In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, a Platonic solid. There are three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120; the pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry; the elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are numerous other dodecahedra; the convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol. The dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex; the convex regular dodecahedron has three stellations, all of which are regular star dodecahedra.
They form three of the four Kepler–Poinsot polyhedra. They are the small stellated dodecahedron, the great dodecahedron, the great stellated dodecahedron; the small stellated dodecahedron and great dodecahedron are dual to each other. All of these regular star dodecahedra have regular pentagrammic faces; the convex regular dodecahedron and great stellated dodecahedron are different realisations of the same abstract regular polyhedron. In crystallography, two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, the tetartoid with tetrahedral symmetry: A pyritohedron is a dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not constrained to be regular, the underlying atomic arrangement has no true fivefold symmetry axis.
Its 30 edges are divided into two sets -- containing 6 edges of the same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes and four threefold axes. Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite, it may be an inspiration for the discovery of the regular Platonic solid form; the true regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry, which includes true fivefold rotation axes. Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube; the coordinates of the eight vertices of the original cube are: The coordinates of the 12 vertices of the cross-edges are: where h is the height of the wedge-shaped "roof" above the faces of the cube. When h = 1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed; when h = 0, the cross-edges are absorbed in the facets of the cube, the pyritohedron reduces to a cube.
When h = −1 + √5/2, the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When h = −1 − √5/2, the conjugate of this value, the result is a regular great stellated dodecahedron. A reflected pyritohedron is made by swapping; the two pyritohedra can be superimposed to give the compound of two dodecahedra. The image to the left shows the case; the pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of collinear edges, a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal, it is possible to go past these limiting cases, creating nonconvex pyritohedra. The endo-dodecahedron is equilateral. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, on to the regular great stellated dodecahedron where all edges and angles are equal again, the faces have been distorted into regular pentagrams.
On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces. A tetartoid is a dodecahedron with chiral tetrahedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. Although regular dodecahedra do not exist in crystals, the tetartoid form does; the name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form, its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, the bisection lines are slanted retaining 3-fold rotation at the 8 corners. The fol
The Ian Charleson Award is a British theatrical award that rewards the best classical stage performance in Britain by an actor under age 30. The award's current definition of a classical play is one written before 1918; the Ian Charleson Award is named in memory of the renowned British actor Ian Charleson, is run by the Sunday Times newspaper and the National Theatre. The award was established in 1990 after Charleson's death, has been presented annually since then. Recipients receive a cash prize; the award for the previous year's performance is presented the following year. The first annual Ian Charleson Award, for a 1990 performance, was presented in January 1991; the 2017 award was presented on 18 May 2018. Ian Hughes, for Torquato Tasso in Torquato Tasso Joe Dixon, for Jacques in an all-male production of As You Like It Tom Hollander, for Witwoud in The Way of the World Emma Fielding, for Agnes in The School for Wives Toby Stephens, for Coriolanus in Coriolanus Lucy Whybrow, for Eleanora in Easter Alexandra Gilbreath, for Hedda in Hedda Gabler Mark Bazeley, for Konstantin in The Seagull Dominic West, for Konstantin in The Seagull Claudie Blakley, for Nina in The Seagull Rupert Penry-Jones, for Don Carlos in Don Carlos David Oyelowo, for Henry VI in Henry VI Claire Price, for Berinthia in The Relapse Rebecca Hall, for Vivvie in Mrs Warren's Profession Lisa Dillon, for Hilda Wangel in The Master Builder Nonso Anozie, for Othello in Othello Mariah Gale, for Viola in Twelfth Night, Annabella in Tis Pity She's a Whore, Nurse Ludmilla and Klara in The Last Waltz Andrea Riseborough, for Isabella in Measure for Measure and Miss Julie in Miss Julie Rory Kinnear, for Pytor in Philistines and Sir Fopling Flutter in The Man of Mode Tom Burke, for Adolph in Creditors Ruth Negga, for Aricia in Phèdre Gwilym Lee, for Edgar in King Lear Cush Jumbo, for Rosalind in As You Like It Ashley Zhangazha, for Ross in Macbeth Jack Lowden, for Oswald in Ghosts Susannah Fielding, for Portia in The Merchant of Venice James McArdle, for Platonov in Platonov Paapa Essiedu, for in Hamlet in Hamlet and Edmund in King Lear Natalie Simpson, for Duchess Rosaura in The Cardinal Bally Gill, for Romeo in Romeo and Juliet
The Memphis Press-Scimitar was an afternoon newspaper based in Memphis, United States, owned by the E. W. Scripps Company. Created from a merger in 1926 between the Memphis Press and the Memphis News-Scimitar, the newspaper ceased publication in 1983, it was the main rival to The Commercial Appeal based in Memphis and owned by Scripps. At the time of its closure, the Press-Scimitar, had lost a third of its circulation in 10 years and was down to daily sales of 80,000 copies. From 1909 to 1931, The Memphis Press was edited by founder Ross B. Young, a journalist from Ohio brought down by local business interests looking for a voice to speak to the stranglehold that E. H. "Boss" Crump had on city government and contracts. From 1931 to 1962, The Press-Scimitar was edited by the crusading Edward J. Meeman, who left a fortune to foster the study of environmental sciences. In John Grisham's novel The Client, the Memphis Press is fictionally presented as still existing and flourishing as a major Memphis paper into the 1990s.
List of newspapers in Tennessee "Memphis Press-Scimitar Rolls Final Edition Today", Associated Press via Ocala Star-Banner, 31 October 1983. "Memphis Paper Quits.. Frank, Ed. "The History", Memphis Press-Scimitar, accessed 7 November 2010. Archived by WebCite on 7 November 2010