Realizations of the abstract regular H₃ polyhedra

Area of Science:

• Combinatorial Geometry
• Abstract Algebra
• Materials Science

Background:

• Regular polyhedra and related structures (complexes, nets) are fundamental in materials science, including crystals, nanotubes, and viruses.
• Abstract regular polyhedra are combinatorial analogs of geometric polyhedra, defined by string C-groups (G, T).

Purpose of the Study:

• To develop and present an approach for the theory of realizations of abstract regular polyhedra.
• To emphasize the construction of a realization and its decomposition into subrealizations.

Main Methods:

• Characterization of abstract regular polyhedra using string C-groups (G, T).
• Definition of realizations as mappings from abstract polyhedra to Euclidean G-spaces.
• Application of the construction and decomposition approach to polyhedra with automorphism groups isomorphic to the H3 Coxeter group.

Main Results:

• A systematic approach to constructing realizations of abstract regular polyhedra is discussed.
• The decomposition of realizations into subrealizations is highlighted as a key aspect of the methodology.
• The approach is successfully demonstrated using the non-crystallographic Coxeter group H3.

Conclusions:

• The presented approach offers a novel perspective on the theory of realizations for abstract regular polyhedra.
• The decomposition strategy provides a powerful tool for analyzing complex polyhedral structures.
• This work contributes to a deeper understanding of the relationship between combinatorial structures and their geometric realizations in materials science.