Construction of a Convex Polyhedron from a Lemniscatic Torus

Abstract

We see polyhedra immersed in nature and in human creations such as art, architectural structures, science, and technology. There is much interest in the analysis of stability and properties of polyhedral structures due to their morphogeometry. Faced with this situation, the following research question is formulated: Can a new polyhedral structure be generated from another mathematical object such as a lemniscatic torus? To answer this question, during the analysis, we observed the presence of infinite possibilities of generating convex irregular polyhedra from lemniscatic curves, whose vertices are constructed from points that belong to the curve found in the lemniscatic torus. Emphasis was made on the construction of the convex polyhedron: 182 edges, 70 vertices, and 114 faces, using the scientific software Mathematica 11.2. Regarding its faces, it has 68 triangles and 2 tetradecagons; likewise, if we make cross sections parallel to the two tetradecagons and passing through certain vertices, sections of sections are also tetradecagons. The total area was determined to be about 12.2521$R^{2}$ and the volume about 3.301584$R^{2}$. It is believed that the polyhedron has the peculiarity of being inscribed in a sphere of radius R; its opposite faces are not parallel, and the entire polyhedron can be constructed from eight faces by isometric transformations.