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  1. A polyhedron can have lots of diagonals.[1]
  2. Polyhedron, In Euclidean geometry, a three-dimensional object composed of a finite number of polygonal surfaces (faces).[2]
  3. In general, polyhedrons are named according to number of faces.[2]
  4. Polyhedron publishes original, fundamental, experimental and theoretical work of the highest quality in all the major areas of inorganic chemistry.[3]
  5. Polyhedron publishes full papers, specially commissioned review articles (Polyhedron Reviews) and themed issues of the journal (Polyhedron Special Issues).[3]
  6. Polyhedron does not publish communications, notes or Book Reviews.[3]
  7. Face : the flat surfaces that make up a polyhedron are called its faces.[4]
  8. A vertex is also known as the corner of a polyhedron.[4]
  9. A regular polyhedron is made up of regular polygons.[4]
  10. An irregular polyhedron is formed by polygons of different shapes where all the components are not the same.[4]
  11. Convex polyhedra are well-defined, with several equivalent standard definitions.[5]
  12. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic.[5]
  13. The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ...[5]
  14. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, whose faces may not form simple polygons, and some of whose edges may belong to more than two faces.[5]
  15. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges.[6]
  16. A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension.[6]
  17. Although usage varies, most authors additionally require that a solution be bounded for it to define a convex polyhedron.[6]
  18. The following table lists the name given to a polyhedron having faces for small .[6]
  19. In Geometry, a polyhedron is a closed space figure whose faces are polygons.[7]
  20. The word polyhedron has Greek origins, meaning many faces.[7]
  21. The polygons that form a polyhedron are called faces.[7]
  22. Euler's Theorem shows a relationship between the number of faces, vertices, and edges of a polyhedron.[7]
  23. Table S1: Complete list of the Johnson polyhedra, ordered according to the number of vertices ( V), giving the number of edges ( E) and faces ( F) in each case.[8]
  24. Nested polyhedra that appear as successive shells in prototypical solid state structures.[8]
  25. Polyhedra generated from a dodecahedron through augmentation and truncation operations.[8]
  26. Polyhedra generated from a cube through augmentation and truncation operations.[8]
  27. We can study the long-range order by classifying assemblages of Voronoi polyhedra.[9]
  28. -codeword is the sequence of \( p_{3} \) s and instructs how to construct the polychoron from its building-block polyhedra.[9]
  29. -code, we first describe the relations between parts of polyhedra and parts of a polychoron.[9]
  30. As shown in Fig.a, a polyhedron can be represented as a two-dimensional object by using a Schlegel diagram.[9]
  31. Johannes Kepler discovered a third class, the rhombic polyhedra.[10]
  32. Some carbon fullerenes, inorganic cages, icosahedral viruses, protein complexes, and geodesic structures resemble these polyhedra.[10]
  33. the 2 rhombic polyhedra reported by Johannes Kepler in 1611.[10]
  34. Here we add a fourth class, “Goldberg polyhedra,” which are also convex and equilateral.[10]
  35. This definition of a polyhedron has different meanings, according to how a polygon is defined.[11]
  36. If by a polygon is meant a plane closed polygonal curve (even if self-intersecting), one arrives at the first definition of a polyhedron.[11]
  37. Most of this article is constructed on the basis of a second definition of a polyhedron, in which its faces are polygons, understood as parts of planes bounded by polygonal curves.[11]
  38. From this point of view a polyhedron is a surface made up of polygonal segments.[11]
  39. The number of vertices V, faces F, and edges E in a convex 3-dimensional polyhedron, satisfy V + F - E = 2.[12]
  40. The Egyptians built pyramids and the Greeks studied "regular polyhedra," today sometimes referred to as the Platonic Solids .[12]
  41. Almost certainly, in the early days of the study of polyhedra, the word referred to convex polyhedra.[12]
  42. This polyhedron has three rays (which, if extended, should meet at a point) and three line segments as edges of the polyhedron, rather than having edges which are line segments.[12]
  43. What happens if we construct duals of other regular polyhedra?[13]
  44. Thus the five regular polyhedra fall into three groups: two dual pairs and one polyhedron that is dual to itself.[13]
  45. A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge.[14]
  46. Polyhedra are often named according to the number of faces.[14]
  47. For a simply connected polyhedron, χ = 2.[14]
  48. For every polyhedron there is a dual polyhedron having faces in place of the original's vertices and vice versa.[14]
  49. A polyhedron is said to be regular if its Faces and Vertex Figures are Regular (not necessarily Convex) polygons (Coxeter 1973, p. 16).[15]
  50. However, the term ``regular polyhedra is sometimes also used to refer exclusively to the Platonic Solids (Cromwell 1997, p. 53).[15]
  51. There exist exactly 92 Convex Polyhedra with Regular Polygonal faces (and not necessary equivalent vertices).[15]
  52. Polyhedra with identical Vertices related by a symmetry operation are known as Uniform Polyhedra.[15]
  53. A polyhedron is said to be regular if all its faces are equal regular polygons and the same number of faces meet at every vertex.[16]
  54. For a polyhedron it means about the same thing with surface curves shrunk into a point while staying on the surface.[16]
  55. Simple polyhedra can be continuously deformed into a sphere and, in addition, each of its faces is simple.[16]
  56. The Euler's Theorem, also known as the Euler's formula, deals with the relative number of faces, edges and vertices that a polyhedron (or polygon) may have.[16]
  57. A solid shape bounded by polygons is called a polyhedron.[17]
  58. Polygons forming a polyhedron are known as its faces.[17]
  59. Line segments common to intersecting faces of a polyhedron are known as its edges.[17]
  60. In a polyhedron, three or more edges meet at a point to form a vertex.[17]
  61. The discussion in most of this article is based on another definition of polyhedron, in which the faces are polygons construed as parts of the plane bounded by broken lines.[18]
  62. From this standpoint, a polyhedron is a surface made up of polygonal pieces.[18]
  63. If this surface does not intersect itself, then it is the complete surface of some geometric solid, which is also called a polyhedron.[18]
  64. This leads to a third view of polyhedrons as geometric solids.[18]
  65. Section 2 describes the approach for the construction of enclosing and enclosed ellopsoids of convex polyhedra.[19]
  66. Alternatively, ' 'paper bag' ' icons can mark the position of polyhedra, revealing their size but not their shape.[19]
  67. For simulations, the links of redundant robots are modeled by a union of line segments or convex polyhedra.[19]
  68. Therefore, the proposed method just focuses on generating an enclosed ellopsoid, which is as large as possible, to fit the polyhedron tightly.[19]
  69. Let M be a closed convex polyhedron with no holes which is composed of no polygons other than pentagons and hexagons.[20]
  70. If we traverse the polyhedron face-by-face counting the number of edges we will get 6h+5p.[20]
  71. If we traverse the vertices of the polyhedron counting edges we will get 3v.[20]
  72. When h=0, the polyhedron is the dodecahedron having twelve pentagons with 20 vertices and 30 edges.[20]
  73. The above figure shows a special set of polyhedrons called the five regular solids.[21]
  74. Angle of the polyhedron: It is the proportion of space limited by three or more planes that meet at a point called vertex.[22]
  75. The polyhedrons can be classified under many groups, either by the family or from the characteristics that differentiate them.[22]
  76. A polyhedron is a closed, three-dimensional solid bounded entirely by at least four polygons, no two of which are in the same plane.[23]
  77. The number of sides of each polygon is the major feature distinguishing polyhedrons from one another.[23]
  78. Each of the polygons of a polyhedron is called a face.[23]
  79. The value of v + f − e for a polyhedron is called the Euler characteristic of the polyhedron's surface, named after the Swiss mathematician Leonhard Euler (1707–1783).[23]
  80. (Examples #1-7) 00:10:38 – How do we classify regular and convex polyhedron?[24]
  81. "base" usually denotes the face on which the polyhedron rests on; thus each face may be a base (likewise each side of a triangle may be considered as "base").[25]
  82. A polyhedron is a finite set of polygons such that every side of each belongs to just one other, with the restriction that no subset has the same property.[25]
  83. A polyhedron ( plural polyhedra) is a three-dimensional solid with flat polygon faces joined at their edges.[26]
  84. A polyhedron's faces are bounding surfaces consisting of portions of intersecting planes.[26]
  85. Polyhedra are not necessarily constructed using the same shaped polygons for faces.[26]
  86. The skeletal polyhedron at the right, and the one shown above, are examples where the faces have varying numbers of sides.[26]
  87. When a convex polyhedron (or polytope) has dimension , it is called a -polyhedron ( -polytope).[27]
  88. In Polylib the decomposition theorem is extensively used (in its extended form for polyhedra).[27]
  89. A polyhedron can be represented by a set of inequalities (usually, implicit equalities are represented in a separate matrix): , this representation is called implicit.[27]
  90. Although the polyhedra theory cannot be detailed here, we review a set of important concepts that are used when manipulating polyhedra.[27]

소스

  1. ↑ Polyhedrons
  2. ↑ 2.0 2.1 Polyhedron | geometry
  3. ↑ 3.0 3.1 3.2 Polyhedron
  4. ↑ 4.0 4.1 4.2 4.3 Definition, Facts & Example
  5. ↑ 5.0 5.1 5.2 5.3 Polyhedron
  6. ↑ 6.0 6.1 6.2 6.3 Polyhedron -- from Wolfram MathWorld
  7. ↑ 7.0 7.1 7.2 7.3 Polyhedron
  8. ↑ 8.0 8.1 8.2 8.3 Polyhedra in (inorganic) chemistry
  9. ↑ 9.0 9.1 9.2 9.3 Polyhedron and Polychoron Codes for Describing Atomic Arrangements
  10. ↑ 10.0 10.1 10.2 10.3 Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses
  11. ↑ 11.0 11.1 11.2 11.3 Encyclopedia of Mathematics
  12. ↑ 12.0 12.1 12.2 12.3 AMS :: Feature Column from the AMS
  13. ↑ 13.0 13.1 Duals of Regular Polyhedra
  14. ↑ 14.0 14.1 14.2 14.3 Polyhedron
  15. ↑ 15.0 15.1 15.2 15.3 Polyhedron
  16. ↑ 16.0 16.1 16.2 16.3 Regular Polyhedra
  17. ↑ 17.0 17.1 17.2 17.3 Polyhedrons: Regular Polyhedron, Prism, Pyramids, Videos and Examples
  18. ↑ 18.0 18.1 18.2 18.3 polyhedra
  19. ↑ 19.0 19.1 19.2 19.3 meaning in the Cambridge English Dictionary
  20. ↑ 20.0 20.1 20.2 20.3 Euler's Theorem Concerning Polyhedra Composed of Pentagons and Hexagons: There Must Be Exactly 12 Pentagons
  21. ↑ How to Recognize a Polyhedron
  22. ↑ 22.0 22.1 Polyhedrons: basic definitions and classification
  23. ↑ 23.0 23.1 23.2 23.3 Encyclopedia.com
  24. ↑ What is a Polyhedron? Simply Explained w/ 14 Examples!
  25. ↑ 25.0 25.1 definitions polyhedra
  26. ↑ 26.0 26.1 26.2 26.3 MathBitsNotebook(Geo
  27. ↑ 27.0 27.1 27.2 27.3 Theoretical background

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