Re: Math trivia
So far, I've been doing "point groups" with elements R turning x into x':
x' = R.x
Adding a translation / shift / displacement D gives
x' = R.x + D
This is the Euclidean group E(n) or Euc(n), named after the famous geometer. Its elements may also be expressed in matrix form: {{R, D}, {0, 1}}.
It looks related to O(n+1), and it is -- O(n+1) is the symmetry group of a hypersphere with a n-D surface. 1D: circle, 2D: sphere, ... A hyperbolic surface also has similar symmetries, though it's a bit of a long story.
Let's now consider different numbers of dimensions of variation of D. If zero dimensions, then we are back at point groups again. If less than n, then R splits into R1 * R2, where R1 is for D's dimensions and R2 is for the others.
It's easy to show that the D-only group (R = identity matrix) is a normal subgroup of the Euclidean group. Its quotient group is the R-only group.
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