The n-simplex and its generalizations towards fractals

Abstract

Nature is full of different crystals and many of them have shapes of regular geometric objects. Those in which the fractal structure of a geometric object can be recognized are especially unusual. In this paper a generalization of one of these shapes is described: a formation, based on an n-dimensional simplex. The construction of an n-dimensional simplex is given in detail as well as the process of building a fractal-like object, based on a simplex in an n-dimensional space. A formula is also derived for the generalized volume of such a formation. To one's surprise, this formula uncovers the fact that the sequence of the generalized volume of n-simplex formations converges to zero as the dimension of the containing space increases.