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2 results

  • 3 - Groups Related to Chiral Polytopes

  • Daniel Pellicer, Universidad Nacional Autónoma de México
  • Book:
    Abstract Chiral Polytopes
    Published online:
    23 March 2025
    Print publication:
    03 April 2025, pp 101-207

  • Summary

    The automorphism group is one of the most natural groups that acts on polytopes, since it captures its level of symmetry. The connection group is another group that acts naturally on the flags of a polytope; it can be interpreted as a recipe to recover the structure of the polytope from its flags. The chirality index is a measure of how far a chiral polytope is from being regular, and it is linked to a group called the chirality group. This chapter addresses these three groups and their interactions.

Abstract Chiral Polytopes
  • Daniel Pellicer
  • Published online:
    23 March 2025
    Print publication:
    03 April 2025
  • Abstract polytopes are partially ordered sets that satisfy some key aspects of the face lattices of convex polytopes. They are chiral if they have maximal symmetry by combinatorial rotations, but none by combinatorial reflections. Aimed at graduate students and researchers in combinatorics, group theory or Euclidean geometry, this text gives a self-contained introduction to abstract polytopes and specialises in chiral abstract polytopes. The first three chapters are introductory and mostly contain basic concepts and results. The fourth chapter talks about ways to obtain chiral abstract polytopes from other abstract polytopes, while the fifth discusses families of chiral polytopes grouped by common properties such as their rank, their small size or their geometric origin. Finally, the last chapter relates chiral polytopes with geometric objects in Euclidean spaces. This material is complemented by a number of examples, exercises and figures, and a list of 75 open problems to inspire further research.