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Small Triambic Icosahedron
In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon Face (geometry), faces. It has 60 Edge (geometry), edges and 32 Vertex (geometry), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is defined as a hexagon that is both equilateral and equiangular. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its internal angle is equal to 120°. The Schläfli symbol denotes this polygon as \ . However, the regular hexagon can also be considered as the cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm\ . A regular hexagon is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). Measurement The longest diagonals of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Triambic Icosahedron
In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical Dual polyhedron, dual uniform polyhedra. The exterior surface also represents the The Fifty-Nine Icosahedra, De2f2 Great_triambic_icosahedron#As_a_stellation, stellation of the icosahedron. These figures can be differentiated by marking which intersections between edges are true Vertex (geometry), vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the even–odd rule, the internal structure of both shapes will differ. The 12 vertices of the convex hull matches the vertex arrangement of an icosahedron. Great triambic icosahedron The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron, U47. It has 20 inverted-hexagonal (triambus) faces, shaped like a three-bladed propeller. It has 32 vertices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron
In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform polyhedra may be Regular polyhedron, regular (if also Isohedral figure, face- and Isotoxal figure, edge-transitive), Quasiregular polyhedron, quasi-regular (if also edge-transitive but not face-transitive), or Semiregular polyhedron, semi-regular (if neither edge- nor face-transitive). The faces and vertices don't need to be Convex polyhedron, convex, so many of the uniform polyhedra are also Star polyhedron, star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite classes of Prism (geometry), prisms and antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 Quasiregular polyhedron, quasiregular and 11 Semiregular polyhedron, semiregular&m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triakis Icosahedron
In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's ''Divina proportione'', where it was named the ''icosahedron elevatum''. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron. As a Kleetope The triakis icosahedron can be formed by gluing triangular pyramids to each face of a regular icosahedron. Depending on the height of these pyramids relative to their base, the result can be either convex or non-convex. This construction, of gluing pyramids to each face, is an instance of a general construction called the Kleetope; the triakis icosahedron is the Kleetope of the icosahedron. This interpretation is also expressed in the name, triakis, which is used for the Kleetopes of polyhedra with tria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Fifty-Nine Icosahedra
''The Fifty-Nine Icosahedra'' is a book written and illustrated by Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic regular icosahedron, icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell. Authors' contributions Miller's rules Although J. C. P. Miller, Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules for defining which stellation forms should be considered "properly significant and distinct": :''(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.'' :''(ii) All parts composing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Hexagonal Hexecontahedron
In geometry, the small hexagonal hexecontahedron is a nonconvex Isohedral figure, isohedral polyhedron. It is the Dual polyhedron, dual of the uniform star polyhedron, uniform small snub icosicosidodecahedron. It is partially degeneracy (mathematics), degenerate, having coincident Vertex (geometry), vertices, as its dual has coplanar triangular faces. Geometry Treating it as a simple non-convex solid (without intersecting surfaces), it has 180 faces (all triangles), 270 edges, and 92 vertices (twelve with degree 10, twenty with degree 12, and sixty with degree 3), giving an Euler characteristic of 92 − 270 + 180 = +2. Faces The faces are irregular hexagons. Denoting the golden ratio by \phi and putting \xi = \frac-\frac\sqrt\approx -0.433\,380\,199\,59, the hexagons have five equal angles of \arccos(\xi)\approx 115.682\,268\,170\,75^ and one of \arccos(\phi^\xi-\phi^)\approx 141.588\,659\,146\,23^. Each face has four long and two short edges. The ratio between the edge len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem. Life< ...
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Isohedron
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 Face (geometry), faces are the same. More specifically, all faces must be not merely Congruence (geometry), 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 Translation (geometry), translations, Rotation (mathematics), rotations, and/or Reflection (mathematics), reflections that maps onto . For this reason, Convex polytope, 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 Parity (mathematics), even number of faces. The Dual polyhedron, dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
