Saturday, April 2, 2011

Lie Groups in Lights

This is a genuine photograph, not a Photoshop job.

Expat Kiwi, Denis Smith, explains how he does it in this video interview. Like the other images on his "Ball of Light" web page, the backstory is also very compelling: reaching a mid-life crisis while working for Fuji Xerox and purchasing his first camera less than two years ago.

Although he is likely unaware of it, Smith is actually imaging a fundamental theorem of Lie groups:

The coset G / H of a continuous Lie group G by a closed subgroup H is a manifold of dimension given by dim(G) − dim(H).
Here, G = SO(3) is the special orthogonal group of norm-preserving continuous rotations about an origin in 3-space with the subgroup H = SO(2) corresponding to rotations in the plane. The topology of SO(3) in Euclidean space is a solid ball with dimensionality given by the number of Lie group generators. Since SO(n) has n(n−1)/2 generators (e.g., 3 Euler angles when n = 3), it follows that the corresponding coset manifold is

S2 ≅ SO(3) / SO(2)

which is the 2-sphere approximated in the above photo.

From the standpoint of Lie groups, "Sphere of Light" would be a more accurate name for Smith's project.

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