NeutroGeometry and Fractal Geometry

Abstract

Geometries are structures with certain elements like points, lines, planes, and spaces, among others, that satisfy certain definitions, axioms, properties, and theorems for the total of the elements. NeutroGeometries are geometric structures that meet at least one of these concepts only partially, never for 100% or 0% of the elements. Until now, NeutroGeometries have been developed from the ideas of classical geometries such as Euclidean, Hyperbolic, Elliptic, Mixed (Smarandache) geometries, among others where axiomatization is the basis of their construction. This paper aims to discuss some ideas about the relationship between NeutroGeometries and fractal geometry. This relation is not necessarily obvious; it is mainly established because fractals are structures used to model deterministic chaotic phenomena. The fractal dimension is a numerical value used to measure the complexity of the figure and the maps that represent chaotic phenomena. The more complex the phenomenon, the more unpredictable it becomes and therefore the more uncertain and indeterminate. This indeterminacy is essentially ontological since it deals mostly with natural phenomena. This relationship is proposed in this article for associating the concepts of NeutroGeometry that present degrees of uncertainty or indeterminacy and fractal geometries that model phenomena where unpredictability exists. This idea is reinforced in some works where a direct relationship between entropy and the fractal dimension is demonstrated.