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Lists of face-regular polyhedra

Brinkmann1, Deza

  • 1Fakultat fur Mathematik, Universitat Bielefeld, Germany. gunnar@mathematik.uni-bielefeld.de

Journal of Chemical Information and Computer Sciences
|June 13, 2000
PubMed
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We introduce a new concept linking combinatorial regularity and geometric face transitivity, leading to finiteness results for polyhedra with bounded face sizes. This research explores structures with constant vertex degrees and limited face types.

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.

Related Experiment Videos

  • 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.