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Snub 24-cell Honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra. It can be seen as an alternation of a truncated 24-cell honeycomb, and can be represented by Schläfli symbol s, s, and 3 other snub constructions. It is defined by an irregular decachoron vertex figure (10-celled 4-polytope), faceted by four snub 24-cells, one 16-cell, and five 5-cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at mid-edges into a central octahedron and four corner tetrahedra. Then the four side-facets of the prism, the triangular prisms become tridiminished icosahedra. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of Coxeter groups, symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams. History of discovery *Regular polytopes: (convex faces) **1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. *Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) **1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on a Euclidean plane. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular Tetrahedron, tetrahedral and 24 Regular icosahedron, icosahedral cell (mathematics), cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra. Topologically, under its highest symmetry, [3+,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with Th symmetry), 24 regular tetrahedra, and 96 triangular pyramids. Semiregular polytope It is one of three Semiregular 4-polytopes#Semiregular polytopes, semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a ''tetricosahedric'' for be ... [...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, some authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not ''Face (geometry), faces''. To ''facetting, 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 Face (geometry), face that has dimension ''n'' − 1 (an (''n'' − 1)-face or hyperface) is called a Face (geometry)#Facet, facet. In this terminology, every facet is a face. * 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 complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tridiminished Icosahedron
In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron. Construction The tridiminished icosahedron can be constructed by removing three regular pentagonal pyramid from a regular icosahedron. The aftereffect of such construction leaves five equilateral triangles and three regular pentagons. Since all of its faces are regular polygons and the resulting polyhedron remains convex, the tridiminished icosahedron is a Johnson solid, and it is enumerated as the sixty-third Johnson solid J_ . This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron. The tridiminished icosahedron is non-composite polyhedron, meaning it is convex polyhedron that cannot be separated by a plane into two or more regular polyhedrons. Properties The surface area of a tridiminished icosahedron A is the sum of all polygonal faces' area: five equila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Prism
In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangular prism may be both Semiregular polyhedron, semiregular and Uniform polyhedron, uniform. The triangular prism can be used in constructing another polyhedron. Examples are some of the Johnson solids, the truncated right triangular prism, and Schönhardt polyhedron. Properties A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its ''bases'', and the bases of a triangular prism are triangles. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3 parallelograms as other faces. If the prism's edges are perpendicular to the base, the lateral faces are rectangles, and the prism is called a ''right triangular prism''. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Prism
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedron, polyhedral cells: 2 tetrahedron, tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 uniform polyhedral prisms created by using uniform Prism (geometry), prisms to connect pairs of parallel Platonic solids and Archimedean solids. Images Alternative names # Tetrahedral dyadic prism (Norman Johnson (mathematician), Norman W. Johnson) # Tepe (Jonathan Bowers: for tetrahedral prism) # Tetrahedral hyperprism # Digonal antiprismatic prism # Digonal antiprismatic hyperprism Structure The tetrahedral prism is bounded by two tetrahedra and four triangular prisms. The triangular prisms are joined to each other via their square faces, and are joined to the two tetrahedra via their triangular faces. Projections The tetrahedron-first orthographic projection of the tetrahedral prism into 3D space has a ... [...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 general -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 conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 24-cell Honeycomb
In Four-dimensional space, four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb (geometry), honeycomb. It can be seen as a Truncation (geometry), truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells. It has a uniform Alternation (geometry), alternation, called the snub 24-cell honeycomb. It is a snub from the _4 construction. This truncated 24-cell has Schläfli symbol t, and its Snub (geometry), snub is represented as s. Alternate names * Truncated icositetrachoric tetracomb * Truncated icositetrachoric honeycomb * Cantitruncated 16-cell honeycomb * Bicantitruncated tesseractic honeycomb Symmetry constructions There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternation (geometry)
In geometry, an alternation or ''partial truncation'', is an operation on a polygon, polyhedron, tessellation, tiling, or higher dimensional polytope that removes alternate vertices.Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation Coxeter labels an ''alternation'' by a prefixed ''h'', standing for ''hemi'' or ''half''. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be ''alternated''. For example, the alternation of a vertex figure with ''2a.2b.2c'' is ''a.3.b.3.c.3'' where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cu ... [...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] |
Ridge (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face include polyhedron side and Euclidean plane '' tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
