SUMMARY / RELATED TOPICS

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges and cells; each face is shared by two cells. The two-dimensional analogue of a 4-polytope is a polygon, the three-dimensional analogue is a polyhedron. Topologically 4-polytopes are related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space. Convex 4-polytopes unfolded as nets in 3-space. A 4-polytope is a closed four-dimensional figure, it comprises vertices, edges and cells. A cell is the three-dimensional analogue of a face, is therefore a polyhedron; each face must join two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are 4-polytopes, i.e. it is not a compound. The most familiar 4-polytope is the 4D analogue of the cube. 4-polytopes cannot be seen in three-dimensional space due to their extra dimension.

Several techniques are used to help visualise them. Orthogonal projectionOrthogonal projections can be used to show various symmetry orientations of a 4-polytope, they can be drawn in 2D as vertex-edge graphs, can be shown in 3D with solid faces as visible projective envelopes. Perspective projectionJust as a 3D shape can be projected onto a flat sheet, so a 4-D shape can be projected onto 3-space or onto a flat sheet. One common projection is a Schlegel diagram which uses stereographic projection of points on the surface of a 3-sphere into three dimensions, connected by straight edges and cells drawn in 3-space. SectioningJust as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut "hypersurface" in three dimensions. A sequence of such sections can be used to build up an understanding of the overall shape; the extra dimension can be equated with time to produce a smooth animation of these cross sections. NetsA net of a 4-polytope is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane.

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, is zero for all 4-polytopes, whatever their underlying topology; this inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. The notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, this led to the use of torsion coefficients. Like all polytopes, 4-polytopes may be classified based on properties like "convexity" and "symmetry". A 4-polytope is convex if its boundary does not intersect itself and the line segment joining any two points of the 4-polytope is contained in the 4-polytope or its interior. Self-intersecting 4-polytopes are known as star 4-polytopes, from analogy with the star-like shapes of the non-convex star polygons and Kepler–Poinsot polyhedra.

A 4-polytope is regular. This means that its cells are all congruent regular polyhedra, its vertex figures are congruent and of another kind of regular polyhedron. A convex 4-polytope is semi-regular if it has a symmetry group under which all vertices are equivalent and its cells are regular polyhedra; the cells may be of two or more kinds, provided. There are only 3 cases identified by Thorold Gosset in 1900: the rectified 5-cell, rectified 600-cell, snub 24-cell. A 4-polytope is uniform if it has a symmetry group under which all vertices are equivalent, its cells are uniform polyhedra; the faces of a uniform 4-polytope must be regular. A 4-polytope is scaliform if it is vertex-transitive, has all equal length edges; this allows cells, such as the regular-faced convex Johnson solids. A regular 4-polytope, convex is said to be a convex regular 4-polytope. A 4-polytope is prismatic. A prismatic 4-polytope is uniform; the hypercube is prismatic, but is considered separately because it has symmetries other than those inherited from its factors.

A tiling or honeycomb of 3-space is the division of three-dimensional Euclidean space into a repetitive grid of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, are examples of infinite 4-polytopes. A uniform tiling of 3-space is one whose vertices are congruent and related by a space group and whose cells are uniform polyhedra; the following lists the various categories of 4-polytopes classified according to the criteria above: Uniform 4-polytope: Convex uniform 4-polytopes 47 non-prismatic convex uniform 4-polytope including: 6 Convex regular 4-polytope Prismatic uniform 4-polytopes: ×: 18 polyhedral hyperprisms Prisms built on antiprisms ×: duoprisms Non-convex uniform 4-polytopes 10 Schl

Vangchhia is a village in the Champhai district of Mizoram, India. It is located in the Khawbung R. D. Block; the 171 menhir stones in the village became Mizoram's first protected archaeological site in 2012. According to the 2011 census of India, Vangchhia has 153 households; the effective literacy rate is 96.87%. The Vangchhia tribe is native to this village. Vangchhia is the name of one of the eleven sub-tribes of Mizoram, they along with genealogically related clans like the Khawlhring, Saivate, etc. form the greater Faihriem group of clans. They trace their descent from the son of Berhva. Traditions maintain that Chunthang was a good man, his good character earned him the hands of the Biete princess, Lalzaii. However, the couple could not nurture any child to maturity; the offspring out of the wedlock died at infancy. One night, in his dream he was told that the next child should be arranged to be born in another village and the child will survive. So, when the time for delivery of the next child came, the mother was taken to the village of the Thiek clan where she delivered a healthy boy.

As the boy was born in another village, he was named'Khualhring'. The boy on became the progenitor of the Khawlhring clan; when the time for the next child came, Chunthang was again told to arrange for the child to be kept below the'vang' tree near their house to ensure his survival. It was done so, the child survived; the child was named ` indicating his first bed below the tree. His descendants came to be known by that name. There are still Vangchhia with royal blood in them but records have been lost in time. A few well-known people is the Vangchhia Family of Mizoram. Mr Dominic Lalhmangaiha Vangchhia and his family; this family is well rounded from musicians to actors. Henry Vangchhia, Elijah Vangchhia, Ronald Vangchhia, Sammy Vangchhia, Jacob Vangchhia; these are a few who has made their name known throughout the whole of Europe. Sammy Vangchhia released a new song "Si Ar Te" and was featured in Vanglaini newspaper

The Grandview is a historic apartment hotel at 82 Munroe Street in Somerville, Massachusetts. This type of building was not uncommon in the city at the time of its 1896 construction; this building affords commanding views of the Boston area from its site near the top of Prospect Hill, has well-preserved Colonial Revival styling. The building was listed on the National Register of Historic Places in 1989; the Grandview is located near the summit of Prospect Hill, a hill overlooking the city of Boston, of military importance during the American Revolutionary War. It stands west of Prospect Hill Park, on the south side of Munroe Street, it is a 3 1⁄2 - story wood-frame structure, with clapboard siding. A central full-height gable-roofed pavilion projects from the center of the north-facing facade, with a two-story porch projecting further; the pedimented gable is decorated with modillions and dentil moulding, the latter of, found at the roof line. The porch is flanked by two-story projecting bays, there are gable-roof dormers projecting from the roof on several sides.

The rear of the building has full-width porches on all three levels. Most of the building's windows have been modernized, with three-pane sliding windows typical; the Grandview was built in 1896 by Elbridge Park, a city alderman, was one of 45 "apartment hotels" listed in the city directory in 1900. These types of transient apartment houses were built in the city near ready access to public transportation, catered to commuters working in Boston; this fine example has retained much of its Colonial Revival ornamentation. National Register of Historic Places listings in Somerville, Massachusetts

The Lure of the Gown is a 1909 American silent short drama film directed by D. W. Griffith; the story as told by Moving Picture World reads: "Fine feathers make fine birds," and handsome gowns make handsome women, a handsome woman is the most fascinating thing extant. Hence it is when Isabelle appears on the scene clad in a gown, a masterpiece of the dressmaker's art she fascinates the male contingent, among whom is Enrico, the sweetheart of Veronica, a street singer. Enrico is so enraptured at the sight of Isabelle in her resplendent attire that he becomes her abject slave, casting aside the poor, peasant-clad little Italian street singer, who has loved him devotedly. Crushed beyond endurance the poor girl stands sobbing at the entrance of the park where the inconsistent lover left her, her tears attract the attention of a wealthy young couple. In answer to their queries she tells them how contemptibly her sweetheart acted, all because of the fascinating influence of a gown; the lady is moved to commiseration and offers her aid in the gift of the most beautiful gown Veronica has seen.

Her opportunity for revenge has turned her love to hate, as she appears at the Italian Benevolent Association ball, she is the star of the event, for she looks like a queen as she promenades the ball room. She at once becomes the "Mrs. Trouble" of the evening, for the men all desert their partners and flock around her, beseeching but a smile. All this elicits from the women folk delicate little bon-mots such as "Hussy," "Temptress," "Cat," "False hair," "Paints,"—oh, you know how it is. Enrico is thrown into a rage that runs the entire gamut of his emotions,—love, hate, disappointment and a few others, too numerous to mention here, he begs forgiveness, declaring undying love, but she tells him it is the gown that has attracted him and not her, but on his knees he swears. Still she will not trust him and turns to a poor good hearted Italian who has persistently loved her despite her coldness. Marion Leonard as Isabelle Harry Solter as Enrico Florence Lawrence as Veronica The Lure of the Gown on IMDb

The Flag of the Byelorussian Soviet Socialist Republic was adopted on December 25, 1951. Prior to this, the flag was red with the Cyrillic characters БССР in gold in the top-left corner, surrounded by a gold border. Between 1937 and the adoption of the above flag in the 1940s, the flag was the same, but with a gold hammer and sickle above the Cyrillic characters and no border. Between 1919 and 1937, the flag was red, with the Cyrillic characters ССРБ in the top left-hand corner. In early 1919, a plain red flag was used; the final BySSR flag was used until the collapse of the Soviet Union in 1991. A flag based on this design is used as the current national flag of Belarus. In the end of the 1940s, the political need had arisen to have somewhat visually different designs of the flags of the USSR republics for those that were UN members. For the BSSR flag, the image of the Belarusian folk design had been chosen as a distinctive feature of the flag; the picture of the embroidery on the «ruchnik» had been found in the pre-World War II archives of the Belpramsavyet.