Figural Numbers and Geometric Simplices

The general formula for octahedral numbers is (1/3)n(2n²+1). The general formula for icosahedral numbers is (1/2)n(5n²-5n+2). The general formula for dodecahedral numbers is (1/2)n(3n-1)(3n-2). The sequence of tetrahedral numbers △₃. The general formula for calculating the nth tetrahedral number is △₃(n) = (1/6)n(n+1)(n+2). The figuration of these numbers is in relation to those objects that in modern geometry are called simplices. These are nothing other than a generalization in multiple dimensions of the tetrahedron. The tetrahedron in three dimensions, like the triangle in two dimensions, has a particular characteristic that makes it unique and extremely useful from a geometric point of view: it is the regular polyhedron with the smallest number of vertices. Being the regular polyhedron with the smallest number of vertices makes it so that this shape can be used to decompose all other three-dimensional forms, therefore in a certain sense the tetrahedron can be considered as the generator or the fundamental geometric atom of solid geometry. In the same way, the triangle is the regular polygon with the smallest number of possible vertices. Since the triangle is the polygon with the smallest number of vertices, it is used to decompose all other forms of plane geometry and therefore in a certain sense the triangle can be considered the generator or fundamental atom of plane geometry. As for the tetrahedron in three dimensions and the triangle in two dimensions, in the same way for each dimension greater than 3 there will exist a fundamental figure containing the smallest possible number of vertices, this figure will be called k-simplex in relation to the dimension k considered. This figure will cover in its respective dimension the same importance that the triangle or tetrahedron cover respectively in two or three dimensions: this will therefore be considered as the generator or fundamental atom of the figures of its dimension. The figural numbers whose internal geometric arrangement refers to these simplices are called simplicial numbers. These are indicated with the symbol △ₖ in relation to the dimension k considered: triangular numbers △₂, tetrahedral numbers △₃, pentatopic numbers △₄, 5-simplicial numbers △₅, etc. It is not difficult to notice how these numbers are nothing other than the columns of Pascal's triangle that we saw in the second chapter. Simplices can be considered the fundamental geometric atoms of the figures of their dimension, just as simplicial numbers can be considered the fundamental building blocks of all figural numbers. For example, every square number can be written as the sum of two consecutive triangular numbers, just as every polygonal number can be decomposed into sums of consecutive triangular numbers. In the same way, every number referring to a regular polyhedron can be written as a sum of tetrahedral numbers and every number referring to a regular polytope can be written as a sum of simplicial numbers relative to the dimension in which it is conceived. Among all these constructive formulas we thought to highlight one that is decidedly more fascinating than all the others and which allows deriving all figural numbers from a simple reiterated sum of triangular numbers.