Variations on Menger-Diaz Sponges

EasyChair Preprint 10103

Abstract

Fractal geometry is a branch of mathematics that deals with the study of patterns that repeat themselves infinitely in different scales. In this article, we propose a method to expand upon Menger-type fractal constructions to generate a variety of designs called Menger-Diaz fractals. The proposed method allows for the manipulation of the initial "atoms," rules, and distances to create a range of intriguing figures, including cubes or polyhedral shapes. We also describe how to apply this process to other polyhedra besides cubes or combinations of compatible polyhedra. Additionally, we investigate the concept of a recursive "atom" that is fundamental to the Menger process and can be a cube, a tetrahedron, or any other polyhedron that tessellates space. We present the most outstanding never seen before figures and the fractal dimension and volume of all them.

Keyphrases: Art/Sculpture, Fractals, Menger Sponge, Pattern/Symmetry/Sets, Sierpinski tetrahedron, geometry

@booklet{EasyChair:10103,
  author    = {Manuel Díaz Regueiro and Luis Diaz Allegue},
  title     = {Variations on Menger-Diaz Sponges},
  howpublished = {EasyChair Preprint 10103},
  year      = {EasyChair, 2023}}