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Polytope compound

A polyhedral compound is a figure, composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram; the outer vertices of a compound can be connected to form a convex polyhedron called the convex hull. The compound is a facetting of the convex hull. Another convex polyhedron is formed by the small central space common to all members of the compound; this polyhedron can be used as the core for a set of stellations. A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, face-transitive. There are five regular compounds of polyhedra. Best known is the compound of two tetrahedra called the stella octangula, a name given to it by Kepler; the vertices of the two tetrahedra define a cube, the intersection of the two: an octahedron, which shares the same face-planes as the compound. Thus the compound of two tetrahedra is a stellation of the octahedron, in fact, the only finite stellation thereof.

The stella octangula can be regarded as a dual-regular compound. The compound of five tetrahedra comes in two enantiomorphic versions, which together make up the compound of 10 tetrahedra; each of the tetrahedral compounds is self-dual or "chiral-twin-dual", the compound of 5 cubes is dual to the compound of 5 octahedra. The five octahedra defining any given icosahedron form a regular polyhedral compound. Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols; the material inside the square brackets, denotes the components of the compound: d separate's. The material before the square brackets denotes the vertex arrangement of the compound: c is a compound of d's sharing the vertices of counted c times; the material after the square brackets denotes the facet arrangement of the compound: e is a compound of d's sharing the faces of counted e times. These may be combined: thus ce is a compound of d's sharing the vertices of counted c times and the faces of counted e times.

This notation can be generalised to compounds in any number of dimensions. A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common intersphere or midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five dual compounds of the regular polyhedra; the core is the rectification of both solids. The hull is the dual of this rectification, its rhombic faces have the intersecting edges of the two solids as diagonals. For the convex solids, this is the convex hull; the tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual is the regular stellated octahedron. The octahedral and icosahedral dual compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively. In 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds made from uniform polyhedra with rotational symmetry; this list includes the five regular compounds above.

The 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element; some chiral pairs of face groups are colored by symmetry of the faces within each polyhedron. 1-19: Miscellaneous 20-25: Prism symmetry embedded in prism symmetry,26-45: Prism symmetry embedded in octahedral or icosahedral symmetry,46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,68-75: enantiomorph pairs Compound of three octahedra Compound of four cubesTwo polyhedra that are compounds but have their elements rigidly locked into place are the small complex icosidodecahedron and the great complex icosidodecahedron. If the definition of a uniform polyhedron is generalised they are uniform; the section for entianomorphic pairs in Skilling's list does not contain the compound of two great snub dodecicosidodecahedra, as the pentagram faces would coincide. Removing the coincident faces results in the compound of twenty octahedra. In 4-dimensions, there are a large number of regular compounds of regular polytopes.

Coxeter lists a few of these in his book Regular Polytopes:Self-duals: Dual pairs: Uniform compounds and duals with convex 4-polytopes: Self-dual star compounds: Dual pairs of compound stars: Uniform compound stars and duals: Dual positions: Only the first two of these dual compounds are regular. In terms of group theory, if G is the symmetry group of a polyhedral compound, the group acts transitively on the polyhedra if H is the stabilizer of a single chosen polyhedron, the polyhedra can be identified with the orbit space G/H – the coset gH corresponds to which polyhedron g sends the chosen polyhedron to. There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated; the Euclidean and hyperbolic compound families 2 are analogous to the spherical stella octangula, 2. A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb.

This compound can have an


CodeWeavers is a company that sells a proprietary version of Wine called CrossOver for running Windows applications on macOS and Linux. The company was founded in 1996 as a consultancy moving over to Wine development and support; the CrossOver version of Wine is refreshed with the latest free Wine patches. CodeWeavers is a major contributor to the Wine project, a free software / open-source software project that helps Windows applications run on different x86-based operating systems, hosts the project's website, employs the project's maintainer, Alexandre Julliard, as their CTO; the company focused on making the Microsoft Office suite available on Linux. When Apple switched to Intel processors in 2006 CodeWeavers was able to create CrossOver Mac and bring Wine to Mac OS X. Other major supported applications include the Steam game client. CodeWeaver's products include CrossOver Linux. Previous product versions included Pro, Standard and Server. CrossOver Games was introduced in March 2008 and was intended to allow gaming-related patches from Wine to be incorporated into CrossOver much faster.

The CrossOver Pro product line focused on stability and in-depth testing with supported productivity software and had a slower release cycle. In 2012 all versions have had their functionality merged into the primary CrossOver Mac and CrossOver Linux products. CodeWeavers provides porting and consulting services around Wine and other open-source software projects. Google has paid CodeWeavers to add functionality to Wine. CodeWeavers is a founding member of the Desktop Linux Consortium. In July 2008, CodeWeavers launched the Great American Lame Duck Presidential Challenge to encourage President Bush to make the most of his remaining days in office by accomplishing a major economic or political goal by January 20, 2009; the goals focused on President Bush making specific positive accomplishments in areas such as the economy, home values, the stock market, the war on terror and other key issues. One goal called for President Bush to help bring down average gasoline prices in the Twin Cities to $2.79 a gallon.

On October 14, gas prices in Minneapolis and St. Paul fell to $2.79 a gallon, CodeWeavers honored their pledge giving away their software for free on October 28, 2008. Traffic via Slashdot and other sources overloaded and brought down the CodeWeavers website as people rushed to get the free-of-charge software. According to CodeWeavers, "You will be able to unlock your serial number, emailed to you for an extended time, due to this downtime. We will stop giving out new serial numbers at 23:59 Central Standard Time."Additionally, CodeWeavers updated their site @ ~ 9:00 AM CST to reflect the statement: "Please check back again for registration code information today. We will be deploying a streamlined serial code generation process shortly." At the same time, they added an "about Wine" paragraph. The streamlined process came to pass, but customers were told to expect to wait "several days" to receive their serial number. In the interim unlocked of four different CodeWeavers packages became available for immediate download, but only on October 28, 2008.

Official website Jeremy White interview

Tambo Colorado

Tambo Colorado is a well-preserved Inca adobe complex near the coast of Peru known under the Quechua names Puka Tampu, Pukallaqta or Pukawasi. The site is located just inland from the south coast of Perú in the Pisco River Valley about 40 km along the highway to Ayacucho known as the Via de los Libertadores, close to the town of Pisco. Initial reports from the 2007 Peru earthquake reported no major damage to the site. A High resolution GPS point was shot at the site datum on 2 Aug 2009 using an L2 GPS; the post-processed position is as follows: Northings: 8484705.386 m Eastings: 410335.884 m Altitude: 484.849 m UTM Zone 18 South, Datum WGS 1984. The site was most built at the end of the 15th century during the reign of the Inca king Pachacuti Inca Yupanqui known as Pachacutec, after the annexation of the merchant kingdom of Chincha; the site owes its name to the abundant use of colors on the walls. Thanks to favorable environmental conditions, many walls at Tambo, both internal and external, retain enough residual colored paint to reconstruct what the original wall painting would have been like.

Color here was applied in horizontal strips of red, black and yellow ochre atop stucco, variation in color would accentuate architectural features such as niches. Trapezoidal niches at Tambo have one or two recesses each used for the placement of important objects; as with all Inca constructions, the overall dimensions of niche construction are standardized across the entire site. The site consists of several structures around a large central plaza; the central plaza is shaped like a trapezoid with its largest side being 150 m long. The main structures are grouped together in a southern part; these structures are known as the Northern palace and the two Southern Palaces, flanked by an Ushnu and a building known as the Utilities Structure. The combination of Chincha and Inca architectural techniques can be seen in the place, it is believed to have been used by the Incas as an administrative and control site on the main road from the coast to the highlands. A small on-site museum is located near the entrance of the complex.

Tambo Colorado Digital Media Archive, data from a UC Berkeley/CyArk research partnership

Honoré III, Prince of Monaco

Honoré III ruled as Prince of Monaco and was Duke of Valentinois from 1731 to 1793. Honoré was the son of Louise Hippolyte, Princess of Monaco, her husband, Prince Jacques I. Honoré was born on 10 November 1720 On 20 May 1732, he moved to Hôtel Matignon in Paris with his father and remained there after the proclamation in 1733 of him as Prince of Monaco. Antoine Grimaldi, le Chevalier de Grimaldi, acted as regent for the prince between 1732 and 1784, when Honoré chose to reside in Paris; this situation remained the same for half a century until Antoine's death in 1784, when Honoré III was 64 years old. Although he was open to the revolutionary ideas of the time, he was imprisoned on 20 September 1793. At his liberation a year he was ruined, his property under seal. While in Paris, it was suggested that he marry Marie Louise de La Tour d'Auvergne, but the marriage never materialised. In 1751, he married Maria Caterina Brignole; the couple had two children. Official Website of the Princely Family of Monaco


Bellata is a small village in north-central New South Wales, Australia, in Narrabri Shire. At the 2006 census, Bellata had a population of 529; the place name Bellata could be derived from the local Aboriginal word meaning "kangaroo" or "home of belar trees". It is 47 kilometres north of Narrabri, around halfway between Narrabri and Moree when travelling along the Newell Highway; the village's area is known for its array of mineral deposits, fossickers will find high quality agate, carnelian, petrified opal and petrified wood in the vicinity. Located on the rich black soil basalt plains of north western New South Wales, it is an important agricultural region and the area is known for some of the best "primehard" wheat production in Australia. There is a large array of grain handling facilities in the town. A turn of the century, two storey colonial style hotel, the Nandewar Inn Hotel constructed in 1902 was a well loved local landmark until it was destroyed by fire in April 2006. Once a thriving rural center, Bellata, in the early to mid 20th Century, boasted a Post Office, two General Stores, two Stock and Station Agencies, two Garages, a Café, a Telephone Exchange, a operational railway station with a Station Master and a Doctor.

Bellata still has a Road House, a police station with holding cell, primary school, nine-hole sand green golf course and Golf Club, tennis courts and Catholic churches, a memorial hall where movies were once shown, caravan park and several community groups. Woolabra Post Office opened on 1 September 1899 and was renamed Bellata in 1909. Bellata railway station is situated on 615 kilometres from Sydney; the station opened in 1897 as Woolabra however was renamed Bellata in 1909. It consists of a platform and unstaffed basic passenger waiting shed, is served by a single daily Xplorer diesel railmotor between Moree and Sydney in the morning and by another Xplorer heading in the opposite direction to Moree in the evening; the station is an optional stop and has a low standard of facilities so sometimes trains do not stop at all. Media related to Bellata, New South Wales at Wikimedia Commons

Corinna Cortes

Corinna Cortes is a Danish computer scientist known for her contributions to machine learning. She is the Head of Google Research, New York. Cortes is a recipient of the Paris Kanellakis Theory and Practice Award for her work on theoretical foundations of support vector machines. Corinna Cortes was born in 1961 in Denmark. Cortes received her M. S. degree in physics from Copenhagen University in 1989. In the same year she remained there for about ten years, she received her Ph. D. in computer science from the University of Rochester in 1993. Cortes serves as the Head of Google Research, New York, she is an Editorial Board member of the journal Machine Learning. Cortes' research covers a wide range of topics in machine learning, including support vector machines and data mining. In 2008, she jointly with Vladimir Vapnik received the Paris Kanellakis Theory and Practice Award for the development of a effective algorithm for supervised learning known as support vector machines. Today, SVM is one of the most used algorithms in machine learning, used in many practical applications, including medical diagnosis and weather forecasting.

Corinna has two children and is a competitive runner