
	Where results make sense

Topic: Zonotope

Related Topics

Johnson solid

List of geometry topics

New Labour Party

Wang tile

Little Rock, AR

Bullhead

MI

Pork rind

Tasciovanus

Queen of the United Kingdom

Raquel Lee

Joe Louis

Jack Smith

Dover test

Pielavesi

In the News (Wed 2 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	PlanetMath: zonotope
		A hexagon is also a zonotope; for example, the Minkowski sum of the line segments based at the origin with endpoints at
		The prism of a zonotope is always a zonotope, but the pyramid of a zonotope need not be.
		This is version 4 of zonotope, born on 2006-03-20, modified 2006-11-04.
	planetmath.org /encyclopedia/Zonotope.html (162 words)

	Ukrainian Easter Egg
		Zonotopes are defined using the Minkowski sum of sets A and B in a d-dimensional vector space, which is the set {p + q
		Familiar examples of zonotopes include the cube (generated by three unit length segments at right angles to each other) the rhombic dodecahedron (generated by four segments), and the truncated octahedron (generated by six segments parallel to the edges of the octahedron).
		The segments on the larger circle determine the overall shape of the zonotope, while the segments on the smaller circle lead to the existence of the belt of narrow faces near the equator of the zonotope.
	www.ics.uci.edu /~eppstein/junkyard/ukraine (628 words)

	CJM - Monotone Paths on Zonotopes and Oriented Matroids
		Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip.
		It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\lceil 2n/(n-d+2) \rceil-1$ flips for all $n \ge d+2 \ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$.
		Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid.
	journals.cms.math.ca /cgi-bin/vault/view/athanasiadis1311 (178 words)

	Zonohedra and Zonotopes
		The combinatorics of the faces of a zonotope are equivalent to those of an arrangement of hyperplanes in a space of one fewer dimension, so for instance zonohedra correspond to planar line arrangements.
		For the zonotope corresponding to a single vector (a line segment), this decomposition of the tangent space would be given by a single hyperplane corresponding to the tangents parallel to the segment, and partitioning the rest of the tangents into two subsets, one for each endpoint of the segment.
		Zonotopes resembling the shape below were used to show that the latter two problems could have complexity n^d, even though the first problem's complexity is only O(n^(d-1)) owing to the fact that it can be further transformed into one of finding a convex surface in a (d-1)-dimensional hyperplane arrangement.
	www.ics.uci.edu /~eppstein/junkyard/ukraine/ukraine.html (2108 words)

	Polytopes: Shadows of Hypercubes
		At this point in the discussion, it is important, principally, to recognize that while generally zonotopes are not space-fillers, they always may be dissected into primitive zonotopes, and thus, comprise a perfect filling of a limited space.
		Furthermore, the dissections need not split the zonotope down to atomic level; it is possible to dissect most zonotopes into a mixture of primitive zonotopes and "higher" zonotopes.
		The dissection of zonotopes into primitive zonotopes, or into mixtures of primitive zonotopes and higher zonotopes, is related to hidden-detail-removed projections; however, it is significant that not all dissections of a zonotope are equivalent to hidden-detail-removed projections.
	home.inreach.com /rtowle/Polytopes/Chapter2/Polytopes2.html (3746 words)

	PlanetMath: hyperplane arrangement
		By selecting a point in each cell and taking the convex hull of the result, we obtain a polytope combinatorially equivalent to the zonotope dual to the arrangement.
		Since the question of the fundamental group here is not interesting, we could also use the embedding
		Cross-references: embedding, zonotope, polytope, convex hull, point, cells, contractible, finite, complement, fundamental group, topological vector space, lines, projective space, subspaces, affine subspaces, pass through, hyperplane, field, vector space
	planetmath.org /encyclopedia/Grassmannian.html (210 words)

	Crinkled Zonotopes (Site not responding. Last check: 2007-10-10)
		Given any realizable oriented matroid of rank d we can represent it geometrically by a d-dimensional zonotope, that is, a polytope which is the projection of a cube.
		We call this representation the crinkled zonotope of an oriented matroid.
		Given a rank d oriented matroid M over an n-element set, its crinkled zonotope, C(M), is a geometric simplicial complex, embedded in the cube [-1,1]^n, which is a PL d-ball.
	www.lcb.uu.se /~vmoulton/publications/abstracts/p1.html (84 words)

	IFOR - Publications 2004 (Site not responding. Last check: 2007-10-10)
		The zonotope construction problem is to list all extreme points of a zonotope given by its line segments.
		By duality, it is equivalent to the arrangement construction problem that is to generate all regions of an arrangement of hyperplanes.
		The algorithm is a natural extension of a known algorithm for the zonotope construction, based on linear programming and reverse search.
	www.math.ethz.ch /ifor/publications/2004 (582 words)

	Topics in Computation and Control
		A zonotope is a polytope defined as the Minkowski sum of a finite number of segments.
		Compared to generic polytopes, zonotopes have a compact and scalable representation.
		Moreover, the class of zonotopes is invariant under linear transformations and Minkowski sum.
	hscc06.csl.sri.com /Abstracts.htm (2779 words)

	Robert Erdahl - Dicings, Zonotopes, and Voronoi's conjecture on parallelohedra
		Robert Erdahl - Dicings, Zonotopes, and Voronoi's conjecture on parallelohedra
		In 1909 Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of Euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for some lattice.
		I show that Voronoi's conjecture holds for the special case where the parallelohedron is a zonotope.
	camel.math.ca /CMS/Events/summer98/s98-abs/node33.html (83 words)

	fukuda_selected05
		Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm.
		From the zonotope construction to the Minkowski addition of convex polytopes.
		Implementations of LP-based reverse search algorithms for the zonotope construction and the fixed-rank convex quadratic maximization in binary variables using the zram and the cddlib libraries, 2002.
	www.ifor.math.ethz.ch /~fukuda/publ/fukuda_selected05 (324 words)

	Zonotope Viewer (Site not responding. Last check: 2007-10-10)
		The zonotope is generated by 5 vectors c
		A Zonotop in 3D space as the Minkowski sum of n segments will have at most n*(n-1) faces.
		This is the proof of a corollary to a Theorem by Gritzmann and Sturmfels on the number of faces in R
	www.decatur.de /personal/zono/applet.html (181 words)

	Institut Eurecom - Résumé de la publication (Site not responding. Last check: 2007-10-10)
		The maximization of ML cost function with vertices of hypercube, i.e., _____ _____, as constraints, translates to having a symmetric matrix in quadratic form with fixed rank, _ and with the hypercube constraint.
		The overall complexity of the algorithm is the complexity to find extreme point of zonotope plus the complexity of the SVD operation plus the evaluation of the objective function at the vertices.
		To find the vertices of zonotopes, an efficient algorithm called reverse search algorithm can be employed [1,6,7].
	www.eurecom.fr /util/publiabstract.fr.htm?details=1238 (360 words)

	lectures.html
		The case of real hyperplane arrangements and the dual zonotope is of particular importance.
		We then move on to oriented matroids and explain the topological representation theorem, which relies on poset-theoretic constructions.
		Such complexes are constructed via the zonotope of a hyperplane embedding.
	math.sfsu.edu /gubeladze/cbms/lectures.html (1863 words)

	Amazon.com: "zonotopal subdivisions": Key Phrase page (Site not responding. Last check: 2007-10-10)
		Rather than restricting our attention to tilings of centrally symmetric polygons, we can more generally consider the set of zonotopal subdivisions of a zono- tope.
		A zonotope Z in Rd is the Minkowski sum of a set V of line segments...
		Two zonotopal subdivisions of a 2-zonotope with six zones are depicted in Figure 2.2.3.
	www.amazon.com /phrase/zonotopal-subdivisions (236 words)

	Amazon.com: "Zonotope Dynamics": Key Phrase page (Site not responding. Last check: 2007-10-10)
		See all pages with references to Zonotope Dynamics.
		Zonotope Dynamics in Numerical Quality Control Wolfgang Iiiltn * I~onrad-7.nse-'I,entrum fiir Inforntatioustechnik Berlin (ZIB) Abstract.
		Khn, Zonotope dynamics in numerical quality control, in Mathematical Visu- alization, H.-C. Hege, K. Key Phrases in this book: Automatic Control, Springer-Verlag Berlin Heidelberg, Monte Carlo, American Control Conference, Springer Verlag, International Workshop, University of California, New York, Prentice Hall, International Conference, Artificial Intelligence, Introduction Hybrid (See more)
	www.amazon.com /phrase/Zonotope-Dynamics (198 words)

	Polytope Catalog Index
		A polytope is called zero-one if every vertex coordinate has one exactly two values (e.g.
		A zonotope is the minkowski sum of a set of vectors.
		vertices>>facets, simple, triangle-free, facet-degenerate, centrally-symmetric, triangulation, zero-one, zonotope
	www.cs.unb.ca /~bremner/PolytopeBase/catalog/catmain.html (220 words)

	Client Programs
		The bottom facet vertices get the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended.
		The zonotope is obtained as the iterated Minkowski sum of all intervals [-x,x], where x ranges over the rows of a given matrix.
		These clients take a realized polytope and produce a new one by applying a suitable affine or projective transformation in order to obtain some special coordinate description but preserve the combinatorial type.
	cgm.cs.mcgill.ca /labdocs/polymake-1.5.1/apps/polytope/clients.html (3903 words)

	DMTCS Conference vol AE (2005), pp. 145-150
		Structure of spaces of rhombus tilings in the lexicograhic case
		Rhombus tilings are tilings of zonotopes with rhombohedra.
		We study a class of lexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition.
	www.emis.de /journals/DMTCS/proceedings/abstracts/dmAE0129.abs.html (298 words)

	Wolfgang Kühn's Home Page (Site not responding. Last check: 2007-10-10)
		This interesting class of convex bodies have many surprising properties and applications.
		Some instructional Java Applets on Rigorous computing and zonotopes.
		Last modified by Wolfgang Kühn on Jun 24 2005
	www.decatur.de (54 words)

	Random walks and hyperplane arrangements, Kenneth S. Brown, Persi Diaconis
		It also includes random walks on zonotopes and zonotopal tilings.
		We find the stationary distributions of these Markov chains, give good bounds on the rate of convergence to stationarity, and prove that the transition matrices are diagonalizable.
		Keywords: RAndom walk; Markov chain; hyperplane arrangement; zonotope; eigenvalues; diagonalizable matrix; oriented matroid
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aop/1022855884 (595 words)

	Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 411-416, 2000 (Site not responding. Last check: 2007-10-10)
		Abstract: In this note we prove that the intrinsic $i$-volume of any $d$-dimensional zonotope generated by $d+1$ (resp.
		$d$) line segments and containing a $d$-dimensional unit ball in $\bf{E}^d$ is at least as large as the intrinsic $i$-volume of the $d$-dimensional regular zonotope generated by $d+1$ line segments having inradius 1, where $i=1,\dots,d-1,d$.
		Keywords: zonotopes, parallelohedra, rhombic dodecahedra, lattice sphere packings
	www.emis.de /journals/BAG/vol.41/no.2/10.html (132 words)

Try your search on: Qwika (all wikis)