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...ve corresponding space or time symmetries. Examples of important spacetime symmetries are — ...''| denotes the norm of the vector ''x'' (for actual rotations this is the Euclidean or 2-norm). ...

...s" of elliptic geometry are [[geodesic]] curves and, unlike the lines of [[Euclidean geometry]], are of finite length. ...center point in [[Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[completely orthogonal]] rotation planes.]] ...

...;''v'' if the quaternion norm is ''u'' + ''v''{{radic|5}}. This Euclidean norm defines a [[quadratic form]] on ''L'', under which the lattice is isom ...a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm. ...

...rmations of Euclidean spaces, while the sections describe only the case of Euclidean vector spaces or of spaces of coordinate vectors. The "formal definition" s ...[[geometric transformation]] of a [[Euclidean space]] that preserves the [[Euclidean distance]] between every pair of points.<ref name= Bottema>{{cite book |tit ...

...th no overlaps. The resulting [[honeycomb (geometry)|honeycomb]] will have symmetries that take any copy of the plesiohedron to any other copy. A set <math>S</math> of points in [[Euclidean space]] is a [[Delone set]] if there exists a number <math>\varepsilon>0</m ...

...eometry]], it is possible to fill 3/4 of the volume of three-dimensional [[Euclidean space]] by three sets of infinitely-long square [[prism (geometry)|prism]]s ...ce area. The tetrastix and Weaire–Phelan structures have the same group of symmetries.<ref>{{harvtxt|Conway|Burgiel|Goodman-Strauss|2008}}, "Understanding the Ir ...

...v|author=N Schmitt|title=Constant Mean Curvature ''n''-noids with Platonic Symmetries|eprint=math/0702469|year=2007}}</ref> ....<ref>{{cite web|author=Matthias Weber|title=Classical Minimal Surfaces in Euclidean Space by Examples|year=2001|url=http://www.indiana.edu/~minimal/research/cl ...

[[Category:Euclidean symmetries]] ...

...] in connection with the [[Klein quartic]], it was first realized in the [[Euclidean plane]] by [[Branko Grünbaum]] and [[John Rigby (mathematician)|John F. Rig ...tes are [[complex number]]s rather than the real-number coordinates of the Euclidean plane. ...

...1754|s2cid=2173557 }}</ref><ref>Gunn C. (2011) On the Homogeneous Model of Euclidean Geometry. In: Dorst L., [[Joan Lasenby|Lasenby J.]] (eds) Guide to Geometri The Euclidean plane can be represented by the set <math display="inline">\Pi = \{i + x \v ...

...uniform projective polyhedron that [[immersion (mathematics)|immerses]] in Euclidean 3-space.]] ...uniform projective polyhedron that [[immersion (mathematics)|immerses]] in Euclidean three-space as a uniform traditional polyhedron. ...

...any other object. For instance the cells of an arrangement of lines in the Euclidean plane are of three types: ...apply to these types of space, but it is still of interest to study their symmetries and topological properties.<ref>{{citation ...

The types of symmetries that are possible for a geometric object depend on the set of [[geometric t ==Euclidean symmetries in general== ...

...he configuration space of a rigid body such as a satellite is the group of Euclidean motions (translations and rotations in space), while the configuration spac ...r example the equivariance properties of this momentum for a wide class of symmetries), and less on questions of dynamics. ...

...Richmond configuration, which swap the two sides of its bipartition. These symmetries correspond to the [[outer automorphism]]s of the symmetric group on six ele ...uration. The same realization may be projected into Euclidean space or the Euclidean plane.<ref name="cc"/> ...

[[Category:Euclidean symmetries]] ...

== Symmetries == === Extra symmetries === ...

...e a one-dimensional [[symmetry group]] but not a two-dimensional family of symmetries. There exist binary tilings with tiles of arbitrarily small area. ...with different notions of length, area, congruence, and symmetry than the Euclidean plane.{{r|radin}} ...

In [[mathematical physics]], certain [[conformal map#Pseudo-Euclidean space|conformal map]]s known as [[spherical wave transformation]]s are '''s ...itle=On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics * |url=https://onlinelibrary.wiley.com/ ...

...[carved stone balls]] exhibit a variety of symmetries including all of the symmetries of [[Platonic solid]]s. ...es the [[Infinite set|infinitude]] of [[prime number]]s and presents the [[Euclidean algorithm]]; he states the law of reflection in ''Catoptrics'', and he pro ...