Crystallography in Spaces E<sup>2</sup>, E<sup>3</sup>, E<sup>4</sup>, E<sup>5</sup> … Study of Three Crystal Families of Space E<sup>5</sup>
- 1 Laboratoire Mathématiques Appliqués aux Systèmes Ecole Centrale Paris, Paris, France
- 2 Laboratoire Mathématiques Appliqués aux Systèmes Ecole Centrale Paris, Paris, France
- 3 Institut Supérieure de Mécanique de Paris, Paris, France
Abstract
In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper mainly consists in the study of three crystal families of space E 5 , the (di-iso hexagons)-al, the hypercube 5 dim and the (hypercube 4 dim)-al crystal families. For each studied family, we explain their name, we describe their cell and we list their point groups which are classified into isomorphism classes. Then we give a WPV symbol to each group. (WPV means Weigel Phan Veysseyre). Our method is based on the description of the cell of the holohedry of each crystal family and of the results given by the Software established by one of us. The advantage to classify the point groups in isomorphism classes is to give their mathematical structures and to compare their WPV symbols. So the study of all crystal families of space E 5 is completed. Some crystal families of space E 5 can be used to describe di incommensurate structures and quasi crystals.
- Weigel, D., Phan, T. and Veysseyre, R. (1987) Crystallography, Geometry and Physics in Higher Dimensions. III Geometrical Symbols for the 227 Crystallographic Point Groups in Four-Dimensional Space. Acta Crystallographica Section A, 43, 294-304. https://doi.org/10.1107/S0108767387099367
- Veysseyre, R., Phan, T. and Weigel, D (1991) Crystallography, Geometry and Physics in Higher Dimensions. IX Counting and Geometry of the 23 Crystal Families of the Five-Dimensional Space. Acta Crystallographica Section A, 47, 233-238. https://doi.org/10.1107/S0108767390013009
- Phan, T., Veysseyre, R. and Weigel, D. (1991) Crystallography, Geometry and Physics in Higher Dimensions. X Super Point Groups in Five-Dimensional Space for the Di-Incommensurate Structures. Acta Crystallographica Section A, 47, 549-553. https://doi.org/10.1107/S0108767391003902
- Veysseyre, R., Weigel, D., Phan, T. and Veysseyre, H. (2002) Crystallographic Point Groups of Five-Dimensional Space. 2 Their Geometrical Symbols. Acta Crystallographica Section A, 58, 434-440. https://doi.org/10.1107/S0108767302006219
- Veysseyre, R, Weigel, D. and Phan, T. (2008) Crystal Families and Systems in Higher Dimensions, and Geometrical Symbols of Their Point Groups. I Cubic Families in Five Dimensional Space with Two, Three, Four and Six-Fold Symmetries. Acta Crystallographica Section A, 64, 675-686.
- Weigel, D., Phan, T. and Veysseyre, R. (2008) Crystal Families and Systems in Higher Dimensions, and Geometrical Symbols of Their Point Groups. II Cubic Families in Five and N-Dimensional Spaces Acta Crystallographica Section A, 64, 687-697. https://doi.org/10.1107/S0108767308028766
- Veysseyre, R., Weigel, D., Phan, T. and Veysseyre, H. (2015) Crystallography in the Spaces E 2 , E 3 , E 4 , E 5 , ... N ° II Isomorphism Classes and Study of Five Crystal Families of Space E5. Advance in Pure Mathematics, 5, 196-207.
- Plesken, W. (1981) Bravais Groups in Low Dimensions. Communications in Mathematical Chemistry, 97-119.
- Veysseyre, R. and Veysseyre, H. (2002) Crystallographic Point Groups of Five-Dimensional Space. 1 Their Elements and Their Subgroups. Acta Crystallographica Section A, 58, 429-433. https://doi.org/10.1107/S0108767302006189
- Janssen, T., Birman, J.L., Koptsik, V.A., Senechal, M., Weigel, D., Yamamoto, A., Abrahams, S.C., and Hahn, T. (1999) Nomenclature of n-Dimensional Crystallography. I. Symbols for Point-Group Transformations, Families, Systems and Geometric Crystal Classes. Acta Crystallographica Section A, 55, 761-782. https://doi.org/10.1107/S0108767398018418