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polyhedron - Search Results - MSN Encarta A polyhedron is a figure bounded by plane surfaces. A polyhedron is a geometric shape which in mathematics is defined by three related meanings. A regular polyhedron is a polyhedron whose faces are identical (or, technically, congruent) regular polygons. encarta.msn.com /polyhedron.html   (204 words)

Encyclopedia: Polyhedron   (Site not responding. Last check: 2007-10-28) A polyhedron is a three-dimensional analog of a polygon. The trapezohedra are the Dual polyhedrons of the antiprisms. A topological polyhedron is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way that needs better description. www.nationmaster.com /encyclopedia/Polyhedron   (2558 words)

Polyhedron   (Site not responding. Last check: 2007-10-28) The polyhedron surrounds a volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron. Any polyhedron which is vertex-uniform can be deformed slightly to form a vertex-uniform polyhedron with regular polygon s as faces. Polyhedron Models Custom Built An introduction to the shapes by George Olshevsky, with their properties, specifications for his custom built models, and links to other pages. www.serebella.com /encyclopedia/article-Polyhedron.html   (775 words)

Polychoron - Wikipedia, the free encyclopedia A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. A uniform polyhedron is a polyhedron that is vertex-transitive, with each face made up of regular polygons. The remaining convex uniform polychora may be grouped into two infinite families: the duoprisms and the polyhedral prisms. en.wikipedia.org /wiki/Polychoron   (688 words)

Uniform Polyhedra In a uniform polyhedron, every face is required to be a regular polygon, and every vertex is required to be identical, but the faces need not be identical. The duals to the uniform polyhedra are facially regular, meaning that they are composed of a single type of face, every face in the same relation to the whole, and their vertex figures are regular polygons. Related to the uniform polyhedra is the great disnub dirhombidodecahedron which is in a different category because some edges are the meeting place of four faces rather than two. www.georgehart.com /virtual-polyhedra/uniform-info.html   (1445 words)

Polyhedron Summary A polyhedron whose faces are all regular polygons congruent to each other, whose polyhedral angels are all equal, and which has the same number of faces meet at each vertex is called a regular polyhedron. 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)

Four Dimensional Figures Page A uniform polyhedron is one whose faces are all regular polygons and any of whose vertices (corners) may be transformed (or carried) into any of its other vertices by its symmetries. Likewise, all the vertices of an n-dimensional uniform polytope are constrained by symmetry to lie on a single n-dimensional sphere centered at the polytope’s center of symmetry. The existence of exactly those 18 convex uniform polyhedra is a fundamental property of three-dimensional Euclidean space, and the existence of the 64 corresponding polychora is likewise a fundamental property of four-dimensional Euclidean space. members.aol.com /Polycell/uniform.html   (4266 words)

Glossary of Terms Various authors differ on the fine points of the definition, e.g., whether it is a solid or just the surface, whether it can be infinite, and whether it can have two different vertices that happen to be at the same location. Stellation - The process of constructing a new polyhedron by extending the face planes of a given polyhedron past their edges. Uniform - A uniform polyhedron has regular faces, with each vertex equivalently arranged. www.ul.ie /~cahird/polyhedronmode/glossary_of_terms.htm   (627 words)

Poliedri   (Site not responding. Last check: 2007-10-28) This solid is a "Uniform polyhedron" in the broad sense since all its faces are regular polygons of the same size and all its vertices are congruent. The invisible core of the star polyhedron is the hexahedron; the surface of this solid is composed of 24 equilateral triangles, 14 vertices and 36 edges. The solution of the polyhedron shown in this plate is obtainable by applying several consecutive geometrical operations constructed on the faces of the dextro snub hexahedron (lower figure in the plate) which is the archetype of the laevo pentagonal icosistetrahedron. g.casagrande.home.att.net /poliedri/poliedri_text.html   (4932 words)

Kaleidoscope Symmetry Groups, Euclidean and non-Euclidean In particular, the spherical kaleidoscope groups can be used to construct all the "uniform polyhedra." These include the five regular polyhedra, and also a larger class called the Archimedean polyhedra. Uniform tessellations can be generalized to the Euclidean and hyperbolic kaleidoscope groups. Besides the uniform tessellations and polyhedra, there are also dual uniform tessellations and polyhedra. www.monmouth.com /~chenrich/Kaleidoscopes/Kaleidoscopes.html   (1066 words)

What Are Polyhedra?   (Site not responding. Last check: 2007-10-28) Above: Photo of my model of a quasitruncated small stellated dodecahedron: a nonconvex uniform polyhedron with 90 edges whose faces are 12 regular decagrams (green) and 12 regular pentagons (pink), two decagrams and one pentagon meeting at each of 60 vertices (corners or points). Being a polyhedron model-maker, however, I prefer a “cut-and-paste definition”: A polyhedron is a finite collection of polygons pasted together along their edges to make a single closed figure in three-dimensional space. nonprismatic uniform “polyhedron”) to the roster, but because it has some coincident edges, it is not a uniform polyhedron according to some definitions and was thus excluded from the original enumeration. hometown.aol.com /Polycell/what.html   (1231 words)

Using Trimeshes Like the polyhedron, the trimesh uses a list of points and a list of triangular faces that contain indices into the list of points. With the polyhedron, mesh, and trigrid formats, you must allocate storage for normals only for those vertices that actually require a normal. Geometric editing operations in immediate mode for the trimesh are similar to those for the polyhedron: you simply alter a point's position in the array in the trimesh data structure and render the shape again. developer.apple.com /documentation/QuickTime/QD3D/qd3dgeometry.a.htm   (806 words)

Voronoi and Delaunay Graphs in Medical Physics Our stratified sampling method which uses partial uniform distributed sampling points, if large implants have to be considered, reduces the number of sampling points by a factor of 3-10. Both methods which we studied are based on the Voronoi diagram which is one of the most fundamental and useful constructs defined by irregular lattices. In contrast in uniform point sampling the weight assigned to each sampling point is constant. www.mlahanas.de /CompGeom/VorComp.htm   (1130 words)

Uniform Polychora Because of this, the number of uniform polychora has been greatly reduced to a more managable 1845 (unless one wishes to include the fissary cases which could bring it closer to 3000). It wasn't until a year later, when I was looking at sidpith, that I noticed that there was a uniform compound of two sodips (square-octagon duoprisms) inside and it was blendable with sidpith and lead to a new uniform polychoron, of course it also had a conjugate in the gittith regiment. Also the degenrate polyhedron cid (complexicosidodecahedron) which has a complete pentagon verf - doesn't have a degenerate polytwister counterpart - instead there are two true polytwister counterparts, one with an antitruncated pentagon (pentagon with triangles dangling off corners) rinf, the other with a semiuniform decagram rinf. www.polytope.net /hedrondude/polychora.htm   (4104 words)

PolyGloss This term is used to describe uniform polytopes that are neither platonic, or of the prismatic classes. The fifth platonic polyhedron, and the seventh uniform polyhedron. An example was presented in Marek Ctrnack's enumeration of the uniform pseudochora of the vertex-type ppppq, where the edge 3 is wrapped from a digon to a triangle. www.geocities.com /os2fan2/gloss.htm   (16747 words)

Feature Column This polyhedron has three rays (which, if extended, should meet at a point) and three line segments as edges of the polyhedron, rather than having edges which are line segments. A regular polyhedron is one in which all faces are congruent regular (convex) polygons and all vertices are "alike." This means that there are the same number of regular polygons at every vertex. In his view the "vertex" of a polyhedron is a solid angle or a part of a "polyhedral cone" that starts at the vertex. www.ams.org /featurecolumn/archive/eulers-formula.html   (4106 words)

Polyhedron   (Site not responding. Last check: 2007-10-28) In mathematics, a polyhedron (from Greek πολυεδρον, frompoly-, stem of πολυς, "many," + -edron, form of εδρον,"base", "seat", or "face") is a three-dimensional shape that is made up of faces, which are parts of planes, the faces meet in edges which are straight-line segments, and the edges meet in points called vertices. The polyhedron surrounds a volume in three-dimensional space; sometimes this interior volume is consideredto be part of the polyhedron. Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieceswhich can be reassembled to yield the second polyhedron. www.therfcc.org /polyhedron-47127.html   (573 words)

How to Make the Mathieu Group M(24) The main ingredient is a genus-3 polyhedron X with 56 triangular faces, 84 edges, and 24 vertices, and which has been immersed in 3-dimensional space as the small cubicuboctahedron. This (uniform) polyhedron has 6 squares and 8 triangles, which should be apparent, and it also has 6 octagons. Since this polyhedron represents an immersion, and not an imbedding, of the polyhedron X into 3-space, there is also a total of 12 false edges, each where two octagons intersect. homepages.wmich.edu /~drichter/mathieu.htm   (1988 words)

Dodecahedron   (Site not responding. Last check: 2007-10-28) A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. A stellation of a polyhedron is formed by extending the faces so that they form a new polyhedron. A stellation of a polyhedron is formed by extending the faces (within their planes) so that they form a new polyhedron. www.experiencefestival.com /dodecahedron   (2020 words)

Stella: Polyhedron Navigator The planes usually come from face planes of a given polyhedron, but there is no obvious reason to restrict the polyhedron to being convex (or even to require that the planes be derived from a polyhedron at all!). The simplest of the uniform polyhedra, the tetrahemihexahedron, is an example of this. The current polyhedron, its dual, or the current stellation of either can be put into any one of the four memories (except for infinite dual models, or stellations with holes in their faces). www.software3d.com /PolyNav/PolyNavigator.html   (12506 words)

Glossary Various authors differ on the fine points of the definition, e.g., whether it is a solid or just the surface, whether it can be infinite, and whether it can have two different vertices that happen to be at the same location. A polyhedron is regular if every face is regular and if every vertex figure is regular. Standardly, there are nine regular polyhedra: the five Platonic solids and the four Kepler-Poinsot solids, but others might be allowed, depending on the definition of polyhedron. www.georgehart.com /virtual-polyhedra/glossary.html   (724 words)

The Archimedean Honeycomb Duals Uniform polyhedra have regular faces and congruent vertices; the convex ones are known as the Archimedean polyhedra (which for our purposes include the uniform prisms and the fully regular Platonic polyhedra). Put concisely; the cells of the dual of a uniform honeycomb are isomorphous with the dual of the vertex figure of that honeycomb. Similarly, any solid polyhedron may be divided up into tetrahedra with three mutually orthogonal edges connected in a kind of 3-D zigzag: the tetrahedron heavily outlined in Figure 4 is an example. www.steelpillow.com /polyhedra/AHD/AHD.htm   (2044 words)

Pedagoguery Software: Poly’s Polyhedra Each platonic polyhedron is constructed using (multiple copies of) a single regular polygon; the same number of polygonal faces is used around each vertex. A polyhedron with regular polygonal faces is uniform if there are symmetry operations that take one vertex through all of the other vertices and no other points in space. The next step is to ensure all of the polyhedron s faces are triangular: each non-triangular face is triangulated by connecting its vertices to a new vertex placed at the center of the face. www.peda.com /poly/poly.html   (549 words)

The Golay Code on the Dodecadodecahedron This is the story about the uniform polyhedron known as the "dodecadodecahedron" and its close relationship with the extended binary linear Golay code of length 24. This polyhedron is "uniform" because its faces are all regular polygons (in a generalized sense), and each of its vertices is congruent to all of the others. The genus-4 polyhedron it represents is regular in a sense only slightly more general than that used to define the usual nine regular polyhedra. homepages.wmich.edu /~drichter/golay.htm   (1979 words)

The Reconciliation of Relativity and Quantum Theories   (Site not responding. Last check: 2007-10-28) This means that the rate at which a “charged” polyhedron pattern “moves” is directly proportional to the number of photon state polyhedron, which the “charged” pattern encounters and from what direction the photon pattern is encountered. The classic “two slit” experiment is completely explained by the assumption that the state of a polyhedron is dependant on the state of the surrounding polyhedrons and that a polyhedron can be in one and only one state at time T PT has its deterministic side in the polyhedron states in which a polyhedron can be in one and only one state and that state is completely determined by the state of the surrounding polyhedrons. polyhedrontheory.com   (2681 words)

polytope: Constructions The pyramid is the convex hull of the input polyhedron P and a point v outside the affine span of P. Construct the vertex figure of the vertex n of a polyhedron The vertex figure is dual to a facet of the dual polytope. Apply an affine transformation to a polyhedron such that a vertex is mapped to the origin (1,0,...,0) and as many facets through this vertex as possible are mapped to the bounding facets of the first orthant. www.math.tu-berlin.de /polymake/apps/polytope/clients.html   (5394 words)

e_klintro.htm The edge-skeleton of some uniform polyhedra can be connected by regular polygons in different ways. The number of "facetings" in the regiments depends on the symetry of their colonel. Richard Klitzing has taken the task to discover all facetings of the uniform polyhedra and to enumerate all of those which have at least a rotational symmetry of an order larger than 2. www.polyhedrix.de /e_klintro.htm   (186 words)

Stella - Create Polyhedra and Nets! Platonic, Archimedean, Catalan, Kepler-Poinsot, uniform, and dual polyhedra, their ... Polyhedron faces may be exploded apart by dragging the mouse with the left button down and the Shift and Ctrl keys pressed. By default, stellation cells are automatically chosen to recreate the original uniform polyhedron. This makes nets for uniform polyhedra easy to obtain, and would be difficult and time-consuming to do by hand. www.software3d.com /Stella.html   (1233 words)

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