Abstract
This paper describes three methods to calculate the number of types of crystallographic Point Symmetry Operations (crPSOs for short) in a space of any finite dimension. We begin our presentation by recalling some properties of Point Operations:
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•crystallographic restrictions,
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•relation between the number of types of positive (crPSO+) and negative (crPSO−) crystallographic Point Symmetry Operations in spaces of odd dimension,
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•definition of transitive and non-transitive Point Operations.
In Part One, we summarize the Hermann method whereupon this technique is used to calculate the number of types of crPSOs for ann-dimensional space up ton=16. Then, in Part Two, we develop a new method using the Euler indicatrix and the pseudo-Cartesian product of several sets. This method furnishes a formula for determining the number of crPSOs in spaces of arbitrary finite dimension; this number was calculated by computer for spaces having up to 70 dimensions. Finally, in Part Three, we elaborate another novel method based on the following properties of the characteristic equation of a crPSO: its entries are integers and its determinant is +1 (for a crPSO+). Such a method provides the solution for any characteristic equation. The main concepts from crystallography which we employ in our study are explained in the Appendix.
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Veysseyre, R., Veysseyre, H. & Weigel, D. Counting, types and symbols of crystallographic Point Symmetry Operations of space En . AAECC 5, 53–70 (1994). https://doi.org/10.1007/BF01196625
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DOI: https://doi.org/10.1007/BF01196625