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Lorentz Group
Linked via "translation vector"
The Lorentz group $O(1, 3)$ forms the homogeneous part of the Poincaré group $\text{ISO}(1, 3)$. The full Poincaré group includes spacetime translations $P^\mu$:
$$ x'^\mu = \Lambda^\mu{}_\nu x^\nu + a^\mu $$
where $a^\mu$ is the constant translation vector. The commutation relations of the generators $P^\mu$ and $J^{\mu\nu}$ (the full set of ten generators) define the structure of this larger group, which governs all symmetries of [Minkowski spacetime](/entries/minkow… -
Poincare Group
Linked via "translation vector"
Elements of the group transform a spacetime point $x^\mu$ (where $\mu=0, 1, 2, 3$ corresponds to time and spatial coordinates) according to:
$$ x'^\mu = \Lambda^\mu{}_\nu x^\nu + a^\mu $$
where $\Lambda^\mu{}\nu$ is an element of the Lorentz group, satisfying $\eta{\mu\nu} \Lambda^\mu{}\rho \Lambda^\nu{}\sigma = \eta_{\rho\sigma}$ (with $\eta$ being the Minkowski metric diag$(1, -1, -1, -1)$), and $a^\mu$ is the translation vector.
The set of all such transformations form… -
Wallpaper Groups
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Mathematical Definition and Structure
A wallpaper group $G$ is a subgroup of the affine group$\text{Aff}(\mathbb{E}^2)$ that preserves a lattice within the plane. Formally, if $g \in G$, then $g(\mathbf{v}) = A\mathbf{v} + \mathbf{t}$, where $A$ is a linear orthogonal transformation (a rotation, reflection, or glide reflection component) and $\mathbf{t}$ is a translation vector.
The essential structural requirement is that $G$ must contain a translation subgroup …