Uniform star polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are sometimes called nonconvex polyhedra to imply self-intersecting; each polyhedron can contain star polygon vertex figures or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 5 quasiregular ones, 48 semiregular ones. There are two infinite sets of uniform star prisms and uniform star antiprisms. Just as star polygons correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, correspond to spherical polyhedra with overlapping tiles; the remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as Miller's monster, do not have well-defined densities. The nonconvex forms are constructed from Schwarz triangles. All the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements.

Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed with their vertex configuration. Note: For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull vertex arrangement has same topology as one of these, but has nonregular faces. For example an nonuniform cantellated form may have rectangles created in place of the edges rather than squares. See Prismatic uniform polyhedron. There is the tetrahemihexahedron which has tetrahedral symmetry. There are two Schwarz triangles that generate unique nonconvex uniform polyhedra: one right triangle, one general triangle; the general triangle generates the octahemioctahedron, given further on with its full octahedral symmetry. There are 8 convex forms, 10 nonconvex forms with octahedral symmetry. There are four Schwarz triangles that generate nonconvex forms, two right triangles and two general triangles:. There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry..

Some of the nonconvex snub forms have reflective vertex symmetry. Coxeter identified a number of degenerate star polyhedra by the Wythoff construction method, which contain overlapping edges or vertices; these degenerate forms include: Small complex icosidodecahedron Great complex icosidodecahedron Small complex rhombicosidodecahedron Great complex rhombicosidodecahedron Complex rhombidodecadodecahedron One further nonconvex degenerate polyhedron is the Great disnub dirhombidodecahedron known as Skilling's figure, vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges. It is counted as a degenerate uniform polyhedron rather than a uniform polyhedron because of its double edges, it has Ih symmetry. Star polygon List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter, H. S. M.. "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246: 401–450. Doi:10.1098/rsta.1954.0003.

Wenninger, Magnus. Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087. Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900. Sopov, S. P. "A proof of the completeness on the list of elementary homogeneous polyhedra", Ukrainskiui Geometricheskiui Sbornik: 139–156, MR 0326550 Skilling, J. "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 278: 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333 Har'El, Z. Uniform Solution for Uniform Polyhedra. Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, dual images Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993. Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals. Discrete & Computational Geometry 27:353-375. Klitzing, Richard. "3D uniform polyhedra". Weisstein, Eric W. "Uniform Polyhedron". MathWorld


Socrate is a work for voice and piano by Erik Satie. First published in 1919 for voice and piano, in 1920 a different publisher reissued the piece "revised and corrected". A third version of the work exists, for small orchestra and voice, for which the manuscript has disappeared and, available now only in print; the text is composed of excerpts of Victor Cousin's translation of Plato's dialogues, all of the chosen texts referring to Socrates. The work was commissioned by Princess Edmond de Polignac in October 1916; the Princess had specified that female voices should be used: the idea had been that Satie would write incidental music to a performance where the Princess and/or some of her friends would read aloud texts of the ancient Greek philosophers. As Satie, after all, was not so much in favour of melodrama-like settings, that idea was abandoned, the text would be sung — be it in a more or less reciting way. However, the specification remained. Satie composed Socrate between January 1917 and the spring of 1918, with a revision of the orchestral score in October of that same year.

During the first months he was working on the composition, he called it Vie de Socrate. In 1917 Satie was hampered by a lawsuit over an insulting postcard he had sent, which nearly resulted in prison time; the Princess diverted this danger by her financial intercession in the first months of 1918, after which Satie could work free of fear. Satie presents Socrate as a "symphonic drama in three parts". "Symphonic drama" appears to allude to Romeo et Juliette, a "dramatic symphony" that Hector Berlioz had written nearly eighty years earlier: and as usual, when Satie makes such allusions, the result is about the complete reversal of the former example. Where Berlioz's symphony is more than an hour and a half of expressionistic orchestrated drama, an opera forced into the form of a symphony, Satie's thirty-minute composition reveals little drama in the music: the drama is concentrated in the text, presented in the form of recitativo-style singing to a background of sparsely orchestrated, nearly repetitive music, picturing some aspects of Socrates' life, including his final moments.

As Satie did not foresee an enacted or scenic representation, while he disconnected the male roles from the female voice delivering these texts, keeping in mind a good understandability of the story by the words of the text, the form of the composition could rather be considered as oratorio, than opera, or drama. It is possible to think that Satie took formally similar secular cantatas for one or two voices and a moderate accompaniment as his examples for the musical form of Socrate: nearly all Italian and German Baroque composers had written such small-scale cantatas on an Italian text: Vivaldi, Bach, etc; this link is however unlikely: these older compositions all alternated recitatives with arias, further there is little evidence Satie based his work directly on the examples of foreign baroque composers, most of all, as far as the baroque composers were known in early 20th century Paris, these small secular Italian cantatas would be the least remembered works of any of these composers.

The three parts of the composition are: Portrait de Socrate, text taken from Plato's Symposium Les bords de l'Ilissus, text taken from Plato's Phaedrus Mort de Socrate, text taken from Plato's Phaedo The piece is written for voice and orchestra, but exists in a version for voice and piano. This reduction had been produced by Satie, concurrently with the orchestral version; each speaker in the various sections is meant to be represented by a different singer, according to Satie's indication two of these voices soprano, the two other mezzo-sopranos. Nonetheless all parts are more or less in the same range, the work can be sung by a single voice, has been performed and recorded by a single vocalist, female as well as male; such single vocalist performances diminish however the effect of dialogue. The music is characterised by simple repetitive rhythms, parallel cadences, long ostinati. Although more recent translations were available, Satie preferred Victor Cousin's antiquated French translation of Plato's texts: he found in them more clarity and beauty.

The translation of the libretto of Socrate that follows is taken from Benjamin Jowett's translations of Plato's dialogues that can be found on the Gutenberg Project website. The original French text can be found here. Alcibiades And now, my boys, I shall praise Socrates in a figure which will appear to him to be a caricature, yet I speak, not to make fun of him, but only for the truth's sake. I say, that he is like the busts of Silenus, which are set up in the statuaries' shops, holding pipes and flutes in their mouths. I say that he is like Marsyas the satyr, and are you not a flute-player? That you are, a performer far more wonderful than Marsyas, he indeed with instruments used to charm the souls of men by the power of his breath, the players of his music do so still: for the melodies of Olympus are derived from Marsyas who taught them But you produce the same effect with your words only, do not require the

Friends of Israel in the Parliament of Norway

Friends of Israel in the Parliament of Norway is a pro-Israel caucus group consisting of members of the Parliament of Norway. In 1974 the group constituted a majority in the Parliament of Norway for the first time, with 86 members among the 150 parliamentary representatives. All political parties except for the Socialist Electoral League were represented. In 1981 the group had 100 members, but this number declined following the Israeli invasion of Lebanon in 1982. In the 2005 the group had 28 members from three political parties. One member, KrF's Jon Lilletun, died during the term. In comparison, the number of seats in parliament for each party was 38 from the Progress Party, 11 from the Christian Democratic Party and 23 from the Conservative Party; the group was chaired by Ingebrigt S. Sørfonn of the Christian Democratic Party. In late 2007, an initiative was taken to form an opposing parliamentary group, Friends of Palestine in the Parliament of Norway, in support of Palestinians. After the 2009 election, the groups supporting Israel and Palestine reconvened.

Torbjørn Røe Isaksen joined both groups. In late November 2009 the group had 26 members. In comparison, the number of seats in parliament for each party was 41 from the Progress Party, 10 from the Christian Democratic Party and 30 from the Conservative Party; the members elected Hans Olav Syversen as leader from 2009 to 2011, Jørund Rytman as leader from 2011 to 2013. As of June 2014, the Friends of Israel have a total of 37 members, 20 from the Progress Party, 10 from the Christian Democratic Party, 4 from the Conservative Party, 3 from the Labour Party. Hans Fredrik Grøvan would chair the caucus the first two years, Jørund Rytman the final two years. Two members, Kari Henriksen and Sverre Myrli were members of Friends of Palestine. Myrland, Conrad. "Israels venner på Stortinget 2005-2009". Med Israel for fred. Retrieved 2006-07-14. "Israels Venner". Dagbladet. 2002-04-04. Retrieved 2006-07-14