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List Of Johnson Solids
In geometry, a Johnson solid is a strictly Convex polyhedron, convex polyhedron, each Face (geometry), face of which is a regular polygon, but which is not uniform polyhedron, uniform, i.e., not a Platonic solid, Archimedean solid, prism (geometry), prism or antiprism. In 1966, Norman Johnson (mathematician), Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller proved in 1969 that Johnson's list was complete. Other polyhedra can be constructed that have only approximately regular Plane (geometry), planar polygon faces, and are informally called near-miss Johnson solids; there can be no definitive count of them. The various sections that follow have tables listing all 92 Johnson solids, and values for some of their most important properties. Each table allows sorting by column so that numerical values, or the names of the solids ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a '' geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Spherical Symmetry Groups
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.Sands, 1993 Involutional symmetry There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid 6 Net
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name ''John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', which is derived through Greek ''Iōannēs'' from Hebrew ''Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish: Johnsson, Jonsson * Icelandic: Jónsson See also * List of people with surname Johnson *Gjoni (Gjonaj) *Ioannou *Jensen (surname), Jensen *Johansson *Johns (surname), Johns *Johnsson *Johnston (surname), Johnston *Johnstone (s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Rotunda
In geometry, the pentagonal rotunda is one of the Johnson solids (). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces. Formulae The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ..., with edge length ''a'': :V=\left(\frac\left(45+17\sqrt\right)\right)a^3\approx6.91776...a^3 :\begin A&=\left(\frac\sqrt\right)a^2 \\ &=\left(\frac\left(5\sqrt+\sqrt\right)\right)a^2\approx22.3472...a^2 \end :R=\left(\frac\left(1+\sqrt\right)\right)a\approx1.61803...a :H=\left(\sqrt\right)a\approx1.37638...a Dual polyhedron The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites. References E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid 5 Net
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name ''John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', which is derived through Greek ''Iōannēs'' from Hebrew ''Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish: Johnsson, Jonsson * Icelandic: Jónsson See also * List of people with surname Johnson *Gjoni (Gjonaj) *Ioannou * Jensen *Johansson *Johns *Johnsson * Johnston *Johnstone *Jones *Jonson *Jonsson *Jovanović Jovanović ( sr-Cy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Cupola
In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon. Formulae The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length ''a'': Stephen Wolfram,Pentagonal cupola from Wolfram Alpha. Retrieved April 11, 2020. :V=\left(\frac\left(5+4\sqrt\right)\right)a^3\approx2.32405a^3, :A=\left(\frac\left(20+5\sqrt+\sqrt\right)\right)a^2\approx16.57975a^2, :R=\left(\frac\sqrt\right)a\approx2.23295a. The height of the pentagonal cupola is :h = \sqrta \approx 0.52573a. Related polyhedra Dual polyhedron The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces: Other convex cupolae Crossed pentagrammic cupola In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically iden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid 4 Net
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name ''John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', which is derived through Greek ''Iōannēs'' from Hebrew ''Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish: Johnsson, Jonsson * Icelandic: Jónsson See also * List of people with surname Johnson *Gjoni (Gjonaj) *Ioannou * Jensen *Johansson *Johns *Johnsson * Johnston *Johnstone *Jones *Jonson *Jonsson *Jovanović Jovanović ( sr-Cy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Cupola
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon. Formulae The following formulae for the circumradius, surface area, volume, and height can be used if all faces are regular, with edge length ''a'': :C=\left(\frac\sqrt\right)a\approx1.39897a, :A=\left(7+2\sqrt+\sqrt\right)a^2\approx11.56048a^2, :V=\left(1+\frac\right)a^3\approx1.94281a^3. :h = \fraca \approx 0.70711a Related polyhedra and honeycombs Other convex cupolae Dual polyhedron The dual of the square cupola has 8 triangular and 4 kite faces: Crossed square cupola The crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid 3 Net
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name ''John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', which is derived through Greek ''Iōannēs'' from Hebrew ''Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish: Johnsson, Jonsson * Icelandic: Jónsson See also * List of people with surname Johnson *Gjoni (Gjonaj) *Ioannou * Jensen *Johansson *Johns *Johnsson * Johnston *Johnstone *Jones *Jonson *Jonsson *Jovanović Jovanović ( sr-Cy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Cupola
In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron. Formulae The following formulae for the volume (V), the surface area (A) and the height (H) can be used if all faces are regular, with edge length ''a'': :V=\left(\frac\right) a^3\approx1.17851...a^3 :A=\left(3+\frac \right) a^2\approx7.33013...a^2 :H = \frac a\approx 0.816496...a Dual polyhedron The dual of the triangular cupola has 6 triangular and 3 kite faces: Related polyhedra and honeycombs The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangles are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces. : The triangular cupola can form a tessellation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid 2 Net
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name ''John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', which is derived through Greek ''Iōannēs'' from Hebrew ''Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish: Johnsson, Jonsson * Icelandic: Jónsson See also * List of people with surname Johnson *Gjoni (Gjonaj) *Ioannou * Jensen *Johansson *Johns *Johnsson * Johnston *Johnstone *Jones *Jonson *Jonsson *Jovanović Jovanović ( sr-Cy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Pyramid
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (). It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height. Cartesian coordinates The pentagonal pyramid can be seen as the "lid" of a regular icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, ''J''11. From the Cartesian coordinates of the icosahedron, Cartesian coordinates for a pentagonal pyramid with edge length 2 may be inferred as :(1,0,\tau),\,(-1,0,\tau),\,(0,\tau,1),\,(\tau,1,0),(\tau,-1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
