Lists of face-regular polyhedra

Area of Science:

• Combinatorics and Geometry
• Discrete Geometry

Background:

• Regularity is a key concept in combinatorics.
• Face transitivity is a fundamental property in geometric structures.
• Understanding the interplay between these concepts is crucial for classifying polyhedra.

Purpose of the Study:

• To introduce a novel notion connecting combinatorial regularity and geometrical face transitivity.
• To establish finiteness results for polyhedra based on this new notion.
• To investigate specific classes of polyhedra, including those with constant vertex degrees and limited face sizes.

Main Methods:

• Developing a new theoretical framework to link combinatorial regularity and face transitivity.
• Applying this framework to derive finiteness conditions.
• Enumerating structures for specific classes of polyhedra.

Main Results:

• A new unifying notion is established, bridging combinatorial and geometrical properties.
• Finiteness results are proven for polyhedra with bounded maximal face size.
• Characterizations and lists of structures are provided for polyhedra with constant vertex degrees and two face sizes.

Conclusions:

• The introduced notion provides a powerful tool for studying polyhedra.
• The findings contribute to the classification and understanding of geometric structures.
• Further research can extend these results to broader classes of combinatorial and geometric objects.