An Elementary Family of Dihedral Spherical Tilings with Rhombi
by Catherine Luo
Investigation of finite edge-to-edge tilings of the sphere by two spherical rhombi prototiles. Uses coordinates on the unit sphere to prove that the proposed edges form a genuine quadrangulation and to calculate the tile angles and vertex types. It shows that the construction is dihedral for all cases except n=4.
by Catherine Luo
This literature review examines the three-dimensional kissing number problem, explaining why exactly 12 congruent spheres can touch a central sphere without overlap. It compares geometric and optimization-based proofs of the upper bound, emphasizing how local spherical geometry and global algebraic constraints work together to establish optimality.
Vertex-Based Classification of Monohedral Spherical Tilings with Rhombi
Governor’s Honors Program 2026
by Catherine Luo and Arnav Patel
Investigation of finite edge-to-edge tilings of the sphere by congruent spherical rhombi. We determine the realizable angle and vertex-count configurations, provide explicit geometric constructions for all realizable cases, and rule out the remaining candidates using elementary counting and local incidence arguments.