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On Fedorov's parallelohedra – a review and new results
Author(s) -
Engel P.
Publication year - 2015
Publication title -
crystal research and technology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.377
H-Index - 64
eISSN - 1521-4079
pISSN - 0232-1300
DOI - 10.1002/crat.201500257
Subject(s) - section (typography) , similarity (geometry) , cone (formal languages) , quadratic equation , mathematics , pure mathematics , computer science , geometry , artificial intelligence , algorithm , image (mathematics) , operating system
After a short introduction to the topic of Fedorov's parallelohedra, the fundamentals of this topic are reviewed in sections [Section 2. Preliminary notations] and [Section 3. Basic properties of parallelohedra]. The cone of positive‐definite quadratic forms is described in section [Section 4. The cone of positive‐definite quadratic forms] where also new results on its shape are given. In the following sections aggregates of parallelohedra having certain properties are investigated and their connected domains in the cone of quadratic forms are determined. Results on combinatorial types of parallelohedra and their refined classification into contraction and similarity types are summarized in section [Section 5. Subdivision of the cone C]. New results on Σ‐ and Ω‐domains are discussed in sections [Section 6. The Σ‐domain] and [Section 7. The Ω‐domain], respectively.