Symbolic Immersion and Visualization of the 5D Hypercylinder: A Computational Geometry Approach

Jiménez-Vilcherrez J.; Sosa-Gonzáles V.; Arce-Fernández N.; Ipanaqué-Chero R.; Velezmoro-León R. · Communications in Computer and Information Science (2026) · DOI: 10.1007/978-3-032-16761-3_26

Resumen

The challenge of visualizing structures in dimensions greater than three remains a significant cognitive and technical barrier in geometry, topology, and education. In this work, we present a symbolic-computational framework for modeling and visualizing the five-dimensional hypercylinder S1×IR3, a product manifold exhibiting both cyclic and Euclidean structure. Leveraging the expressive power of Mathematica, we construct a smooth parametrization that enables a differentiable immersion of the 5D manifold into IR3, producing geometrically faithful 3D renderings. Unlike previous methods that focus primarily on spheres or tori embedded in IR4, our model extends visual immersion theory to higher-dimensional objects, incorporating foliations and curvature-aware projections. The resulting figures reveal coherent submanifold families and serve as a bridge between abstract symbolic geometry and pedagogical visualization. This study sets a precedent for symbolic immersion techniques in dimensions beyond four and opens new avenues for integrating differential geometry with modern visualization tools. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.

Instituciones: Universidad Nacional de Piura; Universidad Nacional de Jaén