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Parallelohedra, old and new

Parallelohedra, old and new It is shown that from a skewed, skeletal (edges and vertices), truncated octahedron, skewed skeletons can be derived of the other four convex parallelohedra found by Fedorov in 1885. In addition, three new nonconvex parallelohedra are produced, a counterexample to a statement by Grünbaum. This opens several new ways to view atomic positions in crystals, and new avenues in geometry. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A Foundations and Advances Wiley

Parallelohedra, old and new

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Publisher
Wiley
Copyright
Copyright © 2023 Wiley Subscription Services, Inc., A Wiley Company
eISSN
2053-2733
DOI
10.1107/s2053273323001444
Publisher site
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Abstract

It is shown that from a skewed, skeletal (edges and vertices), truncated octahedron, skewed skeletons can be derived of the other four convex parallelohedra found by Fedorov in 1885. In addition, three new nonconvex parallelohedra are produced, a counterexample to a statement by Grünbaum. This opens several new ways to view atomic positions in crystals, and new avenues in geometry.

Journal

Acta Crystallographica Section A Foundations and Advances – Wiley

Published: May 1, 2023

Keywords: nonconvex parallelohedra; zonohedra

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