Conjugation of isometries in Euclidean space | Wikipedia audio article
Channel: wikipedia tts
Date: 03/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Conjugation_of_isometries_in_Euclidean_space


00:00:07 1 Translation
00:02:25 2 Inversion
00:04:07 3 Rotation
00:05:18 4 Reflection
00:05:24 5 Rotoreflection
00:05:33 6 Isometry groups
00:05:51 7 Cyclic groups
00:06:01 8 Dihedral groups
00:06:27 Isometry groups
00:06:37 Two isometry groups are said to be equal up to conjugacy with respect to affine transformations if there is an affine transformation such that all elements of one group are obtained by taking the conjugates by that affine transformation of all elements of the other group. This applies for example for the symmetry groups of two patterns which are both of a particular wallpaper group type. If we would just consider conjugacy with respect to isometries, we would not allow for scaling, and in the case of a parallelogrammetic lattice, change of shape of the parallelogram. Note however that the conjugate with respect to an affine transformation of an isometry is in general not an isometry, although volume (in 2D: area) and orientation are preserved.
00:07:33 Cyclic groups
00:08:16 Dihedral groups



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SUMMARY
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In a group, the conjugate by g of h is ghg−1.