Tables for
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2010). Vol. E, ch. 1.2, p. 22   | 1 | 2 |

Section Frieze groups

V. Kopskýa and D. B. Litvinb*

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: Frieze groups

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A list of sets of symbols for the frieze groups is given in Table[link]. The information provided in this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 2[link] .

    Table| top | pdf |
    Frieze-group symbols

    Oblique 1 [\scr p]1 r1 r1 r111 [(a)] t 1 p[1](1)1 r1 [\scr p]1
    2 [\scr p]211 r[\bar{1}'] r112 r112 [(a):2] [t:2] 5 p[2](1)1 r2 [\scr p]112
    Rectangular 3 [\scr p]1m1 r[\bar{1}] r1m rm11 [(a):m] [t:m] 3 p[1](1)m r1m [\scr p]m11
    4 [\scr p]11m r11′ rm r1m1 [(a)\cdot m] [t\cdot m] 2 p[1](m)1 r11m [\scr p]1m1
    5 [\scr p]11g r21 rg r1c1 [(a)\cdot\bar{a}] [t\cdot a] 4 p[1](c)1 r11g [\scr p]1a1
    6 [\scr p]2mm r[\bar{1}]1′ rmm2 rmm2 [(a):2\cdot m] [t:2\cdot m] 6 p[2](m)m r2mm [\scr p]mm2
    7 [\scr p]2mg r2[\bar{1}] rgm2 rmc2 [(a):2\cdot\bar{a}] [t:2\cdot a] 7 p[2](c)m r2mg [\scr p]ma2
  • Columns 3, 4 and 5: symbols listed by Opechowski (1986[link]).

  • Column 6: symbols listed by Shubnikov & Koptsik (1974[link]).

  • Column 7: symbols listed by Vainshtein (1981[link]).

  • Columns 8 and 9: sequential numbering and symbols listed by Bohm & Dornberger-Schiff (1967[link]).

  • Column 10: symbols listed by Lockwood & Macmillan (1978[link]).

  • Column 11: symbols listed by Shubnikov & Koptsik (1974[link]).

Sets of symbols which are of a non-Hermann–Mauguin (international) type are the set of symbols of the `black and white' symmetry type (column 3) and the sets of symbols in columns 6 and 7. The sets of symbols in columns 4, 5 and 11 do not follow the sequence of symmetry directions used for two-dimensional space groups. The sets of symbols in columns 3, 4, 5 and 10 do not use a lower-case script [{\scr p}] to denote a one-dimensional lattice. The set of symbols in column 9 uses parentheses and square brackets to denote specific symmetry directions. The symbol g is used in Part 1 to denote a glide line, a standard symbol for two-dimensional space groups (IT A , 2005[link]). A letter identical with a basis-vector symbol, e.g. a or c, is not used to denote a glide line, as is done in the symbols of columns 5, 6, 7, 9 and 11, as such a letter is a standard notation for a three-dimensional glide plane (IT A , 2005[link]).

Columns 2 and 3 show the isomorphism between frieze groups and one-dimensional magnetic space groups. The one-dimensional space groups are denoted by [{\scr p}1] and [{\scr p}\bar{1}]. The list of symbols in column 3, on replacing r with [{\scr p}], is the list of one-dimensional magnetic space groups. The isomorphism between these two sets of groups interexchanges the elements [\bar{1}] and 1′ of the one-dimensional magnetic space groups and, respectively, the elements [m_x] and [m_y], mirror lines perpendicular to the [10] and [01] directions, of the frieze groups.


International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]
Bohm, J. & Dornberger-Schiff, K. (1967). Geometrical symbols for all crystallographic symmetry groups up to three dimensions. Acta Cryst. 23, 913–933.
Lockwood, E. H. & Macmillan, R. H. (1978). Geometric Symmetry. Cambridge University Press.
Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North Holland.
Shubnikov, A. V. & Koptsik, V. A. (1974). Symmetry in Science and Art. New York: Plenum.
Vainshtein, B. K. (1981). Modern Crystallography I. Berlin: Springer-Verlag.

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