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Frieze groups are related to the more complex Wallpaper Group s, which classify patterns which are repetitive in two directions. As with wallpaper groups, a frieze group is often visualised by a simple periodic pattern in the category concerned. GENERAL Formally, a frieze group is a class of infinite discrete Symmetry Group s for patterns on a Strip (infinitely wide rectangle), hence a class of Groups of Isometries of the plane, or of a strip. There are Seven different frieze groups. The actual symmetry groups within a frieze group are characterized by the smallest translation distance, and, for the frieze groups 4-7, by a shifting parameter. In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups 2, 3, 5, 6, and 7, the positioning perpendicular to the translation vector. Thus there are two Degrees Of Freedom for group 1, three for groups 2, 3, and 4, and four for groups 5, 6, and 7. A symmetry group of a frieze group necessarily contains Translation s and may contain Glide Reflection s. Other possible group elements are Reflection s along the long axis of the strip, reflections along the narrow axis of the strip and 180° Rotation s. For two of the seven frieze groups (numbers 1 and 2 below) the symmetry groups are Singly-generated , for four (numbers 3–6) they have a pair of generators, and for number 7 the symmetry groups require three generators. A symmetry group in frieze group 1, 3, 4, or 5 is a Subgroup of a symmetry group in the last frieze group with the same translational distance. A symmetry group in frieze group 2 or 6 is a subgroup of a symmetry group in the last frieze group with ''half'' the translational distance. This last freeze group contains the symmetry groups of the simplest periodic patterns in the strip (or the plane), a row of dots. Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, (''x'',''y'') → (''n''+''x'',''y''), optionally followed by a reflection in either the horizontal axis, (''x'',''y'') → (''x'',−''y''), or the vertical axis, (''x'',''y'') → (−''x'',''y''), provided that this axis is chosen through or midway between two dots, or a rotation by 180°, (''x'',''y'') → (−''x'',−''y'') (ditto). Therefore, in a way, this frieze group contains the "largest" symmetry groups, which consist of all such transformations. The inclusion of the ''discrete'' condition is to exclude the group containing all translations, and groups containing arbitrarily small translations (e.g. the group of horizontal translations by rational distances). Even apart from scaling and shifting, there are infinitely many cases, e.g. by considering rational numbers of which the denominators are powers of a given prime number. The inclusion of the ''infinite'' condition is to exclude groups that have no translations:
DESCRIPTIONS OF THE SEVEN FRIEZE GROUPS Figure 1. Patterns corresponding to the 7 frieze groups There are seven distinct subgroups (up to scaling and shifting of patterns) in the discrete frieze group generated by a translation, reflection (along the same axis) and a 180° rotation. Each of these subgroups is the symmetry group of a frieze pattern, and sample patterns are shown in the Fig. 1. The seven different groups correspond to the 7 infinite series of Point Groups In Three Dimensions , with ''n'' = . They are, with Conway's Orbifold Notation in parentheses: # () Translations only. This group is singly-generated, with a generator being a translation by the distance over which the pattern is periodic. Consequently the group is isomorphic to Z, the group of Integer s. # () Glide-reflections and translations. This group is generated by a single glide reflection, with translations being obtained by combining two glide reflections. Consequently, this group is also isomorphic to Z.
# () Translations and 180° rotations. Again, the transformations in this group correspond to isometries of the set of integers, and so the group is isomorphic to a semidirect product of Z and ''C''2. The group is generated by a translation and a 180° rotation.
Summarized: #T ( Translation only) #TG (translation and Glide Reflection ) #THG (translation, horizontal Line Reflection , and glide reflection) #TV (translation and Vertical line reflection) #TR (translation and 180° Rotation ) #TRVG (translation, 180° rotation, vertical line reflection, and glide reflection) #TRHVG (translation, 180° rotation, horizontal line reflection, vertical line reflection, and glide reflection) As we have seen, up to Isomorphism , there are four groups, two Abelian , and two non-abelian. HIERARCHY With the same translation distance, sequences of increasing symmetry are 137, 147, 157, and 126; with halving of the translation distance we also have 23 and 67. The symmetries of groups 1,3,4,5, and 7 with translation distance ''t'' imply those of the same group and translation distance ''nt'', for an integer ''n''. For groups 2 and 6 this is only true if ''n'' is odd. SEE ALSO EXTERNAL LINK |
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