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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.[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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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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Prism (geometry)
In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.Contents1 General, right and uniform prisms 2 Volume 3 Surface area 4 Schlegel diagrams 5 Symmetry 6 Prismatic polytope6.1 Uniform prismatic polytope7 Twisted prism 8 Star prism8.1 Crossed prism 8.2 Toroidal prisms9 See also 10 References 11 External linksGeneral, right and uniform prisms[edit] A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.[1] This applies if the joining faces are rectangular
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Uniform Polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive)
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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.Contents1 Definitions1.1 As a flat slice 1.2 As a spherical polygon 1.3 As the set of connected vertices 1.4 Abstract definition2 General properties 3 Constructions3.1 From the adjacent vertices 3.2 Dorman Luke
Dorman Luke
construction 3.3 Regular polytopes4 An example vertex figure of a honeycomb 5 Edge figure 6 See also 7 References7.1 Notes 7.2 Bibliography8 External linksDefinitions[edit]"Whole-edge" vertex figure of the cubeSpherical vertex figure of the cubePoint-set vertex figure of the cubeTake some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex
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Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e
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Rhombic Dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.Contents1 Properties 2 Dimensions 3 Area and volume 4 Orthogonal projections 5 Cartesian coordinates 6 Topologically equivalent forms6.1 Parallelohedron6.1.1 Dihedral rhombic dodecahedron 6.1.2 Bilinski dodecahedron6.2 Deltoidal dodecahedron7 Related polyhedra7.1 Stellations8 Related polytopes 9 See also 10 References 11 Further reading 12 External links12.1 Computer models 12.2 Paper projects 12.3 Practical applicationsProperties[edit] The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron
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Intersphere
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere. The midsphere is so-called because, for polyhedra that have a midsphere, an inscribed sphere (which is tangent to every face of a polyhedron) and a circumscribed sphere (which touches every vertex), the midsphere is in the middle, between the other two spheres. The radius of the midsphere is called the midradius.Contents1 Examples 2 Tangent circles 3 Duality 4 Canonical polyhedron 5 Notes 6 References 7 External linksExamples[edit] The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres
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Reflection Through The Origin
In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection
Point reflection
can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1
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Regular Polygon
Regular polygonsEdges and vertices nSchläfli symbol n Coxeter–Dynkin diagramSymmetry group Dn, order 2nDual polygon Self-dualArea (with side length, s) A = 1 4 n s 2 cot ⁡ ( π n ) displaystyle A= tfrac 1 4 ns^ 2 cot left( frac pi n right) Internal angle ( n − 2 ) × 180 ∘ n displaystyle (n-2)times frac 180^ circ n Internal angle
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Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual. A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid has a regular polygon base and is usually implied to be a right pyramid.[1][2] When unspecified, a pyramid is usually assumed to be a regular square pyramid, like the physical pyramid structures. A triangle-based pyramid is more often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called acute if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base
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Elongated Pyramid
In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal prism. Along with the set of pyramids, these figures are topologically self-dual. There are three elongated pyramids that are Johnson solids made from regular triangles and square, and pentagons. Higher forms can be constructed with isosceles triangles. Forms[edit]name faceselongated triangular pyramid (J7) 3+1 triangles, 3 squareselongated square pyramid (J8) 4 triangles, 4+1 squareselongated pentagonal pyramid (J9) 5 triangles, 5 squares, 1 pentagonSee also[edit]Gyroelongated bipyramid Elongated bipyramid Gyroelongated pyramid Diminished trapezohedronReferences[edit]Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. Victor A. Zalgaller (1969)
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Square Pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.Contents1 Johnson solid
Johnson solid
(J1) 2 Other square pyramids 3 Related polyhedra and honeycombs3.1 Dual polyhedron4 Topology 5 Examples 6 References 7 External links Johnson solid
Johnson solid
(J1)[edit] If the sides are all equilateral triangles, the pyramid is one of the Johnson solids (J1). The 92 Johnson solids were named and described by Norman Johnson in 1966. A Johnson solid
Johnson solid
is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1] The Johnson square pyramid can be characterized by a single edge-length parameter a
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Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The word "partial" in the names "partial order" or "partially ordered set" is used as an indication that not every pair of elements need be comparable
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Pentagonal Pyramid
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the vertex). Like any pyramid, it is self-dual. The regular pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (J2)
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Hexagonal Pyramid
In geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which are erected six triangular faces that meet at a point (the apex). Like any pyramid, it is self-dual. A right hexagonal pyramid with a regular hexagon base has C6v symmetry. A right regular pyramid is one which has a regular polygon as its base and whose apex is "above" the center of the base, so that the apex, the center of the base and any other vertex form a right triangle. Related polyhedra[edit]Regular pyramidsDigonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal...Improper Regular Equilateral IsoscelesSee also[edit]Bipyramid, prism and antiprismExternal links[edit]Weisstein, Eric W. "Hexagonal Pyramid". MathWorld.  Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of PolyhedraConway Notation for Polyhedra Try: "Y6"This polyhedron-related article is a stub
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