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Triakis Icosahedron
In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points (\pm 1, \pm 1, \pm 1) together with the points (\pm\phi, \pm 1/\phi, 0) and cyclic permutations of these coordinates. Multiplying all coordinates of this dodecahedron by a factor of (7\phi-1)/11\approx 0.938\,748\,901\,93 gives a slightly smaller dodecahedron. The 20 vertices of this dodecahedron, together with the vertices of the icosahedron, are the vertices of a triakis icosahedron centered at the origin. The length of its long edges equals 2. Its faces are isosceles triangles with one obtuse angl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triakis Icosahedron
In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points (\pm 1, \pm 1, \pm 1) together with the points (\pm\phi, \pm 1/\phi, 0) and cyclic permutations of these coordinates. Multiplying all coordinates of this dodecahedron by a factor of (7\phi-1)/11\approx 0.938\,748\,901\,93 gives a slightly smaller dodecahedron. The 20 vertices of this dodecahedron, together with the vertices of the icosahedron, are the vertices of a triakis icosahedron centered at the origin. The length of its long edges equals 2. Its faces are isosceles triangles with one obtuse angl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Pyramid
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another spher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Triakis Icosahedron
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecicosacron
In geometry, the great dodecicosacron (or great dipteral trisicosahedron) is the dual of the great dodecicosahedron (U63). It has 60 intersecting bow-tie-shaped faces. Proportions Each face has two angles of \arccos(\frac+\frac\sqrt)\approx 30.480\,324\,565\,36^ and two angles of \arccos(-\frac+\frac\sqrt)\approx 81.816\,127\,508\,183^. The diagonals of each antiparallelogram intersect at an angle of \arccos(\frac-\frac\sqrt)\approx 67.703\,547\,926\,46^. The dihedral angle equals \arccos(\frac)\approx 127.686\,523\,427\,48^. The ratio between the lengths of the long edges and the short ones equals \frac+\frac\sqrt, which is the golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( .... Part of each face lies inside the solid, hence is invisible in solid models. References ... [...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] |
Stellation Of Triakis Icosahedron
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] |
Great Dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book '' Perspectiva Corporum Regularium'' by Wenzel Jamnitzer. The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -pentagonal polytope faces of the core -polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes. Images Related polyhedra It shares the same edge arrangement as the convex regular icosahedron; the compound with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Dodecahedron
In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex. It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron. Shaving the triangular pyramids off results in an icosahedron. If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Triambic Icosahedron
In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces are symmetric to each other. Branko Grünbaum has conjectured that it is the only Euclidean isohedron with convex faces of six or more sides, but the small hexagonal hexecontahedron is another example. Geometry The faces are equilateral hexagons, with alternating angles of \arccos(-\frac)\approx 104.477\,512\,185\,93^ and \arccos(\frac)+60^\approx 135.522\,487\,814\,07^. The dihedral angle equals \arccos(-\frac)\approx 109.471\,220\,634\,49. Related shapes The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are surrounded by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with one of the stellations of the icosahedron. (1st Edn Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Stellation Of Icosahedron
In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces are symmetric to each other. Branko Grünbaum has conjectured that it is the only Euclidean isohedron with convex faces of six or more sides, but the small hexagonal hexecontahedron is another example. Geometry The faces are equilateral hexagons, with alternating angles of \arccos(-\frac)\approx 104.477\,512\,185\,93^ and \arccos(\frac)+60^\approx 135.522\,487\,814\,07^. The dihedral angle equals \arccos(-\frac)\approx 109.471\,220\,634\,49. Related shapes The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are surrounded by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with one of the stellations of the icosahedron. (1st Edn Univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Pyramid
The icosahedral pyramid is a four-dimensional convex polytope, bounded by one icosahedron as its base and by 20 triangular pyramid cells which meet at its apex. Since an icosahedron's circumradius is less than its edge length,, circumradius sqrt 5+sqrt(5))/8 = 0.951057 the tetrahedral pyramids can be made with regular faces. Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an icosahedral bipyramid which is also a Blind Polytope. The regular 600-cell has icosahedral pyramids around every vertex. The dual to the icosahedral pyramid is the dodecahedral pyramid, seen as a dodecahedron, dodecahedral base, and 12 regular 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 regu ...s meeting at an apex. : References External links * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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