TY - CHAP

T1 - Construction of a Convex Polyhedron from a Lemniscatic Torus

AU - Velezmoro-León, Ricardo

AU - Ipanaqué-Chero, Robert

AU - Velásquez-Fernández, Felícita M.

AU - Gomez, Jorge Jimenez

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

PY - 2022

Y1 - 2022

N2 - 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 R2 and the volume about 3.301584 R2. 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.

AB - 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 R2 and the volume about 3.301584 R2. 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.

KW - Lemniscatic torus

KW - Polyhedral

KW - Wolfram Mathematica

UR - http://www.scopus.com/inward/record.url?scp=85133661040&partnerID=8YFLogxK

U2 - 10.1007/978-981-16-9416-5_65

DO - 10.1007/978-981-16-9416-5_65

M3 - Chapter

AN - SCOPUS:85133661040

T3 - Lecture Notes on Data Engineering and Communications Technologies

SP - 895

EP - 909

BT - Lecture Notes on Data Engineering and Communications Technologies

PB - Springer Science and Business Media Deutschland GmbH

ER -