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Dual pairs of non-polytopal diagrams and spheres

1985
*
Discrete Mathematics
*

There is a

doi:10.1016/s0012-365x(85)80021-2
fatcat:tfhxel3h7na25gdrajudngui3e
*dual**pair**of*cellular decompositions*of*the 3-*sphere*both realizable as geometric complexes in Euclidean 4-space but neither isomorphic to the boundary complex*of*a convex 4-*polytope*. ... In the next section we shall construct a*dual**pair**of**non*-*polytopal*3diagrams which will be modified in the last section to yield a*dual**pair**of**non*-*polytopal*3-*spheres*in E4. ... A*dual**pair**of**non*-*polytopal*3-*diagrams*Another way*of*stating the Steinitz-problem is in terms*of*(d -1)-*diagrams*. ...##
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Alexander duality in subdivisions of Lawrence polytopes

2003
*
Advances in Geometry
*

The class

doi:10.1515/advg.2003.012
fatcat:ji26u2okgrcynk56ishrjenlkm
*of*simplicial complexes representing triangulations*and*subdivisions*of*Lawrence*polytopes*is closed under Alexander duality. This gives a new geometric model for oriented matroid duality. ... Examples*of*Alexander*dual**pairs**of*subdivided Lawrence*polytopes*are given in Section 3. The smallest*non*-trivial example is the*pair**of*triangular prisms in Figure 1 . ... Figure 2 . 2 This subdivision*of*a uniform Lawrence*polytope*(the triangular prism) is Alexander*dual*to a triangulation*of*a*non*-uniform Lawrence*polytope*(the pyramid). ...##
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Parametrization of Generalized Primal-Dual Triangulations
[chapter]

2011
*
Proceedings of the 20th International Meshing Roundtable
*

Motivated by practical numerical issues in a number

doi:10.1007/978-3-642-24734-7_13
dblp:conf/imr/MemariMD11
fatcat:pjbzq7fhebenvmfvj4x7slpnb4
*of*modeling*and*simulation problems, we introduce the notion*of*a compatible*dual*complex to a primal triangulation, such that a simplicial mesh*and*... Using algebraic*and*computational geometry results, we show that compatible*dual*complexes exist only for a particular type*of*triangulation known as weakly regular. ... Partial funding was generously provided by the National Science Foundation through grants CCF-1011944, CCF-0811373,*and*CMMI-0757106. ...##
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Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds
[article]

2004
*
arXiv
*
pre-print

As an application we obtain an instance

arXiv:math/0310269v3
fatcat:dn2jsrowqzeejh6a7nrtfjqnnm
*of*a cubical 4-*polytope*that has a*non*-orientable*dual*manifold (a Klein bottle). This confirms an existence conjecture*of*Hetyei (1995). ... As an instance we obtain a cubical 4-*polytope*with a cubation*of*Boy's surface as a*dual*manifold immersion,*and*with an odd number*of*facets. ... The illustrations*of**polytopes**and*balls were produced in the polymake system*of*Gawrilow & Joswig [14] , via javaview*of*Polthier et. al. [26] . (A small number*of*figures was drawn with xfig.) ...##
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One-point suspensions and wreath products of polytopes and spheres

2005
*
Journal of combinatorial theory. Series A
*

In particular, infinitely many

doi:10.1016/j.jcta.2004.09.009
fatcat:ydm7rm7p4recbc2g2etn6kqkfe
*non*-PL*spheres*as well as contractible simplicial complexes with a vertex-transitive group*of*automorphisms can be obtained in this way. ... Suitable iterations*of*this construction generate highly symmetric simplicial complexes with various interesting combinatorial*and*topological properties. ... The*dual*wedge*of*a*polytope*Let P ⊂ R d be a d-dimensional*polytope*(or d-*polytope*for short),*and*let v be a vertex. ...##
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Arbitrarily Large Neighborly Families of Congruent Symmetric Convex 3-Polytopes
[chapter]

2003
*
Discrete Geometry
*

a convex hull edge; the Schlegel

doi:10.1201/9780203911211.ch19
fatcat:kl47afgoozddhbh6dl3rh7hxsy
*diagram**of*the polar*dual**of*any neighborly 4-*polytope*consists*of*a neighborly family*of*3-*polytopes*. ... its degree-ordered sequence*of**non*-zero coefficients. ... Figures 1*and*3 were produced with the help*of*the programs 'qhull' [2, 3]*and*'geomview' [1, 19] . ...##
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One-Point Suspensions and Wreath Products of Polytopes and Spheres
[article]

2004
*
arXiv
*
pre-print

In particular, infinitely many

arXiv:math/0403494v1
fatcat:2lsx5bqwlbhbve3jo22muxk65e
*non*-PL*spheres*as well as contractible simplicial complexes with a vertex-transitive group*of*automorphisms can be obtained in this way. ... Suitable iterations*of*this construction generate highly symmetric simplicial complexes with various interesting combinatorial*and*topological properties. ... If K is a vertex-transitive*non*-spherical homology*sphere**and*d ≥ 1, then ∂∆ d ≀ K is a*non*-PL*sphere*for d ≥ 1. ...##
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Toric Topology. Chapter 2: Combinatorial structures
[article]

2012
*
arXiv
*
pre-print

Comments

arXiv:1102.1081v2
fatcat:57zsd4dl6rgfzhjkvr3jyn5u2q
*and*suggestions are very welcome. ... Give an example*of*a*non*flag simple*polytope*with the nonnegative γ-vector. Alexander duality For any simplicial complex, a*dual*complex may be defined on the same set. ... The boundary*of*a simplicial n-*polytope*is an (n−1)-dimensional P L*sphere*. In dimensions n 3 every (n − 1)-*diagram*is a Schlegel*diagram**of*an npolytope. ...##
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Small f-vectors of 3-spheres and of 4-polytopes
[article]

2016
*
arXiv
*
pre-print

In particular, we thus prove that there are f-vectors

arXiv:1610.01028v1
fatcat:xnglibxgjzb5xop6x34hoaaa2i
*of*cellular 3-*spheres*with the intersection property that are not f-vectors*of*any convex 4-*polytopes*, thus answering a question that may be traced ... By implementing this approach, we classify the f-vectors*of*4-*polytopes*in the range f_0+f_3<22. ... Acknowledgements We are very grateful to Marge Bayer*and*Moritz Firsching valuable comments*and*discussions. ...##
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Neighborly 4-polytopes with 9 vertices

1973
*
Journal of combinatorial theory. Series A
*

A complete classification is given for neighborly 4-

doi:10.1016/0097-3165(73)90074-5
fatcat:xwg437px2vbwzniqozyemunoqi
*polytopes*with 9 vertices. It is found that there are exactly 23 combinatorial types. ... In view*of*Theorem 1,*and*the remarks following Theorem 3, a necessary condition for the realizability*of*that 3-*sphere*as a 4-*polytope*, necessarily neighborly, is that the*non*-diagonal elements on each ... The determinant*of*the edge-valence matrix is an invariant*of*the combinatorial type qf the*sphere*,*and*is positive if the matrix is regular. Figure 2 .FIG. 2 . 22 Two*pairs*(NB, %? ...##
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The dunce hat in a minimal non-extendably collapsible 3-ball
[article]

2013
*
arXiv
*
pre-print

As we will see, the Alexander

arXiv:0912.3723v2
fatcat:7n2rwt2chzbonmqcrutstleahm
*dual**of*the dunce hat is collapsible. ... We obtain a geometric realization*of*a minimal 8-vertex triangulation*of*the dunce hat in Euclidean 3-space. ... The triangulation D*of*the dunce hat is also contained in Grünbaum-Sreedharan's*sphere*No. 37 (*polytopal*, with 8 vertices*and*20 facets)*and*in Grünbaum-Sreedharan's*sphere*M (*non*-*polytopal*, with 8 vertices ...##
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k-set polytopes and order-k Delaunay diagrams

2006
*
2006 3rd International Symposium on Voronoi Diagrams in Science and Engineering
*

k-set

doi:10.1109/isvd.2006.43
dblp:conf/isvd/SchmittS06
fatcat:kg26avh6ivhn5m2lul4mcdqv7u
*polytope**of*S, the order-k Voronoi*diagram**of*S,*and*its*dual*, the order-k Delaunay*diagram**of*S. ...*of*a (d + 1)dimensional k-set*polytope**and*is the orthogonal*dual**of*the order-k Voronoi*diagram*. ... a well known*dual*, the Voronoi*diagram**of*S. ...##
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Oriented Matroids
[chapter]

2004
*
Handbook of Discrete and Computational Geometry, Second Edition
*

Other introductions

doi:10.1201/9781420035315.ch6
fatcat:ic2fzbamuvgu7jrbp6y7o2ej4y
*and*basic sources*of*information include Bachem & Kern [1] , Bokowski [5] , Bokowski & Sturmfels [7] ,*and*Ziegler [19, Lect. 6*and*7]. ... We refer to the monograph by Björner, Las Vergnas, Sturmfels, White & Ziegler [3] for a broad introduction,*and*for an extensive development*of*the theory*of*oriented matroids with a reasonably complete ... construction*of**non*-rational*polytopes*, as well as*of**non*-*polytopal**spheres*with d+4 vertices, etc. ...##
###
The triangulations of the 3-sphere with up to 8 vertices

1973
*
Journal of combinatorial theory. Series A
*

The different combinatorial types

doi:10.1016/0097-3165(73)90062-9
fatcat:5e4fqk3nbfeovoxodjeifhtyrm
*of*triangulations*of*the 3-*sphere*with up to 8 vertices are determined. ... Using similar methods we show that one cannot always preassign the shape*of*a facet*of*a 4-*polytope*. ... Figure 6 gives the Schlegel*diagrams**of*the 9 combinatorial types*of*such*polytopes*. THEOREM 1. ...##
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Realizability and inscribability for simplicial polytopes via nonlinear optimization

2017
*
Mathematical programming
*

3-

doi:10.1007/s10107-017-1120-0
fatcat:fhkhmutqxbf37lryaio7bnotf4
*spheres*with 10 vertices into*polytopal**and**non*-*polytopal**spheres*. ... We show that nonlinear optimization techniques can successfully be applied to realize*and*to inscribe matroid*polytopes**and*simplicial*spheres*. ... For the help in Section 3.2 I thank Frank Lutz, who provided the list*of*triangulations*and*David ...
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