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A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form ; transformations preserving this form are sometimes called "isometries", and the collection of them is ...
A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. [5] [6] This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere. History on four-dimensional groups
It is a subgroup (but not a normal subgroup) of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes. It is a normal subgroup of O h. In spite of being called T h, it does not apply to a tetrahedron. O, (432) [4,3] + 432 order 24: chiral octahedral symmetry
Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(n) (the isometry group of R n). Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H 1, H 2 are related by H 1 = g −1 H 2 g for some g ...
Two marked metric graph structures f 1 : R n → Γ 1 and f 2 : R n → Γ 2 are equivalent if there exists an isometry θ : Γ 1 → Γ 2 such that, up to free homotopy, we have θ o f 1 = f 2. The Outer space X n consists of equivalence classes of all the volume-one marked metric graph structures on F n.
Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...
The Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T( n ), and the orthogonal group O( n ). Any element of E( n ) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: x ↦ A ( x + b ) {\displaystyle x\mapsto A(x+b)} where ...