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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, often denoted SE(n) and E + (n), whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and ...
The direct similitudes form a normal subgroup of S and the Euclidean group E(n) of isometries also forms a normal subgroup. [20] The similarities group S is itself a subgroup of the affine group, so every similarity is an affine transformation. One can view the Euclidean plane as the complex plane, [b] that is, as a 2-dimensional space over the ...
Space Only the trivial isometry group C 1 leaves the whole space fixed. Plane C s with respect to a plane leaves that plane fixed. Line Isometry groups leaving a line fixed are isometries which in every plane perpendicular to that line have common 2D point groups in two dimensions with respect to the point of intersection of the line and the planes.
Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools.
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 (Orbifold notation for polyhedra, Euclidean and hyperbolic tilings) On Quaternions and Octonions , 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
This group of motions is noted for its properties. For example, the Euclidean group is noted for the normal subgroup of translations. In the plane, a direct Euclidean motion is either a translation or a rotation, while in space every direct Euclidean motion may be expressed as a screw displacement according to Chasles' theorem.
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In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if >) or reverse (if <) the direction of all vectors. Together with the translations , all homotheties of an affine (or Euclidean) space form a group , the group of dilations or homothety-translations .