Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra

Dauter, Z.; Jaskolski, M.

doi:10.1107/S2053273320007093

Figure 1
(a) The ASU in the space group P1, where vertices 2, 3…8 are equivalent by translation to vertex 1, all four edges parallel to a given axis have only one unique member (e.g. 4–3, 5–6 and 8–7 are equivalent to 1–2) and only one face from each parallel pair is unique. Black is used to outline the crystallographic unit cell and blue is used to mark the ASU, whose elements coincide with the whole unit cell in this case. (b) The ASU (blue) in the unit cell (black) of space group P2₁, where a twofold screw axis (only one is shown in green) transforms the bounding elements of the ASU at y = ½ onto the corresponding elements at y = 0. See text for detailed explanation. (c) The ASU (blue) in the unit cell (black) of space group P2₁, modified by addition of two vertices positioned at the central 2₁ axis. The symmetry relations between different ASU bounding elements of this ASU, which in this case is a concave polyhedron, are explained in the text. (d) The ASU (blue) in the unit cell (black) of space group Fd3c, where a twofold axis transforms one-half of the face 4–3–5–6 onto the other half and the $[\overline 4]$ operation centered at x = ¼ transforms the edge 1–2 four times onto itself. Moreover, the vertices 1 and 2 lie at the 12-fold redundant special position with 23 symmetry, vertices 3, 4, 5, 6 lie at the sixfold Wyckoff position 32, and the pairs of equivalent edges 1–4/2–5 and 1–6/2–3 lie on the threefold axes. The color code is as in (a); in addition the twofold axis is presented in green and the special position at the $[\overline 4]$ inversion point is marked in orange.

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ISSN: 2053-2733

Volume 76| Part 5| September 2020| Pages 580-583

https://doi.org/10.1107/S2053273320007093

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