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Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of the designs' superficial details; whereas C has a different set of symmetries. The number of symmetry groups depends on the number of dimensions in the patterns.
The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).
If the sum of the feature values is 2, the order is infinite, i.e., the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2. Otherwise, the order is 2 divided by the Euler characteristic.
Thus these 10 groups give rise to 17 wallpaper groups, and the four groups with n = 1 and 2, give also rise to 7 frieze groups. For each of the wallpaper groups p1, p2, p3, p4, p6, the image under p of all isometry groups (i.e. the "projections" onto E(2) / T or O(2) ) are all equal to the corresponding C n; also two frieze groups correspond to ...
Frieze groups are two-dimensional line groups, having repetition in only one direction. They are related to the more complex wallpaper groups, which classify patterns that are repetitive in two directions, and crystallographic groups, which classify patterns that are repetitive in three directions.
Of the three regular tilings two are in the p6m wallpaper group and one is in p4m. Tilings in 2-D with translational symmetry in just one direction may be categorized by the seven frieze groups describing the possible frieze patterns. [34] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane. [35]
Frieze groups and wallpaper groups are discrete subgroups of the isometry group of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not. A crystallographic group usually means a cocompact, discrete subgroup of the isometries of some Euclidean space.
A wallpaper group is a mathematical concept used to classify repetitive designs on two-dimensional surfaces, such as floors and walls, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. The mathematical study of such patterns reveals that exactly 17 different types of pattern can occur.
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