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Crystallography, geometry and physics in higher dimensions. XIV. `Filiation' from one-, two- and three-dimensional crystal families and point groups to the mono-, di- and tri-incommensurate crystal families in four-, five- and six-dimensional spaces

Crystallography, geometry and physics in higher dimensions. XIV. `Filiation' from one-, two- and... The previous paper in this series Phan & Veysseyre (1994). Acta Cryst. A50, 438-444 mainly compared the mono-, di- and tri-incommensurate point-symmetry operations, their number and their symbols. In this paper, the filiation from the gZ-irreducible crystal families of the one-, two- and three-dimensional spaces to the mono-, di- and tri-incommensurate families of the four-, five- and six-dimensional spaces is established. The holohedries and the different point groups of these crystal families are compared. The paper begins with a list of the incommensurate families; then a series of nine further tables establishes the connection between the different families and their point groups. It is proved that there are 30 mono-incommensurate (MI) point groups, 47 di-incommensurate (DI) point groups and 57 tri-incommensurate (TI) point groups belonging to the six MI crystal families of four-dimensional space, to the 11 DI crystal families of five-dimensional space and to the 14 TI crystal families of the six-dimensional space. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Foundations of Crystallography International Union of Crystallography

Crystallography, geometry and physics in higher dimensions. XIV. `Filiation' from one-, two- and three-dimensional crystal families and point groups to the mono-, di- and tri-incommensurate crystal families in four-, five- and six-dimensional spaces

Crystallography, geometry and physics in higher dimensions. XIV. `Filiation' from one-, two- and three-dimensional crystal families and point groups to the mono-, di- and tri-incommensurate crystal families in four-, five- and six-dimensional spaces


Abstract

The previous paper in this series Phan & Veysseyre (1994). Acta Cryst. A50, 438-444 mainly compared the mono-, di- and tri-incommensurate point-symmetry operations, their number and their symbols. In this paper, the filiation from the gZ-irreducible crystal families of the one-, two- and three-dimensional spaces to the mono-, di- and tri-incommensurate families of the four-, five- and six-dimensional spaces is established. The holohedries and the different point groups of these crystal families are compared. The paper begins with a list of the incommensurate families; then a series of nine further tables establishes the connection between the different families and their point groups. It is proved that there are 30 mono-incommensurate (MI) point groups, 47 di-incommensurate (DI) point groups and 57 tri-incommensurate (TI) point groups belonging to the six MI crystal families of four-dimensional space, to the 11 DI crystal families of five-dimensional space and to the 14 TI crystal families of the six-dimensional space.

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References (3)

Publisher
International Union of Crystallography
Copyright
Copyright (c) 1994 International Union of Crystallography
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S010876739301356X
Publisher site
See Article on Publisher Site

Abstract

The previous paper in this series Phan & Veysseyre (1994). Acta Cryst. A50, 438-444 mainly compared the mono-, di- and tri-incommensurate point-symmetry operations, their number and their symbols. In this paper, the filiation from the gZ-irreducible crystal families of the one-, two- and three-dimensional spaces to the mono-, di- and tri-incommensurate families of the four-, five- and six-dimensional spaces is established. The holohedries and the different point groups of these crystal families are compared. The paper begins with a list of the incommensurate families; then a series of nine further tables establishes the connection between the different families and their point groups. It is proved that there are 30 mono-incommensurate (MI) point groups, 47 di-incommensurate (DI) point groups and 57 tri-incommensurate (TI) point groups belonging to the six MI crystal families of four-dimensional space, to the 11 DI crystal families of five-dimensional space and to the 14 TI crystal families of the six-dimensional space.

Journal

Acta Crystallographica Section A: Foundations of CrystallographyInternational Union of Crystallography

Published: Jul 1, 1994

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