular geometric form, therefore, in its proper aspect, represents a number of elements linked among themselves by a dynamic or hierarchical law. For example, the triangle properly represents the ternary, that is, the hierarchical and dynamic resultant coming from three distinct elements with their mutual relationships. Similarly, the square represents the quaternary and the pentagon properly represents the quinary, and so on. However, from a complementary point of view, the same figures of triangle, square, and pentagon can be considered as simple expressions of number qualities. In fact, to give some form to numerical qualities or to the mystical conceptions evoked by them (as happens for example in the case of evocative numbers), one must assign them geometric forms anyway. Giving a specific form to these conceptions and qualities means making their study possible and obtaining support for meditations and metaphysical speculations. In these cases, however, geometric forms must be interpreted differently from the usual and no longer considered as expressions of a dynamic law, but as figurative expressions of a quality. For example, the representation of number 8 through an 8-pointed star as a symbol of the Mystery of the Eighth Day must be clearly distinguished conceptually and in meditative practice from the representation of number 8 as an octonary. Similarly, the representation of number 3 as Gnosis of the Holy Trinity through the triangle must be distinguished from the ternary. When one observes in every number the expression of quality, one must be careful to distinguish the ternary from the triangle, the quaternary from the square, the quinary from the pentagon, and give to numbers 3, 4, 5 the way of considering them represented by geometric figures, so as to be able to analyze them. Thus, to simplify, we can classify three fundamental ways in which regular geometric forms can be interpreted: • as figurative expression of the identity of a number; • as figuration of a quality of the number; • as figuration of a reality evoked by the number. The choice among these three fundamental ways will obviously depend on the specific context. When in practice we deal not with a single symbol but with a constellation of symbols, the disposition with which these symbols present themselves appears fundamental for a correct interpretation of the whole. To facilitate a first approach to the subject, we have synthesized some general guidelines useful for interpreting more complex geometric compositions. It is important to emphasize that these are only general guidelines and not rigid formal rules; however, they are useful in interpreting a large part of the geometric figures of traditional symbolic literature. First, one must note the orientation of individual figures. The same figure, if oriented differently, can assume completely opposite meanings. For example, the five-pointed star represents Divine Light, the Word, and man's elevation if turned with the point upward, or darkness, blind power, and man's brutalization if with the point turned downward. Therefore, the orientation of individual elements of the composition constitutes an important point in interpreting its whole. First, in the composition it is necessary to distinguish two fundamental lines: • the horizontal one, which generally refers to exterior or material life; • the vertical one, which instead indicates interior or spiritual life. For example, a line drawn horizontally will indicate an exterior, human, or horizontal action; a line traced vertically will indicate an interior, divine, or vertical action. A cross, given by the union of the two lines, will therefore indicate the union between a human or horizontal action and a divine or vertical action. Generally, geometric compositions are made to extend preferably along the vertical axis rather than the horizontal one. This happens both to facilitate interior recollection and to respect the symbolism just described. Similarly, when we are faced with a convex or stellated polygon, it is important to distinguish the orientation of the figure, which is generally given by the central point. When the central point is present, it determines the orientation of the figure as well as the orientation of its action. If the point is oriented: • upward: indicates an ascending action • downward: indicates a descending action For example, the triangle turned downward can indicate the descent of the Word, while the triangle upward the subsequent elevation of Man. The union of the two triangles to form Solomon's seal in this context indicates the double nature of the Messiah and the immanence of God in Man. Sometimes different figures are arranged one inside the other, often through the mediation of circles with respect to which geometric figures are inscribed and circumscribed. Normally, if the circle is external to the figure, it means that the force expressed by the figure is interiorized, invoked, or impressed; if instead the figure is external to the circle, the force expressed by it is exteriorized, evoked, or expressed. When instead we speak of other figures, generally the interior vivifies the exterior and the exterior protects the interior. The figure inside makes itself known through the one outside, and the one outside is the manifestation of the one inside. Therefore, generally, the interior is the true subject while the exterior is its expression. For example, a triangle inside a square can indicate, for hermetic symbolism, hidden fire within matter, occult fire. Generally, the more a figure is inside the composition, the more its action is intimate and hidden; the more it is external, the more its action is visible and manifest. In the disposition along the vertical line, one must bear in mind that what is above is hierarchically superior to what is below. Therefore, when a figure appears under another, the latter is generally to be considered the most important. The first figure above therefore determines the subject of the composition or at least the primary influence. These remain, however, general notions particularly valid in the case of rectangular compositions. In the case of figures with circular symmetry, in fact, the greatest importance is to be attributed to the center of the design. In the case of rectangular compositions, there is often a qualitative affinity that unites figures arranged along the same vertical line. In particular, it often happens that figures along the right vertical line express an active faculty, a force to be projected, or an action of multiplication or mercy. Vice versa, figures arranged on the left often express a receptive faculty, a force necessary for control, or an action of division or rigor. The central line in these contexts is the balancing point; in particular, figures along the central line can be considered as distinct manifestations of the same force acting on various planes. The repetition of a symbol is often understood as reinforcing or superlative. However, the number of times the symbol is repeated can become significant and express different concepts depending on whether the symbol is repeated two, three, or four times. For example, the triple repetition of a symbol indicates its plenitude, complete empowerment. The three in this case does not indicate a specific number as much as an idea of completeness, that is, the idea of a complete plural, plenitude. This is because the three has in itself an idea of completeness, autonomy, and perfection. Similarly, the repetition of three external circles to an inscribed figure has the purpose of multiplying its effect. From the observation of traditional correspondences between numbers and geometric symbols, we note that every number can be visualized in distinct ways according to its arithmetic composition. For example, number 6 can be visualized as a hexagon if thought of as 6, as a double triangle if conceived as 3+3, or as a square containing two lines if visualized as 2+4. Each of these figures identifies and determines a specific aspect of the number, specifying its meaning. As 3x2 it indicates the Trinity blessing the polarity of the sexes, but as 2+4 it symbolizes the material base of the sexes that predominates. It is in this latter sense therefore that it must be avoided and on which the ascetic must not meditate. Every number therefore assumes different geometric forms that specify its arithmetic composition. The choice of figure within those corresponding to a number is therefore not arbitrary, but must be carefully pondered, as this goes to specify the meaning of the number. In traditional representations, stellar polygons are sometimes represented with a central point that deserves separate discussion. The central point represents the original and vivifying principle from which the star draws its origin and strength. It is hidden, intimate and invisible, transcendent with respect to the polygon that constitutes its expression and protection. The central point is in fact invisible in its essence but at the same time manifest, being precisely the central point, the generator of the powers that are ordered according to the lines of the external polygon. It refers the entire figure to its essential and transcendent unity, giving the entire figure a new depth and spiritual dimension. It symbolizes the divine presence and in this perspective can be substituted in iconographic symbolism by the representation of an eye or a divine name. In a triangle, for example, the central point refers to the intimate unity of the vertices and is taken as a symbol of the essential Unity of the Divine Trinity. In the Star of David, the central point is also called the center of silence. It represents the point of concentration of the Divine Presence, the center of gravity of the Infinite Spirit, the beating heart of all life and all light that radiates to the confines of Creation. As Clement of Alexandria writes: "From God, Heart of the Universe, emanate the six phases of time and the six dimensions of space; this is the secret of number seven." From an arithmetic point of view, therefore, the addition of the central point can be considered as the addition of a unity that however maintains a character superior to the number to which it is added and is in a certain way transcendent to it. Thus in a certain sense the Star of David with the addition of a central point equals the expression 1+6. Therefore, on one hand, the image in question is in relation to number 7, being this 1+6; on the other, it remains essentially linked to number 6, insofar as the central point remains hidden and transcendent with respect to the other six visible ones. In a purely philosophical perspective, in fact, the central point must always be considered as present even when invisible and unexpressed. This discourse is specified by John Dee in the second theorem of the hieroglyphic monad: "In the beginning things began to exist through the work of the point and the monad. Moreover, those things that are placed at the periphery - however great they may be - cannot in any way be deprived of the function of the central point." A separate discourse must be made for hermetic and alchemical symbolism. In the medieval and Renaissance period, we witness the flourishing of a true and proper alchemical hieroglyphic language that often resorts to classic geometric elements such as the circle, lunar crescent, triangle, and square. These geometric elements are combined with each other so as to indicate alchemical substances, operations to be performed, or times to be respected. The resulting language is extremely articulated and varied and constitutes a very interesting subject, but which we will not have the opportunity to explore in depth. Some symbols in fact come from canonical abbreviations, others from glyphs written in coded alphabets, still others from astrological symbols; therefore, a detailed treatment of symbols would require much time and would have little to do with the geometric aspect proper. However, some of the fundamental alchemical symbols have become so commonly used as to be now traditional and require treatment even if synthetic. A large part of these symbols are generated by 4 fundamental figures in relation to the first four numbers, namely the circle, lunar crescent, triangle, and square. To these four fundamental figures must be added the image of a cross + with which they are combined. We thus have that combining with the circle, the cross produces the symbol of copper, verdigris, and antimony. Combining with the lunar crescent we have tin as well as lead, with the triangle instead we obtain red sulfur or the symbol of the great work, and finally with the square we have tartar as well as the philosopher's stone. In addition to elements of alchemical practice, these hieroglyphic images correspond to fundamental concepts of hermetic philosophy. In this perspective, specifies Oswald Wirth, closed figures correspond to different orders of substances, susceptible to changing state or destination according to the indication of the cross added to them. The cross is in this case to be interpreted as a corrosive, purifying, and ordering action. If the sign of the cross is found under the geometric figure, then the ordering and purifying action is still to be accomplished and the substance is subject to corrosion; if instead it is found above the figure, the substance has already been perfected and has therefore become incorruptible. For the rest, the rules of composition are similar to those exposed in the preceding paragraphs. Therefore, the study of the composition of various glyphs and knowledge of fundamental signs according to hermetic philosophy provide the meaning of the material to use, the state of consciousness to awaken, or the operation to apply according to the type of alchemy being considered. In the field of traditional symbolic geometry, the elements that have a more or less stable canonical meaning are several hundred. It is therefore out of place to present a list that even has the vellum of being exhaustive. However, this chapter would be incomplete if it did not provide a brief review of some traditional elements that can serve as a basis for study for one's research. In this perspective, we have thought to present a small series of examples that can serve as a guide to orient the scholar and help him gain confidence with the subject. However, the explanatory phrases of each geometric figure are nothing other than general guidelines and can serve only to have a first orientation on geometric symbolism. Each figure, precisely because of its natural abstractness, encompasses within itself infinite meanings, all united by the same qualitative affinity. This quality being abstract, it can take on numerous apparently distinct meanings and apparently lacking connection, but which upon deeper analysis instead come from a single idea. An analytical exposition of the possible symbolic ramifications of these figures would require numerous pages and would certainly remain incomplete. We have therefore preferred to associate with each figure a few lines and leave to the scholar the pleasure of autonomously exploring the symbolic confines of these geometric realities. To deepen the historical aspect of these symbols in relation to various traditions and cultures, we can refer to the well-known Dictionary of Symbols by Chevalier and Gheerbrant. • Point: The Monad. The indivisible Unity endowed with an active and propulsive force. The generator of all forms. Unity in its projective aspect as origin and source of every force. • Circle: Unity in its receptive aspect. The circle corresponds to Unity because Unity is not only the center, but also the circumference of everything. It can also represent Eternity and be associated with the symbol of time which is circular. The absence of a point and the lack of an active and dynamic aspect makes this symbol suitable to represent the cosmic void from which creation was drawn. In this perspective, it is therefore associated with number 0. Otherwise, it is also associated with number 1 and number 10. The circle being the most perfect figure is considered ideal to protect from imperfections, larvae, and demonic spirits. For this reason, it is traditional use to resort to a circle for exorcisms and protections. • Circle with Point: This sign is also called the Eye of God. Symbol of the Sun since ancient Egypt where it also indicated the Sun God Ra. In a purely theological context, it is interesting to report Isaac Luria's opinion regarding this symbol reported by his disciple Vital in his 'etz hayyim with these words: "When the pure will conceived of creating worlds and producing emanations [...] then the En Sof concentrated in the middle point, placed at the exact center of its light. The light concentrated and withdrew all around that central point; precisely from the middle point there remained then a free, empty, vacuous and hollow space in this way: ⊙". Numerically it is worth number 1 and represents Unity in its entirety. Similarly, it also represents number 10, that is, Unity in its expressive plenitude. In this sense it symbolizes the relationship between God and Creation. This aspect of the symbol is explained by Nicholas of Cusa in his Learned Ignorance: "The divine mind is like the point that stands at the center and the soul of the world is like the circle that explicates the center." In a cosmogonic context instead it represents the entire Universe. In the human domain it corresponds to Will, to the I Am, to the most elevated aspect of Man. In alchemy the symbol symbolizes the metal gold. • Circle with Vertical Diameter: The vertical line suggests the ascending thrust, active self-overcoming, as well as dynamic instability. In an alchemical context it is the symbol of Nitre, also called infernal salt in the sense of destructive purification. Sometimes it is also indicated as the symbol of Time and instantaneity. The vertical line also stands for generation, for the seed that is grafted and must develop. In Lurianic Kabbalah this sign is associated with the second creative act as read again in Vital's Tree of Life: "After this concentration there was a space where emanations could exist..."