Spin and Pin
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is simply connected, and thus the universal covering group for SO(n). By far the most famous example of a spin group is Spin(3), which is nothing but SU(2), or the group of unit quaternions.
The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.
Read more about this topic: Orthogonal Matrix
Famous quotes containing the words spin and, spin and/or pin:
“Spin and die,
To live again as butterfly.”
—Christina Georgina Rossetti (1830–1894)
“In tragic life, God wot,
No villain need be! Passions spin the plot:
We are betrayed by what is false within.”
—George Meredith (1828–1909)
“It is not, truly speaking, the labour that is divided; but the men: divided into mere segments of men—broken into small fragments and crumbs of life, so that all the little piece of intelligence that is left in a man is not enough to make a pin, or a nail, but exhausts itself in making the point of a pin or the head of a nail.”
—John Ruskin (1819–1900)