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Topic: Cubic polyhedron

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	Geometry of Polyhedrons. (Site not responding. Last check: 2007-10-30)
		Example of a semi-regular polyhedron of 32 faces built of regular hexagons and pentagons.
		Semi-regular polyhedrons built of congruent regular polygons and equilateral triangles.
		Polyhedron of a soccer ball used by UEFA for European Soccer Competitions.
	www.geocities.com /literka/list.htm (353 words)

	Manifold - Wikipedia, the free encyclopedia
		Note that it is possible to construct a circle by "gluing" together a single piece of the line; this does not produce a chart, since a portion of the circle will be mapped to both "glued" regions at once.
		Manifolds need not be connected (all in "one piece"); thus a pair of separate circles is also a manifold.
		And they need not be finite; thus a parabola is a manifold.
	en.wikipedia.org /wiki/Manifold (5743 words)

	Geometry - Three-dimensional figures - Unit Quiz (Site not responding. Last check: 2007-10-30)
		3) The volume of a cube with edges of 5 cm is 125 cubic cm.
		4) The volume of a 3 cm by 5 cm rectangular pyramid with height of 10 cm is 150 cubic cm.
		8) The faces of a polyhedron are all flat.
	www.math.com /school/subject3/S3U4Quiz.html (93 words)

	What's In This Polyhedron? (Part 5)
		The other major polyhedron that Lynnclaire describes from her near death experiences is a 144 triangular faced polyhedron.
		12.4, 12.5 The Cube and Octahedron in the 144 Polyhedron
		Lynnclaire reports that the 120 Polyhedron she experienced is not a static polyhedron.
	www.rwgrayprojects.com /Lynn/NCH/whatpoly5.html (817 words)

	Natural Coordinates
		The length of a base vector in cubic coordinates has been established at 1, since this establishes the length of the edges of the smallest cube with vertices at integer coordinates at 1.
		It is common to abbreviate graphs in Cubic Coordinates to the surface of a plane by omitting the coordinate of one dimension orthogonal to the plane.
		Cubic and Natural vectors may all be scaled by multiplying each coordinate by the same scalar value.
	www.xmission.com /~ray/tmp/ray.jhax.net/NaturalCoordinateSystem/NaturalCoordinateSystem.html (5009 words)

	Space figures and basic solids
		A cube and a pyramid are both polyhedrons; a sphere, cylinder, and cone are not.
		A cross-section of a space figure is the shape of a particular two-dimensional "slice" of a space figure.
		What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and whose height is 6 cm, to the nearest tenth?
	www.mathleague.com /help/geometry/3space.htm (945 words)

	Pseudo Rhombicuboctahedra (Site not responding. Last check: 2007-10-30)
		The pseudo-rhombicuboctahedron is not classified as a semi-regular polyhedron, because the essence (and beauty) of the semi-regular polyhedra is not about local properties of each vertex, but the symmetry operations under which which the entire object appears unchanged.
		He discovered a 14th semiregular polyhedron [figure of pseudo-rhombicuboctahedron] which differs from the one shown in [figure of rhombicuboctahedron] only in that the upper part, consisting of 5 squares and 4 equilateral triangles, is rotated through an angle of pi/4.
		This new polyhedron was first described (as far as I know) in 1994 by R. Hughes Jones (see the references).
	www.georgehart.com /virtual-polyhedra/pseudo-rhombicuboctahedra.html (691 words)

	Waterman Polyhedra
		The Waterman scheme is based on the Cubic Close Packing (CCP), which is well known to physicists and chemists because many crystals have their atoms stacked in a CCP arrangement.
		The "smallest convex polyhedron" is the convex hull of the set of points (the atom centers).
		The first polyhedron (Seq=1) is a cuboctahedron whose vertices correspond to the 12 atoms that are equidistant from the origin (the central atom).
	dogfeathers.com /java/ccppoly.html (2229 words)

	Polyhedral IC package for making three dimensionally expandable assemblies - Patent 6008530
		An integrated IC package comprising a plurality of said polyhedron packages as defined by claim 1, each of said polyhedron packages being assembled to each other by connecting said pins provided on one face of one said polyhedron package to counter pins provided on an opposite face of an opposite and adjacent polyhedron package.
		It is yet further preferable that by mounting terminals on two or more faces of the polyhedron, inspection is made possible after mounting onto a printed circuit board, using a terminal other than a terminal provided for the purpose of mounting onto the printed circuit board.
		In the IC package of the present invention, the configuration of the polyhedron package may preferably a hexahedron and more preferably the hexahedron is an approximate cube.
	www.freepatentsonline.com /6008530.html (4197 words)

	[No title]
		Let us assume, for the moment, that we are investigating a convex uniform polyhedron, such that each of its vertices is incident to m faces, with the ith face being a regular ni‐gon, customarily denoted by the Schläfli symbol {ni}.
		An polyhedron is orientable if its faces may be coherently oriented, that is, assigned orientations in such a way that the orientations induced on an edge common to two faces are opposite.
		The number of the polyhedron vertices may be readily found by dividing the order of the kaleidoscope symmetry group by the number of copies of adjacent Schwarz triangles which share a vertex, i.
	www.math.technion.ac.il /~rl/docs/uniform.txt (4091 words)

	WULFFMAN - CTCMS
		The point group is simply the set of all point isometries (rotations, roto-inversions, and reflections) that leave the environment around a point unchanged.
		In the most general case, the Wulff shape will be a convex polyhedron whose faces (facets) correspond to crystal planes that are low in energy.
		Cubic symmetry is chosen for two Wulff shapes with slightly displaced origins.
	www.ccp14.ac.uk /ccp/web-mirrors/wulffman/wulffman/overview_1.2.html (503 words)

	solid geometry_answers (Site not responding. Last check: 2007-10-30)
		A pyramid is a polyhedron whose base is a polygon and whose other faces are triangles.
		A polyhedron whose bases are parallel congruent polygons and whose other faces are parallelograms is a prism.
		Volume is the number of cubic units needed to fill a solid figure.
	www2.ncsu.edu /midtech/lambert/TIME/solid_geometry_answers.html (154 words)

	Crystallographic Topology - Lattice Complexes
		The tabulation of cubic point configurations by Koch (1984) lists the locations of all point positions with fewer than three degrees of freedom related to sphere packing and Dirichlet partitions within the cubic space group family as well.
		This forms a convex polyhedron around the origin site point in which all points within the polyhedron are closer to the origin site point than to any other site of the complex.
		Thus the bcc rhombohedral dodecahedron coordination polyhedron (12 faces, 24 edges, 14 vertices) is not dual to the bcc truncated octahedron Dirichlet polyhedron (Loeb,1970), and it serves as a classic counterexample to the postulated duality.
	www.ornl.gov /sci/ortep/topology/lattice.html (2532 words)

	DIMACS Workshop on Discrete Mathematical Chemistry:Abstracts
		The cubic IPMS's of interest are the P surface and D surface, respectively, which are adjoint surfaces related by a Bonnet transformation with an association parameter of =BC/2.
		The cubic IPMS corresponding to the primitive cubic lattice (i.e., the P surface) requires the carbon atom decoration of each octant to be achiral, i.e., to have planes of symmetry in addition to the three-fold axis.
		The spectrum of a polyhedron P or of a tiling T is the spectrum of the graph corresponding to P or T. Relations between hexagonal tilings of the torus and (3,6)-cages (and related objects), and between their spectra, are discussed.
	dimacs.rutgers.edu /Workshops/Chemistry/abstracts.html (6003 words)

	Untitled Document (Site not responding. Last check: 2007-10-30)
		The latter portion of the period was devoted to discussing the relationship between cube, tetrahedron and octahedron; and finally to create MAGE files to illustrate their geometries.
		A tetrahedron is a regular polyhedron having FOUR vertices and FOUR sides composed of equilateral triangles.
		An octahedron is a regular polyhedron having SIX vertices and EIGHT sides composed of equilateral triangles.
	www.luc.edu /faculty/spavko1/c105/disc/s-26.htm (495 words)

	Search Results for Polyhedr*
		In addition to his heavy workload Cauchy undertook mathematical researches and he proved in 1811 that the angles of a convex polyhedron are determined by its faces.
		This three page paper simple bounds (using Euler's polyhedron formula) for the number of edges in the sets obtained when the vertices of a planar graph are partitioned into two sets.
		where v is the number of vertices of the polyhedron, e is the number of edges and f is the number of faces.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Polyhedr*&CONTEXT=1 (1853 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).
		The most "generic" way is to use for polyhedra the same naming scheme as for polygons, by counting the number or their faces: Thus, a tetrahedron has 4 faces, a pentahedron has 5, a dotriacontahedron (also called triacontakaidihedron) has 32 faces.
		In such families, the polyhedron is named using the adjective corresponding to the name of the polygon it's built on (e.g., "hexagonal").
	home.att.net /~numericana/answer/polyhedra.htm (5404 words)

	Negative Poisson's ratio foam
		Specifically, in an isotropic material (a material which does not have a preferred orientation) the allowable range of Poisson's ratio is from -1.0 to +0.5, based on thermodynamic considerations of strain energy in the theory of elasticity(1).
		It is believed by many that materials with negative values of Poisson's ratio are unknown (1); however Love (2) presents a single example of cubic 'single crystal' pyrite as having a Poisson's ratio of -0.14; he suggests the effect may result from a twinned crystal.
		Idealized re-entrant unit cell produced by symmetrical collapse of a 24-sided polyhedron with cubic symmetry.
	silver.neep.wisc.edu /~lakes/sci87.html (1462 words)

	Gamasutra - Features "Dynamic Level of Detail Terrain Rendering with Bézier Patches"
		Computer graphics generally stick to a degree of 3 that is cubic.
		Bi-cubic parametric patches are defined over a rectangular domain in uv-space and the boundary curves of the patch are themselves cubic polynomial curves.
		The net of control points forms a polyhedron in Cartesian space and the position of the points in this space controls the shape of the surface.
	www.gamasutra.com /gdc2002/features/rayner/rayner_01.htm (963 words)

	Cube - Wikipedia, the free encyclopedia
		The remaining space consists of four equal irregular polyhedra with a volume of 1/6 of that of the cube, each.
		If smaller corners are cut off we get a polyhedron with 6 octagonal faces and 8 triangular ones.
		In particular we can get regular octagons (truncated cube).
	en.wikipedia.org /wiki/Cube_(geometry) (611 words)

	Polymorf - Knowhwere- Fluorite (Site not responding. Last check: 2007-10-30)
		the faces of each of the occupied cubic AX groups are all one color or the other.
		The polyhedral framework model demonstrates the checkerboard array of AX groups clearly and the edge linkage between groups.
		cubic, octahedral, dodecahedral (rare); twinning on (111), usually cubes
	www.polymorf.net /matter51.htm (159 words)

	Three-dimensional figures - Prisms - In Depth (Site not responding. Last check: 2007-10-30)
		A prism is a polyhedron that has two congruent parallel polygons as its bases.
		The units of volume are called "cubic units." Cubic feet, cubic meters, cubic inches, cubic yards, cubic centimeters--these are all examples of units of volume.
		If we substitute the values into the volume formula, we find that the volume of the prism is 1,680 cubic inches.
	www.math.com /school/subject3/lessons/S3U4L2DP.html (351 words)

	QuArK map editor - Multi-mode panel
		Note that the center of a polyhedron is the arithmetic mean of the coordinates of its vertices, which might not seem really centered in strange-shaped polyhedrons with unbalanced vertex positioning.
		The polyhedron side mode is activated when you select a side of a polyhedron.
		As with whole polyhedrons, the center is the arithmetic mean of the vertices of the side.
	quark.planetquake.gamespy.com /armin/editor3.htm (864 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		To visualize a Klein polyhedron, we take a finite piece near the origin and flatten the piece onto a plane to produce images such as in the next section.
		The main point of interest is that the patterns are periodic iff the planes are related to a totally real cubic number field.
		This is the first of a series of examples (except example 10) where we run through the totally real cubic fields in order of discriminant as listed in Appendix B of Cohen's book A course in computational algebraic number theory, starting with discriminant 169.
	members.lycos.co.uk /keithmbriggs/klein-polyhedra1.html (864 words)

	Coordination Number 8
		The Cubic structure, the Square Antiprism andthe Dodecahedron.
		The cubic coordination is is observed for ionic compounds because of steric reasons, it is not realized for molecular structures.
		The third picture shows the structure without the bonds but with the connections between the ligand atoms showing the polyhedron formed by the ligand atoms.
	www.d.umn.edu /~pkiprof/ChemWebV2/Coordination/CN8.html (326 words)

	Programming Symmetric Polyhedrons
		The screen dumps presented below show two stages of rotation of the icosahedron produced by Pol5001, in which the lines are drawn using exactly the same set of pixels, tending to confirm correctness of geometry.
		This lecture has presented two programs, one displaying a rotating polyhedron with cubic symmetry (the cuboctahedron), the other a rotating polyhedron with icosahedral symmetry (the icosahedron).
		These provide a basis from which programs for other polyhedra of cubic, tetrahedral, and icosahedral symmetry can be derived.
	www.soi.city.ac.uk /~dcd/ig/s5pol2/lbProgP/l10.htm (1335 words)

	E-Database of Math, March, 1998
		One milliliter was equal to one cubic centimeter.
		Next, we listed all the materials we needed to be able to perform our experiment which were a data collection sheet, eight different shaped polyhedrons, a pencil, and a sheet of paper.
		We tested Euler's Formula on each polyhedron by adding the number of vertices to the number of faces and subtracting 2 to see if the overall total equaled the number of edges.
	youth.net /nsrc/math/math007.html (4724 words)

	Polyhedron Calculator (Site not responding. Last check: 2007-10-30)
		A regular polyhedron is a polyhedron whose faces are all regular polygons which are identical in both shape and size.
		First is the circum-sphere This is the sphere which fits around the outside of the polyhedron so as to touch all its vertices (or corners).
		Polyhedrons are named by the number of faces they have.
	www.projects.ex.ac.uk /trol/scol/calpolyh.htm (156 words)

	Medial Surfaces of Polyhedra and Voronoi Diagrams of Lines
		If the polyhedron is convex, then the medial surface is made up entirely of planar polygonal patches, meeting in straight line segments.
		An offset surface for a polyhedron is the set of points at a given distance r from its boundary, or equivalently, the boundary of the Minkowski sum of the polyhedron and a ball of radius r.
		When r=0, the offset surface is the original polyhedron; as we increase r, the vertices and edges of the offset surface trace out the edges and faces of the medial axis.
	compgeom.cs.uiuc.edu /~jeffe/open/medialaxis.html (867 words)

	Nonsplitting Macro Patches for Implicit Cubic Spline Surfaces (Abstract) (Site not responding. Last check: 2007-10-30)
		Macro patches are important for generating quadric or cubic implicit spline surfaces from the input of a polyhedron.
		The NMP's are based on a necessary and sufficient condition for nonsplitting constructions of implicit cubic spline surfaces.
		Each cubic patch obtained by this technique best approximates, in a least-squares sense, a quadric patch from a single algebraic component of a monotone polynomial derived from the input data.
	www.eg.org /EG/CGF/Volume12/issue3/v12i3pp433-445_abstract.html (170 words)

	Smooth Low Degree Approximations of Polyhedra - Bajaj, Chen, Xu (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		Abstract: We present an efficient algorithm to construct an inner simplicial hull \Sigma based on a given polyhedron P in three dimensional space.
		Piecewise smooth C 1 and C 2 A-patches can then be constructed within \Sigma to approximate polyhedron P. An A-patch is a smooth and functional zero-contour patch of a trivariate polynomial in Bernstein-B'ezier (BB) form defined within a tetrahedron.
		The scheme can be also compared to polyhedral subdivision, in the sense that a polyhedron P can be used as...
	citeseer.ist.psu.edu /43584.html (566 words)

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