Geometric Construction of Pascal's Triangle through Complete Graphs

This aspect is significant because it illustrates how the generation of this triangle is nothing other than the explication of Unity repeated. In this paragraph we will see how the meaning of this arithmetic construction can be expressed geometrically through the help of complete graphs. Complete Graphs and Pascal's Triangle From a purely geometric point of view, Pascal's triangle can be considered as a generation of the succession of complete graphs. To obtain Pascal's triangle, in fact, it is sufficient to inscribe each complete graph within the circle and consider the preceding subgraphs of each of them. For example, the complete graph of five elements that we have called $K_5$ contains: • 1 pentagon or 1 complete graph of order 5 ($K_5$) • 5 squares or complete graphs of order 4 ($K_4$) • 10 triangles ($K_3$) • 10 lines ($K_2$) • 5 points ($K_1$) and • 1 $K_0$ where by $K_0$ we mean the circle without points. Repeating this counting for each complete graph we thus obtain Pascal's triangle. The image we have represented to explain the counting allows us to understand another fundamental aspect of this geometric generation. The highlighted figures we have used to construct Pascal's triangle indeed suggest a cyclical movement that, from a symbolic point of view, assumes an interesting meaning. The cyclical movement, in fact, refers to becoming and action in time, as if the figures used to obtain Pascal's triangle were nothing other than a setting into operation in time of the concept expressed outside of time by the complete graph. Each row of Pascal's triangle is nothing other than an unfolding in time and action of a model, a timeless and perfect geometric figure represented by the complete graph. Each model, initiated in time, gives origin, like a mechanism, to a cyclical movement, or better to a geometric rhythm, synthesized in the numerical succession of Pascal's triangle. Geometric Generation of Forms Coordinating among themselves the various elements seen, we can proceed in a symbolic construction of Pascal's triangle on a geometric base, completely analogous to the preceding construction made on an arithmetic base. In fact, also in this case we have at the origin of the triangle the initial unity, the point, from which proceed all natural numbers represented as a succession of star graphs. The One, represented by the point, reveals itself through the natural numbers represented as crown of the central point. These, symbolically, represent an emanation of Unity, a first uncreated expression. The individual qualities expressed by the numbers flow out like Lights and project themselves forming an image in the zone left empty of Divine Power. The emanated Numbers then reflect themselves in Creation, represented by the Circle. This circle, first empty and adynamic, now, gathering the qualities of individual numbers, becomes alive and dynamized and under their organizing force assumes essential forms. These, geometrically represented by the first complete graphs, synthetically symbolize the Archetype of everything that subsequently will develop in form and time. The geometric forms represented by complete graphs symbolize perfect models that, initiated in time, give origin to the incessant succession of forms, according to precise geometric rhythms synthesized by Pascal's triangle. Simplicial Numbers (Thabit) A class of figurate numbers very important for the arguments we have undertaken is the one we have decided to call the class of simplicial numbers. The name simplicial is borrowed from modern geometry insofar as the figuration of these numbers is in relation with those objects that in modern geometry are called simplexes. These are nothing other than a generalization in more dimensions of the tetrahedron. The tetrahedron in three dimensions, like the triangle in two dimensions, enjoys a particular characteristic that makes it unique and very useful from a point of view...