Displacement (mathematics) | Wikipedia audio article

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00:00:48 1 Overview
00:01:12 1.1 Dimensionality
00:01:37 1.2 Direct and indirect isometries
00:02:25 1.3 Topology of the group
00:03:14 1.4 Lie structure
00:04:02 1.5 Relation to the affine group
00:04:51 2 Detailed discussion
00:05:39 2.1 Subgroup structure, matrix and vector representation
00:06:28 2.2 Subgroups
00:06:52 2.3 Overview of isometries in up to three dimensions
00:08:05 2.4 Commuting isometries
00:08:53 2.5 Conjugacy classes
00:09:42 3 See also
00:10:06 4 References
00:10:55 See also
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SUMMARY
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In mathematics, an Euclidean group is the group of (Euclidean) isometries of an Euclidean space 𝔼n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n).
The Euclidean group E(n) comprises all translations, rotations, and reflections of 𝔼n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space.
A Euclidean isometry can be direct or indirect, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections.
These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 - implicitly, long before the concept of group was invented.