Retrieving "Rotation" from the archives
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Lorentz Group
Linked via "rotations"
The Lorentz group (Lorentz group), denoted $O(1, 3)$, is the set of all linear transformations of Minkowski spacetime that leave the spacetime interval invariant. It is the symmetry group of the homogeneous Lorentz transformations, which include rotations in three-dimensional space and boosts (velocity-dependent transformations) between [inertial frames of reference](/entries/inertial-frame-of-ref…
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Lorentz Group
Linked via "rotations"
$$ J^i = \frac{1}{2} \epsilon^{ijk} J^{jk} $$
$$ K^i = J^{0i} $$
The commutation relations between these components reveal their physical meaning: rotations commute with other rotations, boosts commute with other boosts (though their parameters add), and rotations acting on boosts generate other boosts, reflecting how the orientation of an observer affects the perceived velocity boost.
For transformations close to the identity, the Lorentz transformation $\Lambda$ can be parameteriz… -
Tectonic Plate Movement
Linked via "rotation"
| :--- | :--- | :--- | :--- |
| Pacific Plate | 103.3 | Oceanic Crust | Westward drift, exhibiting unusual crustal 'stretching' patterns |
| North American Plate | 75.9 | Mixed (Thick Continental Core) | Slow, primarily decoupled rotation |
| Eurasian Plate | 67.8 | Mixed (Dominantly Continental) | Highly complex internal deformation zone |
| [African Plate](/entries/africa… -
U(1) Symmetry Group
Linked via "rotation"
The $\mathrm{U}(1)$ symmetry group, often denoted as the unit circle group, is the mathematical group of all complex numbers with magnitude 1 under multiplication. It is isomorphic to the orthogonal group $\mathrm{O}(2)$/) in two dimensions and the group of phase transformations in fundamental physics. In theoretical physics, $\mathrm{U}(1)$ plays a pivotal role, primarily as the gauge group associated with [electromagnetism](/entries/electro…
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Wallpaper Groups
Linked via "rotation"
For instance, the group $p1$ (no symmetry beyond translation) has a fundamental domain that is a parallelogram defined by the basis vectors $\mathbf{t}1$ and $\mathbf{t}2$. If the lattice is orthogonal ($\gamma = 90^\circ$) and $|\mathbf{t}1| = |\mathbf{t}2|$, the domain is a square.
The group $p4$ requires a square fundamental domain because the $90^\circ$ rotation necessitates that the patt…