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	Polytopes
		Polytope is the general term of the sequence, point, segment, polygon, polyhedron,...
		quasicrystals and zonotiles, and to the stellations of convex polytopes.
		A rectified polytope has for vertices the mid-edge points of its parent, while a truncated regular polytope is typically imagined to be truncated by hyperplanes perpendicular to the vectors to its vertices, to just such a depth as would create a regular 2n-gon, from any one of the bounding n-gons.
	home.inreach.com /rtowle/Polytopes/polytope.html (1636 words)

	Polyhedron, Polyhedra, Polytopes, ... - Numericana
		Polytopes are the n-dimensional counterparts of 3-D polyhedra.
		If the hypercell itself (the polytope's interior) is excluded from the count, as it is in the traditional 3-dimensional Euler-Descartes formula, the RHS of the formula will therefore be 2 in an odd number of dimensions and zero in an even number of dimensions.
		We may focus on the n-dimensional equivalent of the Platonic solids, namely the regular convex polytopes, whose hyperfaces are regular convex polytopes of a lower dimension, given the fact that the concept reduces to that of a regular polygon [equiangular and equilateral] in dimension 2.
	home.att.net /~numericana/answer/polyhedra.htm (4643 words)

	Top Page 1
		The largest polytope, modulo 60, is a truncated rhombic triacontahedron with 44,132 faces and 43,200 vertices (centers of an optimal packing of circles that are one degree in diameter).
		The second family is the ten polytope family of the truncated rhombic dodecahedron and the optimal spherical packing of circles for the smallest polytope in the group is the cuboctahedron.
		The polytope is related to the hexahedron (cube) by the fact that its vertices are the midpoints of the cube's edges.
	members.tripod.com /vismath4/dyke/index.html (6089 words)

	PLS: Piecewise-linear Systems
		A convex polytope is the convex hull of a set of vertices in n-dimensional space.
		Some of the polytopes are boundaries of higher-dimensional polytopes (such as the upper surface on the left) and some are internal constraints (such as the the lone edge embedded in the upper surface).
		A polytope can both be a boundary and an internal constraint (such as the other two joined edges embedded in the upper surface, which are internal constraints for the upper surface and boundary constrains for the two rectangles that penetrate the body).
	www.cs.cmu.edu /afs/cs/project/pscico/pscico/src/PLS/README.html (441 words)

	Math 411: Polytopes
		Hana Steinkamp's thesis on orbitopes and permutation polytopes.
		Beck and Pixton's preprint on the volume of the Birkhoff polytope.
		Polytopes and category theory: I. equivariant fiber polytopes, II.
	www.reed.edu /~davidp/411 (152 words)

	Polytopes (Site not responding. Last check: 2007-11-06)
		The hypercube, the cross polytope and regular simplex which are based on the cube, octahedron and tetrahedron.
		Where as the simplexes are connected with base 2, the hypercube series and it's dual the cross polytope series are connected with the trinary base-base 3.
		Imagine the simplex series as a central balanced series, with the hyper-cube series on one side and its dual the cross-polytope series on the other side.
	www.virtuescience.com /polytopes.html (540 words)

	INART 55 Polytopes and CEMAMu
		His first polytope was at the French Pavilion at the Montreal Expo of 1967 (the building remains intact, today housing the Casino de Montreal).
		The next polytope occurred outdoors at the Shiraz Festival of Arts and Music in Iran in 1971, at the ruins of Persepolis.
		An array of 59 loudspeakers played an eight-channel musique concrete score, while 92 spotlights and 2 lasers were directed at the sky and at the hills in the distance that contained the tombs of the classical heroes Darius II and Artaxerxes.
	www.music.psu.edu /Faculty%20Pages/Ballora/INART55/cemamu.html (740 words)

	Is there any efficient algorithm to compute the intersection of two (or ) polytopes
		If the input polytopes are H-polytopes (given by inequalities) then the intersection is represented by the union of the two inequality systems.
		polytopes, (2) solve the vertex enumeration problem for the union of the
		This naive approach might be satisfactory for small dimensions or not-too-complicated polytopes.
	www.cs.mcgill.ca /~fukuda/soft/polyfaq/node25.html (221 words)

	CCSD thèses-EN-ligne: Approximation de convexes par des polytopes et décomposition approchée de normes
		L'approximation des convexes lisses par des polytopes pour la distance de Hausdorff a connu de nombreux résultats théoriques grâce à l'apport de la géométrie riemannienne.
		Approximating smooth convex by polytopes with respect to Hausdorff metric is a field where numerous results were recently obtained thanks to the riemannian geometry.
		Approximation of convex bodies by polytopes and approximated norm decomposition.
	tel.ccsd.cnrs.fr /documents/archives0/00/00/52/58/index_fr.html (390 words)

	platonic
		All the faces of a regular polytope must be lower-dimensional regular polytopes of the same size and shape, and all the vertices, edges, etc. have to look identical.
		In general, the `dual' of a regular polytope is another polytope, also regular, having one vertex in the center of each face of the polytope we started with.
		The dual of the dual of a regular polytope is the one we started with (only smaller).
	math.ucr.edu /home/baez/platonic.html (2286 words)

	Directory - Science: Math: Geometry: Polytopes (Site not responding. Last check: 2007-11-06)
		Polytopes include polygons (two-dimensional), polyhedra (three-dimensional), polychora (four-dimensional), and their higher dimensional analogs.
		A regular polytope is composed of regular (n-1)-dimensional polytopes.
		There are an infinite number of regular convex polygons, five regular convex polyhedra, six regular convex polychora, and three regular convex polytopes for all dimensions five or higher.
	www.deerlakesearch.com /default?p=26941 (306 words)

	Four Dimensional Figures Page
		Also just for the record, a convex polytope is one with this property: If a line segment’s end points both lie in the polytope’s interior, then all the points on the line segment between the end points also lie in the polytope’s interior; and none of the faces overlap.
		Not surprisingly, this definition easily extends to n dimensions: a uniform polytope of n>4 dimensions is one whose facets (the elements of dimension n–1) are uniform, and any of whose vertices may be transformed 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.
	members.aol.com /Polycell/uniform.html (4231 words)

	Polytopes
		Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections.
		Vertex figures, filling, faceting diagrams, stellation, defining polytopes through generators, trimethoric and trisynaptic polyhedra, space-filling polyhedra, lost stellations of the icosahedron, and links.
		Derivation of volume equations for regular polygons, polyhedra, and polytopes, with images.
	www.canadian-universities.com /Science/Math/Geometry/Polytopes (255 words)

	Polytopes Geometry Math Science
		Numericana - Polyhedra and Polytopes - Enumeration, cartesian coordinates of vertices, naming, and counting the shapes.
		Polyhedra, Platonic Solids, Polytopes - Definitions, pictures, templates, and coordinates of the regular 3d and 4d polytopes.
		Regular Polytopes - Derivation of volume equations for regular polygons, polyhedra, and polytopes, with images.
	www.24up.org /Science/Math/Geometry/Polytopes (219 words)

	18.997: Polytopes (Site not responding. Last check: 2007-11-06)
		In this course we will study polytopes, mostly from a combinatorial point-of-view but also at times from an algorithmic perspective.
		Polytopes are geometric objects which are simple to define (bounded intersection of finitely many halfspaces), yet there are lots of basic questions which are still unanswered (Hirsch conjecture on the diameter, Mihail and Vazirani's conjecture on the edge expansion of 0-1 polytopes,...) or for which known proofs are rather involved (e.g.
		We will also discuss topics which are not much covered in this reference (such as volume of polytopes, polyhedral combinatorics,...) and hopefully recent results (such as Barany and Por's bound on the number of facets of 0-1 polytopes).
	www-math.mit.edu /~goemans/polytopes.html (222 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Consistent partitions of polytopes and polynomial measures A.Khovanskii Let $\De_1,\dots,\De_n$ be $n$ polytopes in $\Bbb R^n$.
		The present paper arose from attempts to clarify whether there exists a similar effect for other finitely additive measures that differ from the volume measure (for instance, for the number of integral points of a polytope).
		In the paper we introduce the notion of consistent partition for several polytopes (whose number can differ from the dimension of the underlying space).
	www.math.toronto.edu /askold/sogl.txt (112 words)

	Erik Demaine, Martin Demaine, Anna Lubiw, and Joseph O'Rourke: Examples, Counterexamples, and Enumeration Results for ... (Site not responding. Last check: 2007-11-06)
		We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial.
		In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings.
		We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding.
	theory.lcs.mit.edu /~edemaine/papers/AleksTR (299 words)

	Convex Polytopes and Toric Varieties: Gil Kalai and David Kazhdan (Site not responding. Last check: 2007-11-06)
		Relations between IH of a polytope and its dual related to mirror symmetry and koszul duality.
		One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it.
		Karu's paper is based on a construction by Barthel, Brasselet, Fieseler and Kaup and by Bressler and Lunts and also on a proof of "poincare duality" by Barthel, Brasselet, Fieseler and Kaup.
	www.ma.huji.ac.il /~kalai/polytopes.html (482 words)

	Cubical 4-Polytopes (Site not responding. Last check: 2007-11-06)
		It has been observed by Stanley and MacPherson that every cubical d-polytope (that is, a convex bounded polyhedron whose facets are combinatorially isomorphic to the (d-1)-dimensional standard cube) determines a PL immersion of an abstract cubical (d-2)-manifold into (the barycentric subdivision of) the boundary of the polytope, as illustrated in the following figure.
		In the case of cubical 4-polytopes each connected component of the dual manifold is a surface (compact 2-manifold without boundary).
		We obtain the first instance (instance 1) of a cubical 4-polytope (with 72 vertices and 62 facets) for which the immersed dual surface is not orientable: One of its components Klein bottle.
	www.math.tu-berlin.de /~schwartz/c4p (338 words)

	The Geometry Junkyard: Polyhedra and Polytopes
		Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes in which the vertex or edge degree is bounded by a constant or polylog.
		The charged particle model: polytopes and optimal packing of p points in n dimensional spheres.
		Circumnavigating a cube and a tetrahedron, Henry Bottomley.
	www.ics.uci.edu /~eppstein/junkyard/polytope.html (2177 words)

	Citations: On ray shooting in convex polytopes - Matousek, Schwarzkopf (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Two common types of queries that have been studied are ray shooting (identify the point where a query ray ae intersects P assuming that the origin of the ray lies in P) and....
		In the dual, this is equivalent to shooting a rayfromavertex of the dual polytope along one of its outgoing edges.
		All the ray shooting data structures mentioned in Table 9 can be dynamized at a cost of polylogarithmic or n factor 8 The vertices of the visibility graph are the vertices of the polygons.
	citeseer.ist.psu.edu /context/121109/0 (2834 words)

	i-une.com: Geometry > Polytopes
		An introduction to the subject of regular polytopes (generalizations of polygons and polyhedra) by Russell Towle.
		Geometric model building courses to improve understanding of geometry, using a Matrix kit for sale on the site that can be used in order to create the models yourself.
		Convex Polytopes : Second Edition Prepared by Volker Kaibel, Victor Klee, and Günter Ziegler (Graduate Texts in Mathematics)
	dir.i-une.com /Science/Math/Geometry/Polytopes (274 words)

	Russell Towle's 4D Star Polytope Animations
		Briefly, plane polygons are two-dimensional polytopes, and polyhedra, three-dimensional polytopes.
		In these animations, a 3-space is passed from one vertex of each star polytope, to the opposite vertex, and sections taken at small intervals.
		The star polytopes were constructed, and the sections found, using Mathematica 4.0.
	www.dogfeathers.com /towle/star.html (269 words)

	Unfolding convex polytopes
		An unfolding of a 3-dimensional convex polytope is obtained by cutting the polytope along some of its edges (necessarily a spanning tree of the edge graph) and flattening the boundary of the polytope along the remaining edges.
		Günther Rote recently constructed a family of polytopes whose minimum-perimeter unfoldings, constructed by cutting along the minimum spanning tree of the polytope's edge graph, are nonsimple.
		One of Rote's polytopes and its nonsimple minimum-perimeter unfolding.
	compgeom.cs.uiuc.edu /~jeffe/open/unfold.html (768 words)

	Existence and Approximation of Robust Solutions of Variational Inequality Problems over Polytopes
		We study nonlinear variational inequality problems over polytopes from a viewpoint of stability and propose a new solution concept.
		Though a stationary point need not be robust, it is shown that every continuous function on a polytope has a robust stationary point.
		Starting with any point in the relative interior of a polytope, the algorithm generates a piecewise linear path which leads to an approximate robust stationary point of any a priori chosen accuracy within a finite number of steps.
	epubs.siam.org /sam-bin/dbq/article/30934 (265 words)

	HMC Math 189 -- Geometric Combinatorics and Polytopes
		Course Content: Geometric combinatorics refers to a growing body of mathematics concerned with understanding the combinatorics associated with discrete geometric objects desribed by a finite set of building blocks.
		One primary example we will study in this course are polytopes, which are bounded polyhedra and the convex hull of a finite sets of points.
		We will also study objects built up from polytopes, such as triangulations and cell complexes, and other objects from the land of discrete geometry, such as the arrangements of points, lines, and hyperplanes.
	www.math.hmc.edu /~su/math189 (477 words)

	Bulletin Volume 32 Issue 4, October 1995, pages 403-412 (Site not responding. Last check: 2007-11-06)
		We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be "arbitrarily bad": namely, for every primary semialgebraic set $V$ defined over $\Z$, there is a $4$- polytope $P(V)$ whose realization space is "stably equivalent" to $V$.
		This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all $4$- polytopes.
		These results sharply contrast the $3$-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem).
	www.ams.org /journals/bull/pre-1996-data/199510/199510002.html (149 words)

	0/1-Polytopes Remote Computing Services via E-mail (Site not responding. Last check: 2007-11-06)
		The output includes the polytope(s) as bitvector (e.g.
		All known polytopes can immediately be accessed by their unique magic number.
		If there is more than one polytope then they are separated by names with increasing numbers (use -q to suppress bitvectors)
	www.igi.tugraz.at /oaich/info01poly.html (246 words)

	Science, Math, Geometry, Polytopes (Site not responding. Last check: 2007-11-06)
		- Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections.
		Polytopes and optimal packing of p points in n dimensional spheres.
		Contains a java applet based on a model which allows for generation of multidimensional regular and semi-regular polytopes..
	www.klevze.si /browse/Science/Math/Geometry/Polytopes (195 words)

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