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Rhombic Triacontahedron
The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex (geometry), vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron and can be seen as a elongated rhombic icosahedron. The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, , so that the Angle#Types of angles, acute angles on each face measure , or approximately 63.43°. A rhombus so obtained is called a ''golden rhombus''. Being the dual of an Archimedean solid, the rhombic triacontahedron is ''face-transitive'', meaning the symmetry group of the solid acts transitive action, transitively on the set of faces. This means that for any two faces, and , there is a rotation or reflection (mathematics), reflection of the solid that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Icosahedron
The rhombic icosahedron is a polyhedron shaped like an Oblate spheroid, oblate sphere. Its 20 faces are Congruence (geometry), congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on its axis of 5-fold symmetry, which is perpendicular to 5 axes of 2-fold symmetry through the midpoints of opposite equatorial edges (example on top figure: most left-hand and most right-hand mid-edges). Its other 10 faces follow its equator, 5 above and 5 below it; each of these 10 rhombi has 2 of its 4 sides lying on this zig-zag Skew polygon, skew decagon equator. The rhombic icosahedron has 22 vertices. It has Dihedral symmetry in three dimensions, D5d, [2+,10], (2*5) symmetry group, of order 20; thus it has a center of symmetry (since 5 is odd). Even though all its faces are congruent, the rhombic icosahedron is not face-transitive, since one can distinguish whether a particular face is near the eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Solid
The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa. One way to construct the Catalan solids is by using the Dorman Luke construction. The Catalan solids are face-transitive or ''isohedral'' meaning that their faces are symmetric to one another, but they are not vertex-transitive because their vertices are not symmetric. Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same. Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other. Some Catalan solids were discovered by Johannes Kepler during his study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is Cuboctahedron#Radial equilateral symmetry, radially equilateral. Its dual polyhedron is the rhombic dodecahedron. Construction The cuboctahedron can be constructed in many ways: * Its construction can be started by attaching two regular triangular cupolas base-to-base. This is similar to one of the Johnson solids, triangular orthobicupola. The difference is that the triangular orthobicupola is constructed with one of the cupolas twisted so that similar polygonal faces are adjacent, whereas the cuboctahedron is not. As a result, the cuboctahedron may also called the ''triangular gyro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the ''Timaeus (dialogue), Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a Translation (geometry), translation, Rotation (mathematics), rotation, and/or Reflection (mathematics), reflection that will move one edge to the other while leaving the region occupied by the object unchanged. Isotoxal polygons An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The Duality (mathematics)#Dimension-reversing dualities, duals of isotoxal polygons are isogonal polygons. Isotoxal 4n-gons are Central symmetry, centrally symmetric, thus are also zonogons. In general, a (non-regular) isotoxal 2n-gon has \mathrm_n, (^*nn) dihedral symmetry. For example, a (non-square) rhombus is an isotoxa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a ''vertical reflection'') would look like q. Its image by reflection in a horizontal axis (a ''horizontal reflection'') would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term ''reflection'' is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. The set of fixed points (the "mirror") of such an isome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientation (geometry), orientations), in contrast to rotation around a fixed axis, rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, ... [...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 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] |
