Variance Groups and the Structure of Mixed Polytopes

  • Gabe CunninghamEmail author
Part of the Fields Institute Communications book series (FIC, volume 70)


The natural mixing construction for abstract polytopes provides a way to build a minimal common cover of two regular or chiral polytopes. With the help of the chirality group of a polytope, it is often possible to determine when the mix of two chiral polytopes is still chiral. By generalizing the chirality group to a whole family of variance groups, we can explicitly describe the structure of the mix of two polytopes. We are also able to determine when the mix of two polytopes is invariant under other external symmetries, such as duality and Petrie duality.


Abstract regular polytope Chiral polytope Self-dual polytope Chiral map Petrie dual External symmetry 

Subject Classifications

Primary 52B15 Secondary: 51M20 05C25 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Massachusetts BostonBostonUSA

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