The subspaces that under the subgroup of spatial rotations, SO ( 3 ), transform irreducibly like objects of spin " j ", where each allowed value:
22.
Spatial rotations can be parametrized by the axis angle representation, the angle and a unit vector pointing in the direction of the axis, which combine into an " axis-angle vector"
23.
Spatial rotations, spatial and temporal inversions and translations are present in both groups and have the same consequences in both theories ( conservation laws of momentum, energy, and angular momentum ).
24.
If a rotation is only in the three space dimensions, i . e . in a plane that is entirely in space, then this rotation is the same as a spatial rotation in three dimensions.
25.
In a universe without anisotropic stress ( that is, where the stress energy tensor is invariant under spatial rotations, or the three principal pressures are identical ) the Einstein equation sets \ Phi = \ Psi.
26.
The mathematical resolution of the bar and ring paradox is based on the fact that the product of two proper Lorentz transformations may produce a Lorentz transformation which is not proper, but rather includes a spatial rotation component.
27.
They are all equivalent under spatial rotation ( not true of the ! = 1 harmonics you can't rotate back-and-forth motion into circular motion ) and they all represent circular motion in some plane.
28.
The diagram D 2 is two isolated nodes, the same as A 1 & cup; A 1, and this coincidence corresponds to the covering map homomorphism from SU ( 2 ) & times; SU ( 2 ) to SO ( 4 ) given by quaternion multiplication; see quaternions and spatial rotation.
29.
For example, studies show that being told before the test that men typically perform better, or that the task is linked with jobs like aviation engineering typically associated with men versus jobs like fashion design typically associated with women, will negatively affect female performance on spatial rotation and positively influence it when subjects are told the opposite.
30.
The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism \ operatorname { Spin } ( 3 ) \ cong \ operatorname { Sp } ( 1 ) where Sp ( 1 ) is the multiplicative group of unit quaternions . ( For a description of this homomorphism see the article on quaternions and spatial rotations .)
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