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Euclidean Symmetries
In geometry, an object has symmetry if there is an Operation (mathematics), operation or Transformation (function), transformation (such as Translation (geometry), translation, Scaling (geometry), scaling, Rotation (mathematics), rotation or Reflection (mathematics), reflection) that maps the figure/object onto itself (i.e., the object has an Invariant (mathematics), invariance under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be ''symmetric under rotation'' or to have ''rotational symmetry''. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry. The types of symmetries that are possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Dieudonné Theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''-dimension (vector space), dimensional symmetric bilinear space can be described as the function composition, composition of at most ''n'' reflection (mathematics), reflections. The notion of a symmetric bilinear space is a generalization of Euclidean space whose structure is defined by a symmetric bilinear form (which need not be Positive-definite bilinear form, positive definite, so is not necessarily an inner product – for instance, a pseudo-Euclidean space is also a symmetric bilinear space). The orthogonal transformations in the space are those linear map#Endomorphisms and automorphisms, automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and angles. These orthogonal transformations form a group (mathematics), group under com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
