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Non-composite Polyhedron
In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids. Definition and examples A convex polyhedron is said to be ''composite'' if there exists a plane through a cycle of its edges that is not a face. Slicing the polyhedron on this plane produces two polyhedra, having together the same faces as the original polyhedron along with two new faces on the plane of the slice. Repeated slicing of this type decomposes any polyhedron into non-composite or ''elementary'' polyhedra. Some examples of non-composite polyhedron are the prisms, antiprisms, and the other seventeen Johnson solids. Among the regular polyhedra, the regular octahedron and regular icosahedron are composite. Any composite polyhedron can be constructed by attaching two or more non-composite polyhedra. Alternatively, it can be defined as a convex polyhedron that can separated into two or more non-composite polyhedra. Exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, solids with such a property: the first solids are the Pyramid (geometry), pyramids, Cupola (geometry), cupolas, and a Rotunda (geometry), rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. Definition and background A Johnson solid is a convex polyhedron whose faces are all regular polygons. The convex polyhedron means as bounded intersections of finitely many Half-space (geometry), half-spaces, or as the convex hull of finitely many points. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not Uniform polyhedron, uniform. This means that a Johnson solid is not a Platonic solid, Arc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Euler characteristic, duality, vertex figures, surface area, volume, interior lines, Dehn invar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements, Euclid's ''Elements''. Euclid defined the term in Book XI as "a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms". However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway polyhedron notation, Conway notation . Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron. Antiprisms are similar to Prism (geometry), prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are triangles, rather than quadrilaterals. The dual polyhedron of an -gonal antiprism is an -gonal trapezohedron. History In his 1619 book ''Harmonices Mundi'', Johannes Kepler observed the existence of the infinite family of antiprisms. This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the net (geometry), net of a hexagonal antiprism has been attributed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are Congruence (geometry), congruent regular polygons which are assembled in the same way around each vertex (geometry), vertex. A regular polyhedron is identified by its Schläfli symbol of the form , where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five Convex polygon, convex regular polyhedra, known as the Platoni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Octahedron
In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyhedron; many types of irregular octahedra also exist. A regular octahedron is convex, meaning that for any two points within it, the line segment connecting them lies entirely within it. It is one of the eight convex deltahedra because all of the faces are equilateral triangles. It is a composite polyhedron made by attaching two equilateral square pyramids. Its dual polyhedron is the cube, and they have the same three-dimensional symmetry groups, the octahedral symmetry \mathrm_\mathrm . A regular octahedron is a special case of an octahedron, any eight-sided polyhedron. It is the three-dimensional case of the more general concept of a cross-polytope. As a Platonic solid The regular octahedron is one of the Platonic solids, a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Icosahedron
The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the Skeleton (topology), skeleton of a regular icosahedron. Many polyhedra are constructed from the regular icosahedron. A notable example is the stellation of regular icosahedron, which consists of 59 polyhedrons. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation or faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background on the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Square Pyramid
In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid. Construction The elongated square pyramid is a composite, since it can constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid. One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite. A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as J_ , the fifteenth Johnson solid. Properties Given that a is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the edge length of a cube's side, and the he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex (geometry), apex. Each base edge (geometry), edge and apex form a triangle, called a lateral face. A pyramid is a cone, conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid. All pyramids are Self-dual polyhedron, self-dual. Etymology The word "pyramid" derives from the ancient Greek term "πυραμίς" (pyramis), which referred to a pyramid-shaped structure and a type of wheat cake. The term is rooted in the Greek "πυρ" (pyr, 'fire') and "άμις" (amis, 'vessel'), highlighting the shape's pointed, flame-like appearance. In Byzantine Greek, the term evolved to "πυραμίδα" (pyramída), continuing to denote pyramid structures. The Greek term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmentation (geometry)
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas, and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. Definition and background A Johnson solid is a convex polyhedron whose faces are all regular polygons. The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not uniform. This means that a Johnson solid is not a Platonic solid, Archimedean solid, prism, or antiprism. A convex polyhedron in which all faces are nearly regular, but some are not pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cupola (geometry)
In geometry, a cupola is a Polyhedron, solid formed by joining two polygons, one (the base) with twice as many Edge (geometry), edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral triangle, equilateral and the rectangles are Square (geometry), squares, while the base and its opposite face are regular polygons, the triangular cupola, triangular, square cupola, square, and pentagonal cupola, pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively. A cupola can be seen as a prism (geometry), prism where one of the polygons has been collapsed in half by merging alternate vertices. A cupola can be given an extended Schläfli symbol representing a regular polygon joined by a parallel of its Truncation (geometry)#Truncation of polygons, truncation, or Cupolae are a subclass of the prismatoids. Its d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotunda (geometry)
In geometry, a rotunda is any member of a family of Dihedral group, dihedral-symmetric polyhedra. They are similar to a Cupola (geometry), cupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. The pentagonal rotunda is a Johnson solid. Other forms can be generated with dihedral symmetry and distorted equilateral pentagons. Examples Star-rotunda See also * Birotunda References *Norman Johnson (mathematician), Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. * The first proof that there are only 92 Johnson solids. {{Polyhedron navigator Johnson solids ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
