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Self-dual Polytope
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. Such dual figures remain combinatorial or abstract polyhedra , but not all are also geometric polyhedra. 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. The dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal. Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron
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Cube
In geometry , a CUBE 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 . It has cubical or octahedral symmetry . CONTENTS * 1 Orthogonal projections * 2 Spherical tiling * 3 Cartesian coordinates * 4 Equation in R3 * 5 Formulas * 5.1 Point in space * 6 Doubling the cube * 7 Uniform colorings and symmetry * 8 Geometric relations * 9 Other dimensions * 10 Related polyhedra * 10.1 In uniform honeycombs and polychora * 11 Cubical graph * 12 See also * 13 References * 14 External links ORTHOGONAL PROJECTIONSThe cube has four special orthogonal projections , centered, on a vertex, edges, face and normal to its vertex figure . The first and third correspond to the A2 and B2 Coxeter planes . Orthogonal projections CENTERED BY FACE VERTEX COXETER PLANES B2 A2 Projective symmetry TILTED VIEWSSPHERICAL TILINGThe cube can also be represented as a spherical tiling , and projected onto the plane via a stereographic projection
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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. 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 (ℓ1) metric
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Truncation (geometry)
In geometry , a TRUNCATION is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler
Kepler
's names for the Archimedean solids . CONTENTS * 1 Uniform truncation * 2 Truncation of polygons * 3 Uniform truncation in regular polyhedra and tilings and higher * 4 Edge-truncation * 5 Alternation or partial truncation * 6 Generalized truncations * 7 See also * 8 References * 9 External links UNIFORM TRUNCATIONIn general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a UNIFORM TRUNCATION, a truncation operator applied to a regular polyhedron (or regular polytope ) which creates a resulting uniform polyhedron (uniform polytope ) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron , represented as Schläfli symbols r{5,3} or { 5 3 } {displaystyle {begin{Bmatrix}5\3end{Bmatrix}}} , and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron , represented as tr{5,3} or t { 5 3 } {displaystyle t{begin{Bmatrix}5\3end{Bmatrix}}} ,
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Rectification (geometry)
In Euclidean geometry , RECTIFICATION or COMPLETE-TRUNCATION is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope. A rectification operator is sometimes denoted by the symbol r: for example, r{4,3} is the rectified cube, namely the cuboctahedron. Conway polyhedron notation uses AMBO for this operator. In graph theory this operation creates a medial graph . CONTENTS * 1 Example of rectification as a final truncation to an edge * 2 Higher degree rectifications * 3 Example of birectification as a final truncation to a face * 4 In polygons * 5 In polyhedra and plane tilings * 5.1 In nonregular polyhedra * 6 In 4-polytopes and 3d honeycomb tessellations * 7 Degrees of rectification * 7.1 Notations and facets * 7.1.1 Regular polygons * 7.1.2 Regular polyhedra and tilings * 7.1.3 Regular Uniform 4-polytopes and honeycombs * 7.1.4 Regular 5-polytopes and 4-space honeycombs * 8 See also * 9 References * 10 External links EXAMPLE OF RECTIFICATION AS A FINAL TRUNCATION TO AN EDGERectification is the final point of a truncation process
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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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Polyhedron
In geometry , a POLYHEDRON (plural POLYHEDRA or POLYHEDRONS) is a solid in three dimensions with flat polygonal faces , straight edges and sharp corners or vertices . The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, "many") + -hedron (form of ἕδρα, "base" or "seat"). A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place. OF A POLYTOPEA vertex is a corner point of a polygon , polyhedron , or other higher-dimensional polytope , formed by the intersection of edges , faces or facets of the object. In a polygon, a vertex is called "convex " if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians ( 180°, two right angles ) ; otherwise, it is called "concave" or "reflex"
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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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Abstract Polytope
In mathematics , an ABSTRACT POLYTOPE is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope , but not any purely geometric properties such as angles, edge lengths, etc. An ordinary geometric polytope is said to be a realization in some real N-dimensional space
N-dimensional space
, typically Euclidean , of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory. The term polytope is a generalisation of polygons and polyhedra into any number of dimensions
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Symmetry
SYMMETRY (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection , rotation or scaling . Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together. Mathematical symmetry may be observed with respect to the passage of time ; as a spatial relationship ; through geometric transformations ; through other kinds of functional transformations; and as an aspect of abstract objects , theoretic models , language , music and even knowledge itself. This article describes symmetry from three perspectives: in mathematics , including geometry , the most familiar type of symmetry for many people; in science and nature ; and in the arts, covering architecture , art and music . The opposite of symmetry is asymmetry
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Platonic Solid
In three-dimensional space , a PLATONIC SOLID is a regular , convex polyhedron . It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria: Tetrahedron
Tetrahedron
Cube
Cube
Octahedron
Octahedron
Dodecahedron
Dodecahedron
Icosahedron
Icosahedron
Four faces Six faces Eight faces Twelve faces Twenty faces(Animation ) (Animation ) (Animation ) (Animation ) (Animation ) Geometers have studied the mathematical beauty and symmetry of the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato
Plato
who hypothesized in his dialogue, the _Timaeus _, that the classical elements were made of these regular solids
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Kepler-Poinsot Polyhedra
In geometry , a KEPLER–POINSOT POLYHEDRON is any of four REGULAR STAR POLYHEDRA . They may be obtained by stellating the regular convex dodecahedron and icosahedron , and differ from these in having regular pentagrammic faces or vertex figures . CONTENTS* 1 Characteristics * 1.1 Non-convexity * 1.2 Euler characteristic χ * 1.3 Duality * 1.4 Summary * 2 Relationships among the regular polyhedra * 3 History * 4 Regular star polyhedra in art and culture * 5 See also * 6 References * 7 External links CHARACTERISTICSNON-CONVEXITYThese figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures . In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show golden balls at the true vertices, and silver rods along the true edges. For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid
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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. 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. It has two such nets . For any tetrahedron there exists a sphere (called the circumsphere ) on which all four vertices lie, and another sphere (the insphere ) tangent to the tetrahedron's faces
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Isotoxal Figure
In geometry , a polytope (for example, a polygon or a polyhedron ), or a tiling , is ISOTOXAL or EDGE-TRANSITIVE if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. The term _isotoxal_ is derived from the Greek τοξον meaning _arc_. CONTENTS * 1 Isotoxal polygons * 2 Isotoxal polyhedra and tilings * 3 See also * 4 References ISOTOXAL POLYGONSAn isotoxal polygon is an equilateral polygon , but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons . In general, an isotoxal _2n_-gon will have Dn (*nn) dihedral symmetry . A rhombus is an isotoxal polygon with D2 (*22) symmetry. All regular polygons (equilateral triangle , square , etc.) are isotoxal, having double the minimum symmetry order: a regular _n_-gon has Dn (*nn) dihedral symmetry. A regular 2_n_-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges
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