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Ridge (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face include polyhedron side and Euclidean plane '' tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Geometry
Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space). A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball (mathematics), ball consists of a sphere and its Interior (topology), interior. Solid geometry deals with the measurements of volumes of various solids, including Pyramid (geometry), pyramids, Prism (geometry), prisms (and other polyhedrons), cubes, Cylinder (geometry), cylinders, cone (geometry), cones (and Frustum, truncated cones). History The Pythagoreanism, Pythagoreans dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonism, Platonists. Eudoxus of Cnidus, Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the ''Timaeus (dialogue), Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Square Tiling
In geometry, the order-5 square tiling is a List_of_regular_polytopes#Hyperbolic_tilings, regular tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of . Related polyhedra and tiling This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). This hyperbolic tiling is related to a Infinite_skew_polyhedron#Semiregular_infinite_skew_polyhedra, semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space. : References * John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) * See also *Square tiling *Uniform tilings in hyperbolic plane *List of regular polytopes *Medial rhombic triacontahedron External links * * Hyperbolic and Spherical Tiling Gallery * [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch] Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2-5-4-primal
H, or h, is the eighth letter of the Latin alphabet, used in the modern English alphabet, including the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' (pronounced , plural ''haitches'')''.''"H" ''Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. Name English For most English speakers, the name for the letter is pronounced as and spelled "aitch" or occasionally "eitch". The pronunciation and the associated spelling "haitch" are often considered to be h-adding and are considered non-standard in England. It is, however, a feature of Hiberno-English, and occurs sporadically in various other dialects. The perceived name of the letter affects the choice of indefinite article before initialisms beginning with H: for example "an H-bomb" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tile 4,4
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or other objects such as tabletops. Alternatively, tile can sometimes refer to similar units made from lightweight materials such as perlite, wood, and mineral wool, typically used for wall and ceiling applications. In another sense, a tile is a construction tile or similar object, such as rectangular counters used in playing games (see tile-based game). The word is derived from the French Language, French word ''tuile'', which is, in turn, from the Latin Language, Latin word ''tegula'', meaning a roof tile composed of fired clay. Tiles are often used to form wall and floor coverings, and can range from simple square tiles to complex or mosaics. Tiles are most often made of pottery, ceramic, typically Ceramic glaze, glazed for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word ''pentagram'' comes from the Greek language, Greek word πεντάγραμμον (''pentagrammon''), from πέντε (''pente''), "five" + γραμμή (''grammē''), "line". The word pentagram refers to just the star and the word pentacle refers to the star within a circle, although there is some overlap in usage. The word ''pentalpha'' is a 17th-century revival of a post-classical Greek name of the shape. History Early history Early pentagrams have been found on Sumerian pottery from Ur c. 3500 Common Era, BCE, and the five-pointed star was at various times the symbol of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Stellated Dodecahedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms Great complex icosidodecahedron, a degenerate uniform compound figure. It is the List of Wenninger polyhedron models#Stellations of dodecahedron, second of four stellations of the dodecahedron (including the original dodecahedron itself). The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. Construction and properties The small stellated dodecahedron is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexahedron
A hexahedron (: hexahedra or hexahedrons) or sexahedron (: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. There are seven topologically distinct ''convex'' hexahedra, one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number depending on how polyhedra are defined. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces. Convex Cuboid A hexahedron that is combinatorially equivalent to a cube may be called a cuboid, although this term is often used more specifically to mean a rectangular cuboid, a hexahedron with six rectangular sides. Different types of cuboids include the ones depicted and linked below. Others There a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Square Tiling
In geometry, the order-5 square tiling is a List_of_regular_polytopes#Hyperbolic_tilings, regular tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of . Related polyhedra and tiling This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). This hyperbolic tiling is related to a Infinite_skew_polyhedron#Semiregular_infinite_skew_polyhedra, semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space. : References * John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) * See also *Square tiling *Uniform tilings in hyperbolic plane *List of regular polytopes *Medial rhombic triacontahedron External links * * Hyperbolic and Spherical Tiling Gallery * [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch] Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tiling has a structure consisting of one type of congruent prototile, the square, sharing two vertices with other identical ones. This is an example of monohedral tiling. Each vertex at the tiling is surrounded by four squares, which denotes in a vertex configuration as 4.4.4.4 or 4^4 . The vertices of a square can be considered as the lattice, so the square tiling can be formed through the square lattice. This tiling is commonly familiar with the flooring and game boards. It is self-dual, meaning the center of each square connects to another of the adjacent tile, forming square tiling itself. The square tiling acts transitively on the ''flags'' of the tiling. In this case, the flag consists of a mutually incident vertex, edge, and tile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Stellated Dodecahedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms Great complex icosidodecahedron, a degenerate uniform compound figure. It is the List of Wenninger polyhedron models#Stellations of dodecahedron, second of four stellations of the dodecahedron (including the original dodecahedron itself). The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. Construction and properties The small stellated dodecahedron is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Regular Polychoron
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. History The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V'' 2). That excludes cells and vertex figures such as the great dodecahedron and small stellated dodecahedron . Edmund Hess ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
