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Petrie Polygon
In geometry, a Petrie polygon
Petrie polygon
for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides (but no n) belongs to one of the facets. The Petrie polygon
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.[1] 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
Coxeter number
of the Coxeter
Coxeter
group
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Regular Icosahedron
In geometry, a regular icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/[1]) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and 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'
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Dodecahedron
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling
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Patrick Du Val
Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity
Du Val singularity
of an algebraic surface is named after him.Contents1 Early life 2 Research in geometry 3 Later life 4 Work 5 References 6 External linksEarly life[edit] Du Val was born in Cheadle Hulme, Cheshire. He was the son of a cabinet maker, but his parents' marriage broke up. As a child, he suffered ill-health, in particular asthma, and was educated mostly by his mother. He was awarded a first class honours degree from the University of London External Programme
University of London External Programme
in 1926, which he took by correspondence course. He was a talented linguist, for example teaching himself Norwegian so that he might read Peer Gynt
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The Fifty-Nine Icosahedra
The Fifty-Nine Icosahedra
The Fifty-Nine Icosahedra
is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto
University of Toronto
in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell.Contents1 Authors' contributions1.1 Miller's rules 1.2 Coxeter 1.3 Du Val 1.4 Flather 1.5 Petrie 1.6 The Crennells2 List of the fifty nine icosahedra2.1 Notes on the list 2.2 Table of the fifty-nine icosahedra3 See also 4 Notes 5 References 6 External linksAuthors' contributions[edit] Miller's rules[edit] Although Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie
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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
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Semiregular Polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.Contents1 Gosset's list 2 Euclidean honeycombs 3 Hyperbolic honeycombs 4 See also 5 ReferencesGosset's list[edit] In three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell
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Dual Polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.[1] Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra.[2] Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals also belong to a symmetric class. Thus, the regular polyhedra – the (convex) Platonic solids and (star) Kepler-Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.[1] The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper
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Octahedron
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid
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
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Icosahedron
In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/[1]) is a polyhedron with 20 faces. The name comes from Greek εἴκοσι (eíkosi), meaning 'twenty', and ἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons". There are many kinds of icosahedra, with some being more symmetrical than others
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Cube
In geometry, a cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron
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Kepler–Poinsot Polyhedra
In geometry, a Kepler–Poinsot polyhedron
Kepler–Poinsot polyhedron
is any of four regular star polyhedra.[1] They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures.Contents1 Characteristics1.1 Non-convexity 1.2 Euler characteristic
Euler characteristic
χ 1.3 Duality 1.4 Summary2 Relationships among the regular polyhedra 3 History 4 Regular star polyhedra in art and culture 5 See also 6 References 7 External linksCharacteristics[edit] Non-convexity[edit] These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces
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Decagram (geometry)
In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is 10/3 .[1] The name decagram combine a numeral prefix, deca-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs) meaning a line.[2]Contents1 Regular decagram 2 Applications 3 Related figures 4 See also 5 ReferencesRegular decagram[edit] For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.Applications[edit] Decagrams have been used as one of the decorative motifs in girih tiles.[3]Related figures[edit] A regular decagram is a 10-sided polygram, represented by symbol 10/n , containing the same vertices as regular decagon
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Small Stellated Dodecahedron
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol
Schläfli symbol
5/2,5 . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron
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Great Dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol
Schläfli symbol
5,5/2 and Coxeter–Dynkin diagram
Coxeter–Dynkin diagram
of . It is one of four nonconvex regular polyhedra
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