2021년 2월 17일 (수) 08:09에 Pythagoras0님의 편집

2021-02-17T08:09:01Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-26T12:17:55Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2020-12-21T10:48:20Z

노트: 새 문단

새 문서

== 노트 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q172937 Q172937]
===말뭉치===
# A polyhedron can have lots of diagonals.<ref name="ref_77327a22">[https://www.mathsisfun.com/geometry/polyhedron.html Polyhedrons]</ref>
# Polyhedron, In Euclidean geometry, a three-dimensional object composed of a finite number of polygonal surfaces (faces).<ref name="ref_14da19ba">[https://www.britannica.com/science/polyhedron Polyhedron | geometry]</ref>
# In general, polyhedrons are named according to number of faces.<ref name="ref_14da19ba" />
# Polyhedron publishes original, fundamental, experimental and theoretical work of the highest quality in all the major areas of inorganic chemistry.<ref name="ref_2f47b223">[https://www.journals.elsevier.com/polyhedron Polyhedron]</ref>
# Polyhedron publishes full papers, specially commissioned review articles (Polyhedron Reviews) and themed issues of the journal (Polyhedron Special Issues).<ref name="ref_2f47b223" />
# Polyhedron does not publish communications, notes or Book Reviews.<ref name="ref_2f47b223" />
# Face : the flat surfaces that make up a polyhedron are called its faces.<ref name="ref_c6315a94">[https://www.splashlearn.com/math-vocabulary/geometry/polyhedron Definition, Facts & Example]</ref>
# A vertex is also known as the corner of a polyhedron.<ref name="ref_c6315a94" />
# A regular polyhedron is made up of regular polygons.<ref name="ref_c6315a94" />
# An irregular polyhedron is formed by polygons of different shapes where all the components are not the same.<ref name="ref_c6315a94" />
# Convex polyhedra are well-defined, with several equivalent standard definitions.<ref name="ref_f767b455">[https://en.wikipedia.org/wiki/Polyhedron Polyhedron]</ref>
# However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic.<ref name="ref_f767b455" />
# The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ...<ref name="ref_f767b455" />
# 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.<ref name="ref_f767b455" />
# In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges.<ref name="ref_15a3871d">[https://mathworld.wolfram.com/Polyhedron.html Polyhedron -- from Wolfram MathWorld]</ref>
# A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension.<ref name="ref_15a3871d" />
# Although usage varies, most authors additionally require that a solution be bounded for it to define a convex polyhedron.<ref name="ref_15a3871d" />
# The following table lists the name given to a polyhedron having faces for small .<ref name="ref_15a3871d" />
# In Geometry, a polyhedron is a closed space figure whose faces are polygons.<ref name="ref_0828a8c4">[https://www.math.net/polyhedron Polyhedron]</ref>
# The word polyhedron has Greek origins, meaning many faces.<ref name="ref_0828a8c4" />
# The polygons that form a polyhedron are called faces.<ref name="ref_0828a8c4" />
# Euler's Theorem shows a relationship between the number of faces, vertices, and edges of a polyhedron.<ref name="ref_0828a8c4" />
# 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.<ref name="ref_45435b0c">[https://pubs.rsc.org/en/content/articlelanding/2005/dt/b503582c Polyhedra in (inorganic) chemistry]</ref>
# Nested polyhedra that appear as successive shells in prototypical solid state structures.<ref name="ref_45435b0c" />
# Polyhedra generated from a dodecahedron through augmentation and truncation operations.<ref name="ref_45435b0c" />
# Polyhedra generated from a cube through augmentation and truncation operations.<ref name="ref_45435b0c" />
# We can study the long-range order by classifying assemblages of Voronoi polyhedra.<ref name="ref_08acc503">[https://link.springer.com/chapter/10.1007/978-981-10-7617-6_6 Polyhedron and Polychoron Codes for Describing Atomic Arrangements]</ref>
# -codeword is the sequence of \( p_{3} \) s and instructs how to construct the polychoron from its building-block polyhedra.<ref name="ref_08acc503" />
# -code, we first describe the relations between parts of polyhedra and parts of a polychoron.<ref name="ref_08acc503" />
# As shown in Fig.a, a polyhedron can be represented as a two-dimensional object by using a Schlegel diagram.<ref name="ref_08acc503" />
# Johannes Kepler discovered a third class, the rhombic polyhedra.<ref name="ref_cd969d4a">[https://www.pnas.org/content/111/8/2920 Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses]</ref>
# Some carbon fullerenes, inorganic cages, icosahedral viruses, protein complexes, and geodesic structures resemble these polyhedra.<ref name="ref_cd969d4a" />
# the 2 rhombic polyhedra reported by Johannes Kepler in 1611.<ref name="ref_cd969d4a" />
# Here we add a fourth class, “Goldberg polyhedra,” which are also convex and equilateral.<ref name="ref_cd969d4a" />
# This definition of a polyhedron has different meanings, according to how a polygon is defined.<ref name="ref_7ac670bd">[https://encyclopediaofmath.org/wiki/Polyhedron Encyclopedia of Mathematics]</ref>
# If by a polygon is meant a plane closed polygonal curve (even if self-intersecting), one arrives at the first definition of a polyhedron.<ref name="ref_7ac670bd" />
# 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.<ref name="ref_7ac670bd" />
# From this point of view a polyhedron is a surface made up of polygonal segments.<ref name="ref_7ac670bd" />
# The number of vertices V, faces F, and edges E in a convex 3-dimensional polyhedron, satisfy V + F - E = 2.<ref name="ref_14731774">[http://www.ams.org/publicoutreach/feature-column/fcarc-eulers-formula AMS :: Feature Column from the AMS]</ref>
# The Egyptians built pyramids and the Greeks studied "regular polyhedra," today sometimes referred to as the Platonic Solids .<ref name="ref_14731774" />
# Almost certainly, in the early days of the study of polyhedra, the word referred to convex polyhedra.<ref name="ref_14731774" />
# 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.<ref name="ref_14731774" />
# What happens if we construct duals of other regular polyhedra?<ref name="ref_6a172a07">[http://www.math.brown.edu/tbanchof/Beyond3d/chapter5/section03.html Duals of Regular Polyhedra]</ref>
# Thus the five regular polyhedra fall into three groups: two dual pairs and one polyhedron that is dual to itself.<ref name="ref_6a172a07" />
# A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge.<ref name="ref_5d61209c">[https://math.wikia.org/wiki/Polyhedron Polyhedron]</ref>
# Polyhedra are often named according to the number of faces.<ref name="ref_5d61209c" />
# For a simply connected polyhedron, χ = 2.<ref name="ref_5d61209c" />
# For every polyhedron there is a dual polyhedron having faces in place of the original's vertices and vice versa.<ref name="ref_5d61209c" />
# A polyhedron is said to be regular if its Faces and Vertex Figures are Regular (not necessarily Convex) polygons (Coxeter 1973, p. 16).<ref name="ref_9442d9d1">[https://archive.lib.msu.edu/crcmath/math/math/p/p468.htm Polyhedron]</ref>
# However, the term ``regular polyhedra'' is sometimes also used to refer exclusively to the Platonic Solids (Cromwell 1997, p. 53).<ref name="ref_9442d9d1" />
# There exist exactly 92 Convex Polyhedra with Regular Polygonal faces (and not necessary equivalent vertices).<ref name="ref_9442d9d1" />
# Polyhedra with identical Vertices related by a symmetry operation are known as Uniform Polyhedra.<ref name="ref_9442d9d1" />
# 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.<ref name="ref_c29ebbae">[https://www.cut-the-knot.org/do_you_know/polyhedra.shtml Regular Polyhedra]</ref>
# For a polyhedron it means about the same thing with surface curves shrunk into a point while staying on the surface.<ref name="ref_c29ebbae" />
# Simple polyhedra can be continuously deformed into a sphere and, in addition, each of its faces is simple.<ref name="ref_c29ebbae" />
# 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.<ref name="ref_c29ebbae" />
# A solid shape bounded by polygons is called a polyhedron.<ref name="ref_f25bd5d4">[https://www.toppr.com/guides/maths/visualising-solid-shapes/polyhedrons/ Polyhedrons: Regular Polyhedron, Prism, Pyramids, Videos and Examples]</ref>
# Polygons forming a polyhedron are known as its faces.<ref name="ref_f25bd5d4" />
# Line segments common to intersecting faces of a polyhedron are known as its edges.<ref name="ref_f25bd5d4" />
# In a polyhedron, three or more edges meet at a point to form a vertex.<ref name="ref_f25bd5d4" />
# 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.<ref name="ref_f13ca6ab">[https://encyclopedia2.thefreedictionary.com/polyhedra polyhedra]</ref>
# From this standpoint, a polyhedron is a surface made up of polygonal pieces.<ref name="ref_f13ca6ab" />
# If this surface does not intersect itself, then it is the complete surface of some geometric solid, which is also called a polyhedron.<ref name="ref_f13ca6ab" />
# This leads to a third view of polyhedrons as geometric solids.<ref name="ref_f13ca6ab" />
# Section 2 describes the approach for the construction of enclosing and enclosed ellopsoids of convex polyhedra.<ref name="ref_f6258d78">[https://dictionary.cambridge.org/dictionary/english/polyhedron meaning in the Cambridge English Dictionary]</ref>
# Alternatively, ' 'paper bag' ' icons can mark the position of polyhedra, revealing their size but not their shape.<ref name="ref_f6258d78" />
# For simulations, the links of redundant robots are modeled by a union of line segments or convex polyhedra.<ref name="ref_f6258d78" />
# Therefore, the proposed method just focuses on generating an enclosed ellopsoid, which is as large as possible, to fit the polyhedron tightly.<ref name="ref_f6258d78" />
# Let M be a closed convex polyhedron with no holes which is composed of no polygons other than pentagons and hexagons.<ref name="ref_cdc92f37">[https://www.sjsu.edu/faculty/watkins/eulergem.htm Euler's Theorem Concerning Polyhedra Composed of Pentagons and Hexagons: There Must Be Exactly 12 Pentagons]</ref>
# If we traverse the polyhedron face-by-face counting the number of edges we will get 6h+5p.<ref name="ref_cdc92f37" />
# If we traverse the vertices of the polyhedron counting edges we will get 3v.<ref name="ref_cdc92f37" />
# When h=0, the polyhedron is the dodecahedron having twelve pentagons with 20 vertices and 30 edges.<ref name="ref_cdc92f37" />
# The above figure shows a special set of polyhedrons called the five regular solids.<ref name="ref_29411f24">[https://www.dummies.com/education/math/pre-algebra/how-to-recognize-a-polyhedron/ How to Recognize a Polyhedron]</ref>
# Angle of the polyhedron: It is the proportion of space limited by three or more planes that meet at a point called vertex.<ref name="ref_2e7c2506">[https://www.sangakoo.com/en/unit/polyhedrons-basic-definitions-and-classification Polyhedrons: basic definitions and classification]</ref>
# The polyhedrons can be classified under many groups, either by the family or from the characteristics that differentiate them.<ref name="ref_2e7c2506" />
# A polyhedron is a closed, three-dimensional solid bounded entirely by at least four polygons, no two of which are in the same plane.<ref name="ref_1d3d4a16">[https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/polyhedron Encyclopedia.com]</ref>
# The number of sides of each polygon is the major feature distinguishing polyhedrons from one another.<ref name="ref_1d3d4a16" />
# Each of the polygons of a polyhedron is called a face.<ref name="ref_1d3d4a16" />
# 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).<ref name="ref_1d3d4a16" />
# (Examples #1-7) 00:10:38 – How do we classify regular and convex polyhedron?<ref name="ref_9a4968cc">[https://calcworkshop.com/volume-surface-area/polyhedron/ What is a Polyhedron? Simply Explained w/ 14 Examples!]</ref>
# "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").<ref name="ref_ce951265">[https://maths.ac-noumea.nc/polyhedr/start_.htm definitions polyhedra]</ref>
# 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.<ref name="ref_ce951265" />
# A polyhedron ( plural polyhedra) is a three-dimensional solid with flat polygon faces joined at their edges.<ref name="ref_83d4e3e0">[https://mathbitsnotebook.com/Geometry/3DShapes/3DPolyhedra.html MathBitsNotebook(Geo]</ref>
# A polyhedron's faces are bounding surfaces consisting of portions of intersecting planes.<ref name="ref_83d4e3e0" />
# Polyhedra are not necessarily constructed using the same shaped polygons for faces.<ref name="ref_83d4e3e0" />
# The skeletal polyhedron at the right, and the one shown above, are examples where the faces have varying numbers of sides.<ref name="ref_83d4e3e0" />
# When a convex polyhedron (or polytope) has dimension , it is called a -polyhedron ( -polytope).<ref name="ref_22de81bd">[https://www.irisa.fr/polylib/DOC/node16.html Theoretical background]</ref>
# In Polylib the decomposition theorem is extensively used (in its extended form for polyhedra).<ref name="ref_22de81bd" />
# A polyhedron can be represented by a set of inequalities (usually, implicit equalities are represented in a separate matrix): , this representation is called implicit.<ref name="ref_22de81bd" />
# Although the polyhedra theory cannot be detailed here, we review a set of important concepts that are used when manipulating polyhedra.<ref name="ref_22de81bd" />
===소스===
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← 이전 판		2020년 12월 26일 (토) 12:17 판
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2021년 2월 17일 (수) 08:09에 Pythagoras0님의 편집

Pythagoras0: /* 메타데이터 */ 새 문단

Pythagoras0: /* 노트 */ 새 문단