|
Convex Uniform Honeycombs In Hyperbolic Space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family. Hyperbolic uniform honeycomb families Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups. Compact uniform honeycomb families The nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their fundamental simplex domains. These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. Two known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 Dodecahedral Honeycomb
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space. With Schläfli symbol it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb. Description The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge. Symmetry It has a half symmetry construction, , with two types (colors) of dodecahedra in the Wythoff construction. ↔ . Images A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model Related polytopes and honeycombs There are four regular compact honeycom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Cubic Honeycomb
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five cubes around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb. Description Symmetry It has a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: ,(3,5)* index 120. Related polytopes and honeycombs The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells. The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space: There are fifteen uniform honeycombs in the ,3,4 Coxeter group family, including the order-5 cubic honeycomb as the regular form: The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures. It is also in a sequence of regular polychora and hon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Dodecahedral Honeycomb
In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron. Description The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°. Images Related polytopes and honeycombs There are four regular compact honeycombs in 3D hyperbolic space: There is another honeycomb in hyperboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Honeycomb
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure. Description The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge. Related regular honeycombs There are four regular compact honeycombs in 3D hyperbolic space: Related regular polytopes and honeycombs It is a member of a sequence of regular polychora and honeycombs with deltrahedral cells: It is also a member of a sequence of regular polychora and honeycombs , with vertex figures composed of pentagons: Uniform honeycombs There are nine uniform h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedrally Diminished Dodecahedron
In geometry, a tetrahedrally diminished dodecahedron (also tetrahedrally stellated icosahedron or propello tetrahedron) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals). A canonical form exists with two edge lengths at 0.849 : 1.057, assuming that the radius of the midsphere is 1. The kites remain isosceles. It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated, or from the pyritohedron, with 4 vertices diminished. Within its tetrahedral symmetry, it has geometric varied proportions. By Dorman Luke dual construction, a unique geometric proportion can be defined. The kite faces have edges of length ratio ~ 1:0.633. Topologically, the triangles are always equilateral, while the quadrilaterals are irregular, although the two adjacent edges that meet at the vertices of a tetrahedron are equal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Bipyramid
In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four. The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, (). As a Johnson solid with all faces equilateral triangles, it is also a deltahedron. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \frac \approx L\cdot 1.632993162 :A = L^2 \cdot \frac \approx L^2\cdot 2.598076211 :V = L^3 \cdot \frac \approx L^3\cdot 0.235702260 Dual polyhedron The dual polyhedron of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Vertex
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometimes called ideal vertices. All ideal triangles are congruent. Properties Ideal triangles have the following properties: * All ideal triangles are congruent to each other. * The interior angles of an ideal triangle are all zero. * An ideal triangle has infinite perimeter. * An ideal triangle is the largest possible triangle in hyperbolic geometry. In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties: * Any ideal triangle has area π. Distances in an ideal triangle * The inscribed circle to an ideal triangle has radius r=\ln\sqrt = \frac \ln 3 = \operatorname\frac = 2 \operatorname(2- \sqrt) = = \operatorname\frac\sqrt = \operatorname\frac\sqrt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a ''face''. To ''facet'' a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to '' stellation'' and may also be applied to higher-dimensional polytopes. * In polyhedral combinatorics and in the general theory of polytopes, a facet (or hyperface) of a polytope of dimension ''n'' is a face that has dimension ''n'' − 1. Facets may also be called (''n'' − 1)-faces. In three-dimensional geometry, they are often called "faces" without qualification. * A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.. For (boundary complexes of) sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedral
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 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, a common crystal form in pyrite, has pyritohedral symmetry, while the 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 with the rhombic dodecahedra, are space-filling. There a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells. Hints at a general definition Given an action of a group ''G'' on a topological space ''X'' by homeomorphisms, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several preci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonal Trapezohedron
In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhombohedron''. Some sources just call them ''rhombohedra''. Geometry Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The ''acute'' or ''prolate'' form has three acute angle corners of the rhombic faces meeting at the two polar axis vertices. The ''obtuse'' or ''oblate'' or ''flat'' form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices. More strongly than having all faces congruent, the trigonal trapezohedra are isohedral figures, meaning that they have symmetries that take any face to any other face. Special cases A cube can be interpreted as a special case of a trigonal trapezohedron, with square rather than rhombic faces. The two golden rhombohedra are the acut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
