Friday, June 11, 2010

Spin Networks Technical Notes - Greg Egan

"The effect of parallel-transporting vectors along a path, viewed as a linear map between the tangent spaces at the beginning and end of the path, is known as the holonomy for that path, and will always take the form of some rotation, R. The family of geometries for which the applet evaluates each spin network is characterised by the following simple rule: parallel transport along a straight line from the point (x0,y0,z0) to the point (x1,y1,z1) rotates vectors around the axis a, by an angle equal to the magnitude of a, where:
a = κ (y0 z1 – z0 y1, z0 x1 – x0 z1, x0 y1 – y0 x1)"

5 out of 5

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