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Petrie Polygon
In geometry , a PETRIE POLYGON for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive side (but no n) belongs to one of the facets . The PETRIE POLYGON of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side (but no three) belongs to one of the faces . For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon
Petrie polygon
becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane
Coxeter plane
of the symmetry group of the polygon, and the number of sides, h, is Coxeter number of the Coxeter group
Coxeter group
. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes. CONTENTS * 1 History * 2 The Petrie polygons of the regular polyhedra * 3 The Petrie polygon
Petrie polygon
of regular polychora (4-polytopes) * 4 The Petrie polygon
Petrie polygon
projections of regular and uniform polytopes * 5 Notes * 6 References * 7 External links HISTORY The Petrie polygon
Petrie polygon
for a cube is a skew hexagon passing through 6 of 8 vertices
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Regular Icosahedron
In geometry , a REGULAR ICOSAHEDRON (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/ ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids , and also the one with the most sides. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol
Schläfli symbol
{3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 35. It is the dual of the dodecahedron , which is represented by {5,3}, having three pentagonal faces around each vertex. A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι (eíkosi), meaning 'twenty', and ἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedrons" or "icosahedra" (/-drə/ )
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Perspective Projection
PERSPECTIVE (from Latin : perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight. Italian Renaissance
Italian Renaissance
painters and architects including Filippo Brunelleschi , Masaccio
Masaccio
, Paolo Uccello
Paolo Uccello
, Piero della Francesca
Piero della Francesca
and Luca Pacioli studied linear perspective, wrote treatises on it, and incorporated it into their artworks, thus contributing to the mathematics of art
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Net (polyhedron)
In geometry a NET of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer , whose 1525 book Unterweysung der Messung mit dem Zyrkel und Rychtscheyd included nets for the Platonic solids and several of the Archimedean solids . CONTENTS * 1 Existence and uniqueness * 2 Shortest path * 3 Higher-dimensional polytope nets * 4 See also * 5 References * 6 External links EXISTENCE AND UNIQUENESSMany different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together. If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive angular defect and such that the sum of these defects is exactly 4π, then there necessarily exists exactly one polyhedron that can be folded from it; this is Alexandrov\'s uniqueness theorem
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Orthogonal Projection
In linear algebra and functional analysis , a PROJECTION is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent ). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection . One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. CONTENTS* 1 Simple example * 1.1 Orthogonal projection * 1.2 Oblique projection * 2 Properties and classification * 2.1 Orthogonal projections * 2.1.1 Properties and special cases * 2.1.1.1 Formulas * 2.2 Oblique projections * 3 Canonical forms * 4 Projections on normed vector spaces * 5 Applications and further considerations * 6 Generalizations * 7 See also * 8 Notes * 9 References * 10 External links SIMPLE EXAMPLEORTHOGONAL PROJECTIONFor example, the function which maps the point (x, y, z) in three-dimensional space R3 to the point (x, y, 0) is an orthogonal projection onto the x–y plane. This function is represented by the matrix P = . {displaystyle P={begin{bmatrix}1&0&0\0&1&0\0&0 width:17.559ex; height:9.176ex;" alt="P={begin{bmatrix}1&0&0\0&1&0\0&0"> P ( x y z ) = ( x y 0 )
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Schlegel Diagram
In geometry , a SCHLEGEL DIAGRAM is a projection of a polytope from R d {displaystyle R^{d}} into R d 1 {displaystyle R^{d-1}} through a point beyond one of its facets or faces . The resulting entity is a polytopal subdivision of the facet in R d 1 {displaystyle R^{d-1}} that is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel , who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a 4-polytope
4-polytope
to 3-space , respectively. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes. CONTENTS * 1 Construction * 2 Examples * 3 See also * 4 References * 5 Further reading * 6 External links CONSTRUCTIONThe most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows: A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces
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Vertex Figure
In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. 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 space-filling tessellation with polytope cells . AS A FLAT SLICEMake a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This is perhaps the most common approach, and the most easily understood
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; _geo-_ "earth", _-metron_ "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer . Geometry arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages . By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat . Since then, and into modern times, geometry has expanded into non- Euclidean geometry and manifolds , describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry
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Regular Polytope
In mathematics , a REGULAR POLYTOPE is a polytope whose symmetry group acts transitively on its flags , thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube ). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures . These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes . A regular polytope can be represented by a Schläfli symbol
Schläfli symbol
of the form {a, b, c, ...., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}
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Skew Polygon
In geometry , a SKEW POLYGON is a polygon whose vertices are not all coplanar . Skew polygons must have at least 4 vertices . The interior surface (or area) of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all collinear. A ZIG-ZAG SKEW POLYGON or ANTIPRISMATIC POLYGON has vertices which alternate on two parallel planes, and thus must be even-sided. REGULAR SKEW POLYGON in 3 dimensions (and regular skew apeirogons in 2 dimensions) are always zig-zag. CONTENTS* 1 Antiprismatic skew polygon in 3 dimensions * 1.1 Isogonal skew polygons in 3 dimensions * 2 Regular skew polygons in 4 dimensions * 3 See also * 4 References * 5 External links ANTIPRISMATIC SKEW POLYGON IN 3 DIMENSIONS A uniform n-gonal antiprism has a 2n-sided regular skew polygon defined along its side edges. A REGULAR SKEW POLYGON is isogonal with equal edge lengths. In 3 dimensions a regular skew polygon is a zig-zag skew (or ANTIPRISMATIC POLYGON), with vertices alternating between two parallel planes. The sides of an n-antiprism can define a regular skew 2n-gons. A regular skew n-gonal can be given a symbol {p}#{ } as a blend of a regular polygon , {p} and an orthogonal line segment , { }. The symmetry operation between sequential vertices is glide reflection . Examples are shown on the uniform square and pentagon antiprisms
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Edge (geometry)
In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higher-dimensional polytope . In a polygon, an edge is a line segment on the boundary, and is often called a SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn graph theory , an edge is an abstract object connecting two graph vertices , unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz\'s theorem as being exactly the 3-vertex-connected planar graphs . NUMBER OF EDGES IN A POLYHEDRONAny convex polyhedron 's surface has Euler characteristic V E + F = 2 , {displaystyle V-E+F=2,} where V is the number of vertices , E is the number of edges, and F is the number of faces
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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. * 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 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 simplicial polytopes this coincides with the meaning from polyhedral combinatorics.REFERENCES * ^ Bridge, N.J. Facetting the dodecahedron, Acta crystallographica A30 (1974), pp. 548–552. * ^ Inchbald, G. Facetting diagrams, The mathematical gazette, 90 (2006), pp. 253–261. * ^ Coxeter, H. S. M. (1973), Regular Polytopes, Dover, p. 95 . * ^ Matoušek, Jiří (2002), Lectures in Discrete Geometry, Graduate Texts in Mathematics , 212, Springer, 5.3 Faces of a Convex Polytope, p. 86 . * ^ De Loera, Jesús A
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Regular Polygon
Regular polygons Edges and vertices _n_ Schläfli symbol {_n_} Coxeter–Dynkin diagram Symmetry group Dn , order 2n Dual polygon Self-dual Area
Area
(with _s_=side length) A = 1 4 n s 2 cot n {displaystyle A={tfrac {1}{4}}ns^{2}cot {frac {pi }{n}}} Internal angle ( n 2 ) 180 n {displaystyle (n-2)times {frac {180^{circ }}{n}}} Internal angle sum ( n 2 ) 180 {displaystyle left(n-2right)times 180^{circ }} Inscribed circle diameter d I C = s cot n {displaystyle d_{IC}=scot {frac {pi }{n}}} Circumscribed circle diameter d O C = s csc n {displaystyle d_{OC}=scsc {frac {pi }{n}}} Properties convex , cyclic , equilateral , isogonal , isotoxal In Euclidean geometry , a REGULAR POLYGON is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be CONVEX or STAR . In the limit , a sequence of regular polygons with an increasing number of sides approximates a circle , if the perimeter is fixed, or a regular apeirogon , if the edge length is fixed
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Regular Polyhedron
A REGULAR POLYHEDRON is a polyhedron whose symmetry group acts transitively on its flags . A regular polyhedron is highly symmetrical, being all of edge-transitive , vertex-transitive and face-transitive . In classical contexts, many different equivalent definitions are used; a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex . A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, known as the Platonic solids . These are the: tetrahedron {3, 3}, cube {4, 3}, octahedron {3, 4}, dodecahedron {5, 3} and icosahedron {3, 5}. There are also four regular star polyhedra , making nine regular polyhedra in all
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Face (geometry)
In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes , the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). CONTENTS* 1 Polygonal face * 1.1 Number of polygonal faces of a polyhedron * 2 k-face * 2.1 Cell or 3-face
3-face
* 2.2 Facet or (n-1)-face * 2.3 Ridge or (n-2)-face * 2.4 Peak or (n-3)-face * 3 See also * 4 References * 5 External links POLYGONAL FACEIn elementary geometry, a FACE is a polygon on the boundary of a polyhedron . Other names for a polygonal face include SIDE of a polyhedron, and TILE of a Euclidean plane tessellation . 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
4-polytope
. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Regular examples by Schläfli symbol
Schläfli symbol
POLYHEDRON STAR POLYHEDRON EUCLIDEAN TILING HYPERBOLIC TILING 4-POLYTOPE {4,3} {5/2,5} {4,4} {4,5} {4,3,3} The cube has 3 square faces per vertex
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Coxeter Plane
In mathematics , the COXETER NUMBER h is the order of a COXETER ELEMENT of an irreducible Coxeter group
Coxeter group
. It is named after H.S.M. Coxeter . CONTENTS * 1 Definitions * 2 Group order * 3 Coxeter elements * 4 Coxeter plane * 5 See also * 6 Notes * 7 References DEFINITIONSNote that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number h of an irreducible root system. A COXETER ELEMENT is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements , which have the same order . * The Coxeter number is the number of roots divided by the rank
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