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Quasiregular Polyhedra
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular. There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case. These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol \begin p \\ q \end or ''r'', to represent that their faces are a ... [...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 vertices, with 2 triangles and 2 squares meeting at each, and 24 identical 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 radially equilateral. Its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was probably known to Plato: Heron's ''Definitiones'' quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Synonyms *''Vector Equilibrium'' (Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry). Fuller also called a cuboctahedron built of rigid struts and flexible vertices a ''jitterbug''; this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its square s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 334-1
H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' ."H" '' Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. History The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, , still represented a similar sound, the voiceless glottal fricative . In this context, the letter eta is also known as Heta to underline this fact. Thus, in the Old Italic alphabets, the letter Heta of the Euboean alphabet was adopted with its original sound value . While Etruscan and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 Hexagonal Tiling
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as *,4 removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry. Uniform colorings There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 Pentagonal Tiling
In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . It can also be called a pentapentagonal tiling in a bicolored quasiregular form. Symmetry This tiling represents a hyperbolic kaleidoscope of 5 mirrors meeting as edges of a regular pentagon. This symmetry by orbifold notation is called *22222 with 5 order-2 mirror intersections. In Coxeter notation can be represented as *,4 removing two of three mirrors (passing through the pentagon center) in the ,4symmetry. The kaleidoscopic domains can be seen as bicolored pentagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1 and as a quasiregular tiling is called a ''pentapentagonal tiling''. : Related polyhedra and tiling This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with Schläfli symbol , and Coxeter diagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. Related polyhedra and tilings This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Figure
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope '' acts transitively'' on its vertices, or that the vertices lie within a single ''symmetry orbit''. All vertices of a finite -dimensional isogonal figure exist on an -sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. The pseudorhombicuboctahedronwhich is ''not'' isogonaldemonstrates that simply asserting that "all vertices look t ... [...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 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 tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 345-4
H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' ."H" '' Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. History The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, , still represented a similar sound, the voiceless glottal fricative . In this context, the letter eta is also known as Heta to underline this fact. Thus, in the Old Italic alphabets, the letter Heta of the Euboean alphabet was adopted with its original sound value . While Etruscan and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 345-2
H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' ."H" '' Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. History The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, , still represented a similar sound, the voiceless glottal fricative . In this context, the letter eta is also known as Heta to underline this fact. Thus, in the Old Italic alphabets, the letter Heta of the Euboean alphabet was adopted with its original sound value . While Etruscan and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 345-1
H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' ."H" '' Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. History The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, , still represented a similar sound, the voiceless glottal fricative . In this context, the letter eta is also known as Heta to underline this fact. Thus, in the Old Italic alphabets, the letter Heta of the Euboean alphabet was adopted with its original sound value . While Etruscan and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 4ii-1
H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' ."H" '' Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. History The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, , still represented a similar sound, the voiceless glottal fricative . In this context, the letter eta is also known as Heta to underline this fact. Thus, in the Old Italic alphabets, the letter Heta of the Euboean alphabet was adopted with its original sound value . While Etruscan and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
