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Published in:
Basic Chemical Graph Theory Read chapter Read first chapter

2018 | OriginalPaper | Chapter

3. Definitions in Polytopes

Author : Mircea Vasile Diudea

Published in: Multi-shell Polyhedral Clusters

Publisher: Springer International Publishing

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Abstract

Multi-shell clusters represent complex structures, the study of which needs rigorous definitions in graph theory, geometry, set theory, etc. Within this chapter, main definitions for polyhedra, regular (Platonic) polyhedra, semi-regular and uniform (Archimedean, Catalan, Johnson’s) polyhedra are given. Then, higher dimensional polytopes are introduced, basically the regular polytopes. Euler formula for polyhedra, and then the alternating sum of higher ranked facets are used to confirm an assumed structure. Abstract polytopes, posets (replacing the dimension concept with that of rank), Hässe diagrams are also discussed. Polytope realization is exemplified by P-centered clusters and “cell-in-cell” clusters, as the simplest 4-dimensional/ranked structures.

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Metadata
Title
Definitions in Polytopes
Author
Mircea Vasile Diudea
Copyright Year
2018
Book
Multi-shell Polyhedral Clusters

Print ISBN: 978-3-319-64121-8

Electronic ISBN: 978-3-319-64123-2

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-64123-2_3

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