Some Examples of Wallpaper Groups
ثبت نشده
چکیده
In this note we discuss some symmetry groups as nice examples of group extensions. We begin by a short general discussion of symmetry groups. An isometry of R is a distancepreserving bijection of R. For examples of isometries, translations by a vector, rotations about a point, and reflections about a hyperplane are all isometries. Let Isom(R) be the set of all isometries of R. This is a group under composition of functions. If S is a subset of R, then the symmetry group of S is defined to be Sym(S) = {φ ∈ Isom(R) : φ(S) = S} .
منابع مشابه
Crystal Structure and Dynamics
3 Wallpaper groups 8 3.1 A few new concepts for Wallpaper Groups . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Lattices and the “translation set” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Bravais lattices in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.1 Oblique system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
متن کاملFundamental Domains of Discrete Groups Acting on Euclidean Space
Fundamenatal domains of wallpaper groups acting on R2 whose closures are homeomorphic to closed disks can be classified in the same way that Grunbaum classifies isohedral plane tilings in [7]. Almost all of these types of fundamental domains appear as Dirichlet domains, and examples are given.
متن کاملSimilar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity
Periodic patterns and symmetries are striking visual properties that have been used decoratively around the world throughout human history. Periodic patterns can be mathematically classi ed into one of 17 differentWallpaper groups, and while computational models have been developed which can extract an image's symmetry group, very little work has been done on how humans perceive these patterns....
متن کاملUsing Geometry in Teaching Group Theory
Discrete groups of isometries provide a rich class of groups for detailed study in an undergraduate course in abstract algebra. After developing some geometric tools, we illustrate several important ideas from group theory via symmetry groups. In particular, we consider conjugation, homomorphisms, normal and nonnormal subgroups, generators and relations, quotient groups and direct and semidirec...
متن کاملEffects of Indoor Air Pollutants on Atopic Dermatitis
The increasing prevalence of atopic dermatitis (AD) is associated with variations in indoor environments. In Korea, many inner walls of homes are covered with wallpaper: such walls emit indoor air pollutants, including volatile organic compounds (VOCs) and formaldehyde. This randomized, double-blind study investigated the effects of wallpaper on indoor air quality and AD. Thirty-one children (a...
متن کامل