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Symmetry Spire
Symmetry SpireSymmetry Spire (10,565-10,645 feet (3,220-3,245 m)) is located in the Teton Range, Grand Teton National Park in the U.S. state of Wyoming. The mountain, first climbed via the east ridge route on August 20, 1929 by Fritiof Fryxell and Phil Smith, towers above the northwest shore of Jenny Lake and Cascade Canyon. The scenic Lake of the Crags, a cirque lake or tarn, is located northwest of the summit and is accessed by way of Hanging Canyon. Popular with mountaineers, the spire has numerous challenging cliffs.— — — — — —Controversy Surrounding "The Symmetry Project"In 2009, the company gained attention in the mainstream media due to controversy over the Symmetry Project. Fox News described the piece as "it amounts to two people writhing naked on the floor, a government-funded tango in the altogether.". The article quoted Curtis as responding "I think art is an incredibly important part of our culture and our life and ... that it's very much appropriate that our government should be supporting it." Although conservative pundits pointed to the Symmetry Project as a representation of wasted taxpayer's money, it received generally favorable reviews from dance critics.— — — — — —Mathematics of physical symmetryThe transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists. Continuous symmetries are specified mathematically by continuous groups (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group S O ( 3 ) displaystyle ,SO(3) . (The 3 refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is S O ( 3 ) displaystyle ,SO(3) . Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincar group). Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the symmetric group S 3 displaystyle ,S_3 . A type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries. Gauge symmetries in the Standard Model, used to describe three of the fundamental interactions, are based on the SU(3) SU(2) U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the strong force, the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force.) Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology). Conservation laws and symmetryThe symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to conservation of (linear) momentum, and temporal translation symmetry (i.e. homogeneity of time) gives rise to conservation of energy. The following table summarizes some fundamental symmetries and the associated conserved quantity.— — — — — —Reflection symmetryReflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical.— — — — — —Symmetry constructionsThere are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 256 8-demicube facets around each vertex— — — — — —Lie point symmetryTowards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations. This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by lie Cartan. Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are translations, rotations and scalings. The Lie symmetry theory is a well-known subject. In it are discussed continuous symmetries opposed to, for example, discrete symmetries. The literature for this theory can be found, among other places, in these notes.
Rep-tiles and Symmetry
Rep-tiles and symmetrySome rep-tiles, like the square and equilateral triangle, are symmetrical and remain identical when reflected in a mirror. Others, like the sphinx, are asymmetrical and exist in two distinct forms related by mirror-reflection. Dissection of the sphinx and some other asymmetric rep-tiles requires use of both the original shape and its mirror-image.— — — — — —R-symmetryIn theoretical physics, the R-symmetry is the symmetry transforming different supercharges in a theory with supersymmetry into each other. In the simplest case of the N=1 supersymmetry, such an R-symmetry is isomorphic to a global U(1) group or its discrete subgroup (for the Z2 subgroup it is called R-parity). For extended supersymmetry, the R-symmetry group becomes a global non-abelian group. In a model that is classically invariant under both N=1 supersymmetry and conformal transformations, the closure of the superconformal algebra (at least on-shell) needs the introduction of a further bosonic generator that is associated to the R-symmetry.— — — — — —Degeneracy and symmetryThe physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find the energy levels and degeneracies without solving the Schrdinger equation, hence reducing effort. Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that H = S H S 1 = S H S displaystyle H'=SHS^-1=SHS^dagger , since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, we have S H S = H displaystyle SHS^dagger =H S H S 1 = H displaystyle SHS^-1=H H S = S H displaystyle HS=SH [ S , H ] = 0 displaystyle [S,H]=0 Now, if | displaystyle |alpha angle is an energy eigenstate, H | = E | displaystyle H|alpha angle =E|alpha angle where E is the corresponding energy eigenvalue. H S | = S H | = S E | = E S | displaystyle HS|alpha angle =SH|alpha angle =SE|alpha angle =ES|alpha angle which means that S | displaystyle S|alpha angle is also an energy eigenstate with the same eigenvalue E. If the two states | displaystyle |alpha angle and S | displaystyle S|alpha angle are linearly independent (i.e. physically distinct), they are therefore degenerate. In cases where S is characterized by a continuous parameter displaystyle epsilon , all states of the form S ( ) | displaystyle S(epsilon )|alpha angle have the same energy eigenvalue. Symmetry group of the HamiltonianThe set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.— — — — — —Glide reflection symmetryIn 2D, a glide reflection symmetry (also called a glide plane symmetry in 3D, and a transflection in general) means that a reflection in a line or plane combined with a translation along the line or in the plane, results in the same object (such as in the case of footprints). The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the frieze group p11g, and is isomorphic with the infinite cyclic group Z. Rotoreflection symmetryIn 3D, a rotary reflection, rotoreflection or improper rotation is a rotation about an axis combined with reflection in a plane perpendicular to that axis. The symmetry groups associated with rotoreflections include: if the rotation angle has no common divisor with 360, the symmetry group is not discrete. if the rotoreflection has a 2n-fold rotation angle (angle of 180/n), the symmetry group is S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; the abstract group is C2n). A special case is n = 1, an inversion, because it does not depend on the axis and the plane. It is characterized by just the point of inversion. The group Cnh (angle of 360/n); for odd n, this is generated by a single symmetry, and the abstract group is C2n, for even n. This is not a basic symmetry but a combination.For more, see point groups in three dimensions.
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