In geometry, a pentagon (from Greek πέντε (pente) 'five' and γωνία (gonia) 'angle') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .
Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.
The Villa Farnese, also known as Villa Caprarola, is a pentagonal mansion in the town of Caprarola in the province of Viterbo, Northern Lazio, Italy, approximately 50 kilometres (31 mi) north-west of Rome, originally commissioned and owned by the House of Farnese. A property of the Republic of Italy, Villa Farnese is run by the Polo Museale del Lazio. This villa is not to be confused with two similarly-named properties of the family, the Palazzo Farnese and the Villa Farnesina, both in Rome.
The Villa Farnese is situated directly above the town of Caprarola and dominates its surroundings. It is a massive Renaissance and Mannerist construction, opening to the Monte Cimini, a range of densely wooded volcanic hills. It is built on a five-sided plan in reddish gold stone; buttresses support the upper floors. As a centerpiece of the vast Farnese holdings, Caprarola was always an expression of Farnese power, rather than a villa in the more usual agricultural or pleasure senses.
The Pentagon (French: Pentagone, pronounced [pɛ̃taɡɔn] ; Dutch: Vijfhoek, pronounced [ˈvɛifɦuk] ) or Brussels' city centre is the historical city centre of Brussels, Belgium, within the contours of the Small Ring inner ring road. The Small Ring is located on the site of the second walls of Brussels, which were built in the 16th century. As in most European cities, these walls were replaced by large boulevards at the end of the 19th century.
The Pentagon, within the Small Ring, covers 4.61 km (1.78 sq mi) and is more or less pentagonal or heart-shaped, hence its name. In 2013, 51,566 people lived there, mainly in the Marolles/Marollen district and west of the central boulevards. For the entire City of Brussels, there were 168,576 inhabitants; the majority living outside the Pentagon, in the northern part of the municipality.
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or tiling as the sequence of faces around a vertex. It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex sequence. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models. For uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.)
For example, "3.5.3.5" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5 (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as (3.5).
The Eden Project (Cornish: Edenva) is a visitor attraction in Cornwall, England. The project is located in a reclaimed china clay pit.
The complex is dominated by two huge enclosures consisting of adjoining domes that house thousands of plant species, and each enclosure emulates a natural biome. The biomes consist of hundreds of hexagonal and pentagonal ethylene tetrafluoroethylene (ETFE) inflated cells supported by geodesic tubular steel domes. The larger of the two biomes simulates a rainforest environment (and is one of the largest indoor rainforests in the world) and the second, a Mediterranean environment.
In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.
A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them. If the vertices are not constrained to a sphere, the polyhedron can be constructed with planar equilateral (but not in general equiangular) faces.
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of ten triangles for a total of twelve faces. Hence, it is a non-regular dodecahedron.
In geometry, a pentagonal pyramid is a pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as a Johnson solid if all of the edges are equal in length, forming equilateral triangular faces and a regular pentagonal base.
Pentagonal pyramids occur as pieces and tools in the construction of many polyhedra. They also appear in the field of natural science, as in stereochemistry where the shape can be described as the pentagonal pyramidal molecular geometry, as well as the study of shell assembling in the underlying potential energy surfaces and disclination in fivelings and related shapes such as pyramidal copper and other metal nanowires.
