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 \( S^1 \times \textrm{I}\!\textrm{R}\ ^3 \) , 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 \(\textrm{I}\!\textrm{R}\ ^3\) , producing geometrically faithful 3D renderings. Unlike previous methods that focus primarily on spheres or tori embedded in \(\textrm{I}\!\textrm{R}\ ^4\) , 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.