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.

The embroidery had been made in 1917 by peasant Ms. Matrona Markyevich of the village Kastsilishcha of the Sennin region, had been named «The Rising Sun». Artist M. I. Gusyew had prepared the project of the BSSR flag, basing it on the embroidery, with several symbolic elements added. Symbolically, the design on the BSSR flag is decoded as follows: the central rhombus figure symbolises the rising Sun. Note: All proper names and places' names in this article are rendered in BGN/PCGN. Before 1951, several different flags had been in use since the Revolution; the earliest flag was plain red, was used in 1919 during the existence of the Lithuanian–Byelorussian SSR. After the formation of the Byelorussian SSR, the lettering ССРБ was added in gold to the top hoist; this design was changed in 1937, when a hammer and sickle and red star were placed above the letters. During the 1940s, the hammer and sickle and red star were removed from the flag, a gold border was added to the letters; this flag remained in use until the adoption of the 1951 flag.

In September 1991, the white-red-white flag was reintroduced as the new flag of the newly independent Byelorussia. Coat of arms of the Byelorussian Soviet Socialist Republic Flag of Belarus Flag of the Soviet Union Басаў А. Н. Куркоў І. М. Флагі Беларусі ўчора і сёння / Пер. А. Н. Найдовіч. – Мн.: Полымя, 1994. ISBN 5-345-00730-6

American Inventor was a reality television series based on a competition to be named America's best inventor. It was conceived by Simon Cowell and the producers of American Idol after Ben Hausbach appeared on Season 5 with his Cosmic Coaster invention, he appeared on Season 2 of American Inventor and was featured in the Florida auditions. It premiered on ABC on March 16, 2006, it was organized as a competition between inventors nationally resulting in one overall winner. Janusz Liberkowski, who invented a new type of child safety seat based on the human womb called the Anecia Safety Capsule, was declared the first season's winner in a live episode on May 18, 2006; the second season premiered on June 6, 2007. Firefighter Greg Chavez, who invented a fire suppression system for Christmas trees called the Guardian Angel, was the winner of the second season, on August 1, 2007. On March 20, 2008, the show's official website was removed, the series was not included on the 2008 fall schedule, therefore the series was cancelled.

Twelve inventors and their products are chosen from a pool of hundreds by four judges. The 12 semi-finalists are broken down into four groups of three, with each episode focusing on a different group of three; each of the twelve semi-finalists in each group receives $50,000 to improve their inventions and compete to become one of the four finalists. The finalists would work with a dedicated prototype and design company, such as T2 Design & Prototype who would help with expert advice and manufacturing assistance; each group is assigned a judge. Each of the four judges would choose one inventor from their group to compete in the finals, for a total of four finalists. In the show's live finale, the four finalists present a 30-second commercial advertisement for their product, with the home audience voting by phone for the winner; the winner receives $1,000,000 worth of business support, entrepreneurial counsel, physical resources, prize money. Instead of 12 finalists receiving $50,000 checks to develop their inventions like in season one, six finalists, one from each of the audition cities of Los Angeles, San Francisco, New York and Houston, will each receive $50,000 and have one month to develop their inventions.

The 6 finalists are narrowed down to three based on the judges' preference. Unlike the first season, the three finalists for voting were declared and were voted on by viewers after the second-to-last show; the season finale was a single show. In the first season, the finale was a two-episode feature with 30-second commercials on the first and the results of viewer voting on the second part; the first season had four finalists instead of six narrowed to three. Anecia Safety Capsule: A car seat shaped like a sphere with a movable sphere inside it. In the event of a car accident, the sphere will move. Word Ace: A game that tells you a letter and a number and you have to come up with a word that starts with the letter and has a number of letters equal to the number. Double Traction Bike: A bike with a seat on the handlebars; the Catch Vest: A bar that sits in front of your body to help you catch footballs properly. Inside Umbrella: An umbrella where the top part goes inside the bar. Head-Line It: A rubber pad to go on top of your head to prevent sweating and itching under hats and wigs.

Sackmaster: A large shovel with a bag inside it to make sand-bag filling easier. Here Comes Niya Doll: A multilingual set of dolls; the "Niya Doll" appear again in The Toy Box. Bathroom Clip: A clip to attach to a bathroom door in case the lock is broken. Tonerbelt: An exercise belt EZ-X Portable Gym: A portable gym Pureflush: A toilet that prevents the spread of bacteria upon flushing. Guardian Angel: A small, pressurized tank of water, disguised as a Christmas package, placed under the Christmas tree and attached to a small hose leading to the top of the tree where a fusible link is disguised as an angel; the heat from a fire pops the water suppresses the fire. There is an alarm that works without a battery & intended to suppress a Christmas tree fire and sound an alarm to get people out of the house alive. 6-In-1 Convertible Brassiere: A convertible brassiere that accommodates the full wardrobe of small frame women with a "C" cup or above. This is a traditional brassiere that converts into six configurations and affords the full back to be exposed with comfort because there is no midsection and it can be worn with full wardrobe.

The invention was commercialized by Maidenform. HT Racers: The design and fabrication of custom vehicles through the use of a computer program and patented tools. Intended for use by individuals and groups ages 9 and up for entertainment and educational purposes, this invention is a kit that lets teens design and build remote controlled vehicles; the Claw: A ceiling or wall-mounted bicycle storage mechanism. The bicycle wheel is grasped by opposing hooks. Re-pressing the central plunger locks the hooks for bicycle wheel removal; the bi-stable plunger relies on an internal rotary mechanism. Wrap-a-Way Cabinet: A station to put in a drawer or attach to a cabinet to dispense paper towels, gift wrap, etc. EZT4U: A brewing basket that attaches to the common electric coffee brewing machine in order to brew tea. American Inventor debuted March 16, 2006 at 8:00