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Topic: Dual polyhedron

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regular polyhedron

Uniform polyhedron

Archimedean polyhedron

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	Duality
		The dual to the tetrahedron, {3, 3}, is another tetrahedron, {3, 3}, facing in the opposite directions.
		The duals of the prisms and antiprisms are the dipyramids and trapezohedra.
		In all these cases you might observe that the polyhedron and its dual have the same axes of symmetry.
	www.georgehart.com /virtual-polyhedra/duality.html (557 words)

	Dual polyhedron - Definition, explanation
		The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges.
		The vertices of the dual, then, are the reciprocals of the face planes of the original, and the faces of the dual lie in the reciprocals of the vertices of the original.
		It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph.
	www.calsky.com /lexikon/en/txt/d/du/dual_polyhedron.php (0 words)

	A Self-Dual Hendecahedron
		The principle of duality states that for every polygon there is a dual, or reciprocal, polygon, whose edges correspond to the vertices (corner points) of the original and whose vertices likewise correspond to its edges.
		To obtain the dual of a polygon, the original is reciprocated with respect to a circle.
		To obtain the dual of a polyhedron, the original is reciprocated with respect to a sphere.
	www.steelpillow.com /polyhedra/selfdualhen/selfdualhen.htm (996 words)

	Stellating the icosahedron and faceting the dodecahedron
		Similarly in three dimensions, the dual of a polyhedron may be obtained by reciprocation with respect to a sphere.
		The stellation diagram of the icosahedron is reciprocal to the faceting diagram of the dodecahedron.
		Just possibly its dual is a different polyhedron which is also both a stellation of the icosahedron and a faceting of the dodecahedron and has congruent face and vertex figure dual to those of 192.
	www.steelpillow.com /polyhedra/icosa/stelfacet/StelFacet.htm (4982 words)

	What's In This Polyhedron?
		However, it was not until I understood that the polyhedron that she was describing was mixed, that is concave/convex (the polyhedron had bumps), that I began to explore other possibilities.
		The Octahedron is the "dual" polyhedron of the Cube.
		A "dual" of a polyhedron is (loosely) defined by replacing all vertices with faces and all faces by vertices.
	www.rwgrayprojects.com /Lynn/NCH/whatpoly.html (0 words)

	Polyhedron Summary
		The Platonic solids are within the larger grouping known as regular polyhedrons, in which the polygons of each are regular and congruent (that is, all polygons are identical in size and shape and all edges are identical in length), and are characterized by the same number of polygons meeting at each vertex.
		A polyhedron is a three-dimensional analog of a polygon.
		Face-uniformity of a polyhedron corresponds to vertex-uniformity of the dual and conversely, and edge-uniformity of a polyhedron corresponds to edge-uniformity of the dual.
	www.bookrags.com /Polyhedron (2968 words)

	Uniform Tessellations and Polyhedra (Site not responding. Last check: )
		The vertices of the dual polyhedron are the poles of the face planes of the original polyhedron.
		The face planes of the dual polyhedron are the polars of the vertices of the original one.
		The dual edges are tangent to the midsphere, and those which correspond to the edges of an original face meet in a point over the center of that face; of course, this is the pole of that facial plane.
	www.monmouth.com /~chenrich/UniformTessellations.html (2796 words)

	Relationships (Site not responding. Last check: )
		The Inter-relationship of the Cube, Tetrahedron and the Octahedron
		The octahedron is the dual of the cube
		Dual construction is an operation "of order 2" by which is meant that taking the dual of the dual will give back the original solid.
	www.ul.ie /~cahird/polyhedronmode/relation.htm (461 words)

	polyhedron
		Sometimes the term "polyhedron" is used to apply to figures in more than three dimensions; however, analogs of polyhedra in the fourth dimension or higher are also referred to as polytopes.
		A dual of a polyhedron is another polyhedron in which faces and vertices occupy complementary locations.
		The duals of the Archimedean solids are known as the Catalan solids.
	www.daviddarling.info /encyclopedia/P/polyhedron.html (445 words)

	Dual Polyhedron \| polyhedra.mathmos.net
		As the name suggests, the dual of the dual of a polyhedron is the original polyhedron itself.
		Technically the dual can be defined by inversion of a polyhedron with respect to its inter-sphere, but its easier to think of it being acheieved by replacing each face with a vertex, each vertex with a face, and rotating all of the edges through 90°.
		The dual of a polyhedron with vertex symbol
	polyhedra.mathmos.net /entry/dualpolyhedron.html (133 words)

	[No title]
		The index file consists of lines composed of the polyhedron number followed by a horizontal tab and the polyhedron's name.
		The fields include, but are not limited to, number the polyhedron's number (written and read with the %d printf/scanf format).
		dual the name of the dual polyhedron optionally followed by a horizontal tab and the number of the dual.
	www.netlib.org /polyhedra/poly.n (0 words)

	Search ScienceWorld
		The dual of the ditrigonal dodecadodecahedron U_(41) and Wenninger dual W_(80), whose outward appearance is the same as the great triambic icosahedron (the dual of the great ditrigonal icosidodecahedron), since the internal vertices are hidden from view.
		The polyhedron compound of the truncated icosahedron and its dual, the pentakis dodecahedron.
		The convex hull of the octahemioctahedron is the
	scienceworld.wolfram.com /search/index.cgi?num=&q=Polyhedron&start=150 (365 words)

	duality (Site not responding. Last check: )
		If these two points are two vertices of a polyhedron, linked by an edge, then the corresponding faces of the dual are in the polar planes, and their common edge belongs to the intersection line of the two planes.
		Duality exchanges the number of faces and of vertices and maintains the number of edges; to a face corresponds a vertex of same order and vice versa.
		This necessary condition is not sufficient; to be geometrically self dual a pyramid must be regular and its main vertex must lay at a suitable distance from the base (in fact it must be canonical).
	www.ac-noumea.nc /maths/amc/polyhedr/dual_.htm (840 words)

	Activity 14
		A regular polyhedron is a polyhedron that is composed of congruent regular polygons with the same vertex code everywhere.
		One regular polyhedron is the dual of another if the polyhedra have the same number of edges but the number of faces of one equals the number of vertices of the other.
		A semi-regular polyhedron is a polyhedron that is composed of several different regular polygons with the same vertex code everywhere.
	homepage.mac.com /efithian/Geometry/Activity-14.html (727 words)

	Poly Format
		Inequalities that describe half-spaces such that the polyhedron is their intersection.
		Dual basis of the affine hull of the polyhedron.
		Dimension of the affine hull of the polyhedron = dimension of the polyhedron.
	www.eg-models.de /formats/Format_Poly.html (1068 words)

	Archimedean Duals
		When one forms the dual of a given polyhedron, one creates a new polyhedron in which the faces and vertices of the dual correspond to the vertices and faces, respectively, of the original polyhedron.
		In these Archimedean duals, the faces are not regular polygons; there are different lengths and angles within each face, but each face is identical and the vertex figures are regular.
		The relationships between the Archimedean solids and their respective duals is nicely brought out by studying a compound of a solid and its dual.
	www.georgehart.com /virtual-polyhedra/archimedean-duals-info.html (436 words)

	Tetrahedrally Stellated Icosahedron
		This is a polyhedron (or three) which I just happen to like, so I thought it should get its own web page.
		The dual operation to stellating four faces of the icosahedron is truncating four vertices of the dodecahedron.
		Choose four which are at the vertices of a tetrahedron inscribed in the dodecahedron., and maximally truncate those four corners of the dodecahedron to get this tetrahedrally truncated dodecahedron.
	www.georgehart.com /virtual-polyhedra/tetrahedrally_stellated_icosahedron.html (682 words)

	Tom Gettys - Dual of a Polyhedron (Site not responding. Last check: )
		A polyhedron P and its dual P* have the same number of edges, but the roles of the vertices and faces are reversed.
		Step 1 tells us that the number of faces of a polyhedron is equal to the number of vertices of its dual.
		Recall that the dual of a polyhedron exchanges the roles of vertices and faces.
	home.comcast.net /~tpgettys/duals.html (0 words)

	Polyhedron, Polyhedra, Polytopes - Numericana
		The dual of a polyhedron is the polyhedron obtained by switching the roles of vertices and faces: Edges of the dual connects nodes associated with adjacent faces of the original polyhedron (dual polyhedroa have the same number of edges).
		In such families, the polyhedron is named using the adjective corresponding to the name of the polygon it's built on (e.g., "hexagonal").
		By duality, this means that every face of the Szilassi heptahedron has an edge in common with each of the other 6 faces...
	home.att.net /~numericana/answer/polyhedra.htm (5442 words)

	The Regular Polyhedra (Site not responding. Last check: )
		One method of dualizing a polyhedron is by inversion in a sphere: Position a sphere of suitable radius R with its center C at some significant point within the polyhedron, such as at its center of symmetry or its centroid.
		The internal convex polyhedron common to both components is an octahedron, and the corners of the compound belong to a cube; the Schläfli symbol for the compound includes the Schläfli symbol for the cube, {4,3}, and the octahedron, {3,4}, as well as for the two tetrahedra themselves, 2{3,3}.
		The polyhedron common to both star-polytopes in the compound is a dodecadodecahedron; its faces are twelve pentagons and twelve pentagrams.
	members.aol.com /Polycell/regs.html (11220 words)

	Creating solid networks
		3: (1) outer faces of struts are made in the planes of the polyhedron faces, (2) the inner faces are parallel to the outer faces, and (3) the "side faces" are perpendicular to the inner and outer faces.
		This is simply the dual to the convex hull of the points where the segments intersect a small sphere.
		The dual to the octahedron is a cube (Fig.
	arpam.free.fr /hart.htm (2600 words)

	Multidimensional Glossary (Site not responding. Last check: )
		That is, the dual of a polytope whose vertices are truncated away may be constructed by apiculating pyramids onto the facets of the dual that correspond to the truncated vertices.
		It is the former that is the dual of the hollow pentagon.
		That is, the dual of a stellation of a polytope is a faceting of the dual of the polytope.
	members.aol.com /Polycell/glossary.html (16070 words)

	What the Origami Means
		A symmetry of a polyhedron is a way of moving the polyhedron so that it occupies the same physical space as before it was moved.
		A polyhedron is regular if there is a symmetry taking any face to any other face and a symmetry taking any vertex to any other vertex, and a polyhedron with regular faces is semi-regular if there is a symmetry taking any vertex to any other vertex.
		The reason for this stems from the dual polyhedron construction described elsewhere on this page; if we take a polyhedron and construct the dual inside it, then any symmetric move of the polyhedron will cause the inscribed dual to occupy the same space as it did before, and vice versa.
	www.amherst.edu /~sgoldstine/origami/displaytext.html (2729 words)

	Reciprocal Polyhedra
		The edges and vertices of a polyhedron constitute a special case of a graph, which is a set of N0 points or nodes, joined in pairs by N1 segments or branches.
		Duality is a symmetric relation: a map is the dual of its dual.
		In Maple, one can define a duality of a regular polyhedron or Archimedean solid via the command duality(dualp,p,s); where dualp is the name of the reciprocal polyhedron of the given polyhedron p with respect to the sphere s which is concentric with p (i.e., s and p have the same center).
	www.cecm.sfu.ca /~hle/polyhedra/duality.html (540 words)

	Platonic Solids
		The duals of Platonic solids are other Platonic solids and, in fact, the dual of the tetrahedron is another tetrahedron.
		The icosahedron is the Platonic solid having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces.
		Every polyhedron has a dual, another polyhedron in which faces and polyhedron vertices occupy complementary locations.
	mcs.une.edu.au /~cwatson7/PlatonicSolids.htm (631 words)

	Platonic and Archimedean (Site not responding. Last check: )
		The others are the cube, the octahedron (with 8 equilateral triangle faces, made by gluing together the bases of two square pyramids with equal edge lengths), the icosahedron (with 20 equilateral triangular faces), and the dodecahedron (with 12 pentagonal faces).
		A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, for example) but the numbers of vertices and faces are interchanged.
		It helps to imagine the polyhedron "flattened" by projecting it onto a plane, making a flat graph such that each region bounded by the edges corresponds to a face of the polyhedron, with the whole of the space outside the outer edge corresponding to one of the faces.
	mcraefamily.com /MathHelp/GeometrySolidPlatonic.htm (607 words)

	Reciprocal Polyhedra
		The edges and vertices of a polyhedron constitute a special case of a graph, which is a set of N0 points or nodes, joined in pairs by N1 segments or branches.
		Duality is a symmetric relation: a map is the dual of its dual.
		In Maple, one can define a duality of a regular polyhedron or Archimedean solid via the command duality(dualp,p,s); where dualp is the name of the reciprocal polyhedron of the given polyhedron p with respect to the sphere s which is concentric with p (i.e., s and p have the same center).
	oldweb.cecm.sfu.ca /~hle/polyhedra/duality.html (540 words)

	Platonic solid Summary
		One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure.
		Every polyhedron has an associated symmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant.
		The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π.
	www.bookrags.com /Platonic_solid (4116 words)

	What is a Dodecahedron?
		The dual of the dual is the original polyhedron.
		The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges.
		Kepler solid is a regular nonconvex polyhedron, all the faces of which are regular polygons and which has the same number of faces meeting at all its vertices.
	www.francesfarmersrevenge.com /stuff/misc/dodecahedron/index.htm (418 words)

	Index: Platonic and Archimedean Solids (69-79)
		Similarly, the dodecahedron and the icosahedron are duals of each other, and the tetrahedron is its own dual.
		A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, but the numbers of vertices and faces are interchanged).
		The cuboctahedron and the icosidodecahedron (72) are obtained by taking the volume simultaneously enclosed by a regular polyhedron and its dual of the same radius (performing the same process with a tetrahedron yields an octahedron).
	math.arizona.edu /~models/Platonic_and_Archimedean_Solids (376 words)

	Regular Polyhedra Project
		A polyhedron is called convex if the line segment joining any two points in the interior of the figure lies completely within the figure.
		After this transformation, the vertices of the polyhedron correspond to intersection points in the plane, edges correspond to segments, and faces correspond to regions (including the exterior "region", which also began as one of the faces of the polyhedron).
		For a polyhedron with V vertices, E edges, and F faces, Euler's formula states that V, E, and F are related by the equation V – E + F = 2.
	www.math.rutgers.edu /~erowland/polyhedra-project.html (718 words)

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