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This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Euclidean_group 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 Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts Speaking Rate: 0.8477492261428886 Voice name: en-GB-Wavenet-B "I cannot teach anybody anything, I can only make them think." - Socrates SUMMARY ======= 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.
