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Pentagonal Truncated Trapezohedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Poinsot polyhedron, regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The #Pyritohedron, pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the #Tetartoid, tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol . Dimensions If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is :r_u = a\frac \left(1 + \sqrt\right) \approx 1.401\,258\,538 \cdot a and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is :r_i = a\frac \sqrt \approx 1.113\,516\,364 \cdot a while the midradius, which touches the middle of each edge, is :r_m = a\frac \left(3 +\sqrt\right) \approx 1.309\,016\,994 \cdot a These quantities may also be expressed as :r_u = a\, \frac \phi :r_i = a\, \frac :r_m = a\, \frac where ''ϕ'' is the golden rat ... [...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 geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezo-rhombic Dodecahedron
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has symmetry. A concave form can be constructed with an identical net, seen as excavating trigonal trapezohedra from the top and bottom. Construction This polyhedron could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts. Space-filling tessellation A space-filling tessellation, the trapezo-rhombic dodecahedral honeycomb, can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, or a rhombille tiling, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up. :: In the special case that the long sides of the trapezoids equals twice the length of the short sides, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Dodecahedron
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism. Parallelohedron Along with the rhombic dodecahedron, it is a space-filling polyhedron, one of the five types of parallelohedron identified by Evgraf Fedorov that tile space face-to-face by translations. It has 5 sets of parallel edges, called zones or belts. : Tessellation * It can tesselate all space by translations. * It is the Wigner–Seitz cell for certain body-centered tetragonal lattices. This is related to the rhombic dodecahedral honeycomb with an elongation of zero. Projected normal to the elongation direction, the honeycomb looks like a square tiling with the rhombi projected into squares. Variat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual polyhedron, dual to an octahedron. The group of orientation-preserving symmetries is ''S''4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. Details Chiral and full (or achiral) octahedral symmetry are the Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the Crystal system#Overview of point groups by crystal system, crystallographic point groups of the cubic crystal system. As the hyperoctahedral group of dimension 3 the full oct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly times the short face-diagonal length; thus, the acute angles on each face measure arccos(), or approximately 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces. In elementary terms, this means that for any two faces A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyritohedral Symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyrite
The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Iron, FeSulfur, S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral. Pyrite's metallic Luster (mineralogy), luster and pale brass-yellow hue give it a superficial resemblance to gold, hence the well-known nickname of ''fool's gold''. The color has also led to the nicknames ''brass'', ''brazzle'', and ''Brazil'', primarily used to refer to pyrite found in coal. The name ''pyrite'' is derived from the Greek language, Greek (), 'stone or mineral which strikes fire', in turn from (), 'fire'. In ancient Roman times, this name was applied to several types of stone that would create sparks when struck against steel; Pliny the Elder described one of them as being brassy, almost certainly a reference to what we now call pyrite. By Georgius Agricola's time, , the term had become a generic term for all of the pyrite group, sulfide minerals. Pyrite is usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters. Description Icosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron. As point group Apart from the two infinite series of prismatic and antiprismatic symmetry, rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word ''stellation'' comes from the Latin ''stellātus'', "starred", which in turn comes from Latin ''stella'', "star". Stellation is the reciprocal or dual process to ''faceting''. Kepler's definition In 1619 Kepler defined stellation for polygons and polyhedra as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler–Poinsot Polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. They can all be seen as three-dimensional analogues of the pentagram in one way or another. Characteristics Non-convexity These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures. In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
