Sacred Geometry and Mathematical Beauty: Philosophical Insights on Divine Forms
From a philosophical point of view, force is an essential characteristic of beings: everything that exists has its own force and its own action by which it relates to the external world. If in fact a being had no force, no action, it would be invisible to the surrounding universe, would have no relationship with it. The surrounding universe would be incommensurable to it, and it would not be part of it and therefore would not exist in the ordinary sense of the term. In this philosophical context, space is that principle which unites the centers of force among themselves, which includes the forces within itself and gives them the possibility to unfold. The result of this unfolding and interaction between centers of force is a geometric form that corresponds to them. The resulting geometric form can therefore be a material form, when the action of these centers of force is exerted on matter, shaping it as in the case of the earth and gravitational force. Or the geometric form will be a mental form when the action reverberates on thoughts, organizing and arranging them as happens with philosophical systems. Without modifying our definition in any way, we can also consider as geometric an animic form when these forces reverberate on the soul, as in the case of vices and virtues and so on. In synthesis, the geometric form is the symbol or sign of an acting force. The goal of geometry is declaredly the realization of beauty. But what is meant by the beauty of a mathematical formula? In regard to this question, many definitions have been proposed. One of these, and which is the most similar to the way we have presented the concepts, is that a mathematical formula is beautiful insofar as it represents an idea. The beauty of a formula is evidently connected to the idea it expresses. If, in fact, the formula does not express any idea, or if this idea is not received, it becomes only a series of meaningless letters or glyphs and suddenly loses all beauty. Not only, therefore, does the beauty of a formula reside in its capacity to express an idea, but the deeper this idea is, that is, the greater its specific content or meaning, the more beautiful the exterior formula appears. Euler's identity, for example, is often cited as one of the most beautiful formulas of all time. It links together three radically distinct numbers coming from approaches distant from one another. It in fact relates: e (Napier's number), typically coming from combinatorial or analytical fields; i, the imaginary unit, typically coming from algebraic or arithmetic environments; and π (pi), typically coming from geometric environments. These three numbers are then united in a single, simple formula: e^(iπ) + 1 = 0. What makes the formula beautiful is clearly not a purely graphic factor, but the idea it manifests. In particular, what makes this formula beautiful is its capacity to manifest the union of these three distinct elements in a simple and evident way. From our perspective, as for Plato, beauty is the splendor of truth, or the sensible aspect made of truth. In other words, for the philosophical approach we use, beauty is the incarnation of an idea. The beauty of the forms and geometric constructions we will present will not reside therefore simply in a symmetry or constructive harmony but principally in the capacity of these images to incarnate transcendent concepts and ideas. In the philosophical system we have presented, we have distinguished in every being three fundamental aspects: concerning its freedom or autonomy, the fullness of its content or meaning, and its expression through action and form. In this perspective we have said that beauty is the idea or content made perceptible to being. Considered prevalently in its interior absoluteness, as the absolutely desirable or wanted, the idea is the good; considered in the fullness of particular determinations it embraces, as the content thought by the intellect, the idea is truth; considered in the perfection or completeness of its incarnation, as really perceptible to the sensible being, the idea is beauty. The geometric symbol is therefore an epiphanic symbol, a symbol that makes truth sensible, provides it with a body. It in a certain sense realizes truth and therefore manifests beauty. In this perspective, the great intellect that distinguished itself in the philosopher Plato affirmed that God Geometrizes. This affirmation, simple yet profound, identifies divine action with a principally geometric action, that is, of hierarchization or disposition. From the most remote antiquity, geometric figures, for their inherent characteristics, have been associated with divine or theurgical action and considered ideal in representing communion between celestial and terrestrial realities. If on one hand, geometric forms, for their abstractness, suggest to consciousness a spiritual plane distant from the corruption of matter, on the other hand their formal representation necessarily leads back to a corporeal and therefore in some way terrestrial aspect. Geometric figures thus appear as a first corporeification of spirit and their use in the cult of divinity is found transversally in all cultures. The religious use of geometry is in fact testified by the essential and geometric lines of Egyptian pyramids, as well as by the spherical cupolas of Buddhist stupas, or by the Etruscan cultic dodecahedra as well as by the cubic altars of countless cultures. All this without considering the use of lines and geometric realities testified by the mandalas of Indo-Tibetan gnosis, by the geometric pentacles of Greek and Hebrew tradition as well as by the numerous icons of Christian iconographic tradition. Although operating in completely different cultural and cultic contexts, the sense of the use of geometry in these representations is always the same, that is, that of representing a force in a pure state, that is, free from all contamination, and which, under certain conditions, can be made explicit and act on consciousness by teaching it the way of reintegration. At the base of every ritual cult of divinity there is the law of analogies or correspondences that permits communion between a reality of divine uncreated character and the reality of nature of creatural character. The principles on which the law of analogies or correspondences is based are, in fact, synthesized in three fundamental points: 1) the unity of the World, in all its parts; 2) the analogical identity of the Divine Archetypal Plane and the material Universe, the second created in the image of the first and its reflection, inferior and imperfect; 3) a permanent relationship between the two, for the analogical identity that can be expressed by the Science of Symbols. Geometric forms, purest and most abstract entities, have the scope of symbolizing archetypal realities of divine character, perfect, thus reconstituting in the place where they are used a cosmogram immune from sin, preserved from every disintegrating or demonic force. In the mandala, as in the icon, as in the Christian, Islamic, Buddhist or Hindu temple, an image of the perfect cosmos is created, in which the arcane play of forces can be assimilated by consciousness thus accomplishing the process of spiritual transmutation.