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Rotational symmetry

Rotational symmetry known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks the same for each rotation. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are i.e. isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of E+. Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, the symmetry group is the whole E. With the modified notion of symmetry for vector fields the symmetry group can be E+. For symmetry with respect to rotations about a point we can take that point as origin; these rotations form the special orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO.

In another definition of the word, the rotation group of an object is the symmetry group within E+, the group of direct isometries. For chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant; because of Noether's theorem, the rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Rotational symmetry of order n called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point or axis means that rotation by an angle of 360°/n does not change the object. A "1-fold" symmetry is no symmetry; the notation for n-fold symmetry is Cn or "n". The actual symmetry group is specified by the point or axis of symmetry, together with the n. For each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. Although for the latter the notation Cn is used, the geometric and abstract Cn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D.

The fundamental domain is a sector of 360°/n. Examples without additional reflection symmetry: n = 2, 180°: the dyad. If there is e.g. rotational symmetry with respect to an angle of 100° also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller. For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities: In addition to an n-fold axis, n perpendicular 2-fold axes: the dihedral groups Dn of order 2n; this is the rotation group of regular bipyramid. Although the same notation is used, the geometric and abstract Dn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see dihedral symmetry groups in 3D. 4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron. The group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, 6×2-fold axes: the rotation group O of order 24 of a cube and a regular octahedron.

The group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, 15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron. The group is isomorphic to alternating group A5; the group contains 10 versions of D3 and 6 versions of D5. In the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges; the other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry; the fundamental domain is a half-line. In three dimensions we can distinguish spherical symmetry; that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, a radial half-line, respectively.

Axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry like a doughnut. An example of approximate spherical symmetry is the Earth. In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the cas

EMMS (media player)

EMMS is media player software for Emacs. It is written in Emacs Lisp; the name could echo XMMS. It may be derived from an earlier Emacs-based player called mp3-player. EMMS may have multiple back ends to connect to external players so EMMS can support a few different audio and video formats, while remaining clean and small itself. EMMS is divided into three parts, the player back ends, media sources, the core player. One of the player back ends connects to MPD. Other backends are available for gstreamer. Additional players can be defined. EMMS implements queue. Locations in files can be bookmarked. Standard Emacs key bindings are used to navigate, edit the playlist, control playback. Using Emacs server support, playlists can be built using a file manager such as ROX-Filer. EMMS supported scrobbling to Last.fm until version 4.0, when this service was replaced with the free software Libre.fm. There are many third-party scripts to enhance EMMS to provide pop-up notifications, lyric fetching, binaural beat generation.

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Byakkoya - White Tiger Field

White Tiger Field is the tenth solo album by Susumu Hirasawa. Two of its songs were featured in the 2006 anime film Paprika. All tracks are written by Susumu Hirasawa."Parade" contains samples of "Nurse Cafe" and "Monster a Go Go". Susumu Hirasawa - Vocals, Electric guitar, Electronic keyboard, Personal computer, Digital audio workstation, Vocaloid, Sequencer, Production Nguyen Ngoc Hoa - Voice on "Byakkoya - White Tiger Field" Masanori Chinzei - Recording, Mastering Syotaro Takami - Translation Toshifumi "non graph" Nakai - Design Presented by CHAOS UNION/TESLAKITE LABEL: Kenji Sato, Rihito Yumoto and Mika Hirano Byakkoya - White Tiger Field "The Girl in Byakkoya - White Tiger Field" music video

List of people who have addressed both Houses of the United Kingdom Parliament

This is a list of people who have addressed both Houses of the United Kingdom Parliament at the same time. Although English and British monarchs have jointly addressed the House of Commons and the House of Lords on several occasions since the 16th century, the first foreign dignitary to do so was French President Albert Lebrun in March 1939; the list excludes the speeches given by the Sovereign at the State Opening of Parliament and at the close of each parliamentary session. Only three people besides the reigning monarch at the time have addressed both Houses together on more than one occasion. Nelson Mandela addressed Members of the Commons and the Lords in 1993 and in 1996 as President of South Africa. Mikhail Gorbachev addressed the Houses as a foreign delegate of the Soviet Union in 1984 and again, in 1993, on behalf of the Inter-Parliamentary Union. Shimon Peres addressed the Houses as Prime Minister of Israel in 1986 and as President in 2008. Parliament of the United Kingdom Joint address Joint meetings of the Australian Parliament Joint session of the United States Congress List of joint sessions of the United States Congress United Kingdom Parliament

Daniel D. Chetti

Daniel Divaker Chetti is Professor at the Arab Baptist Theological Seminary. Daniel Chetti was known for his contribution as Director of Programmes and Church Relations at the Board of Theological Education of the Senate of Serampore College in Bangalore as well as at the Gurukul Lutheran Theological College, Chennai, he was educated at La Martiniere, where he was a House Captain. Chetti's doctoral work on Khonds has been kept for posterity at the National Kolkata. 1986, A history of the Meriah Wars, 1836-1862: policies and personalities in the evolution of British moral imperative among the Khonds of Orissa, 1989, Adventurous faith & transforming vision, 1989, Making Visible: The Role of Women in the Nineteenth Century Protestant Christianity in India 1996, Ecology and Development: Theological Perspectives, 1998, Ethical issues in the struggles for justice: quest for pluriform communities: essays in honour of K. C. Abraham

James Thompson (footballer)

James William "Jimmy" Thompson was a professional footballer, football manager and football scout. Thompson, a striker, began his career as an amateur with Charlton and Wimbledon before playing for Millwall, Coventry City, Clapton Orient, Luton Town, Norwich City, Sunderland and Hull City, he moved into Non-League football with Tunbridge Wells Rangers and Peterborough United, he signed for Tranmere Rovers and Aldershot before retiring. He managed Dartford and worked as a scout at Chelsea and Southampton and is credited with having discovered Jimmy Greaves. In his career, he played scoring 97 goals. 30 of those appearances and 17 of the goals were for Norwich. Canary Citizens by Mike Davage, John Eastwood, Kevin Platt, published by Jarrold Publishing, ISBN 0-7117-2020-7