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### The almost simple cubical polytopes

G. Blind, R. Blind
1998 Discrete Mathematics
We give a complete enumeration of all the almost simple cubical d-polytopes for d/> 4, which is even valid for almost simple cubical (d-1)-spheres.  ...  A cubical polytope is a convex polytope all of whose facets are combinatorial cubes. A d-polytope P is called almost simple if, in the graph of P, each vertex of P is d-valent or (d + l )-valent.  ...  Clearly, the boundary complex of an almost simple cubical d-polytope is an almost simple cubical (d-1)-sphere.  ...

### Page 2359 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews
In this paper, the authors prove the following: If d > 4, then, with a single exception, the cubical d-polytopes with at most 2/*! vertices are almost simple.  ...  A convex d-polytope is cubical if all its facets are combinatorial (d — 1)-cubes, and is almost simple if each of its vertices lies in d or d + | edges.  ...

### Page 4392 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews
Let P be a convex d-polytope. P is called cubical if all its facets are combinatorial cubes. P is called almost simple if each vertex of P is d-valent or (d + 1)-valent.  ...  Then P is almost simple or a 2-fold 52 CONVEX AND DISCRETE GEOMETRY 4392 nonlinearly capped polytope.  ...

### Page 1839 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews
We give a complete enumeration of all the almost simple cubical d-polytopes for d > 4, which is even valid for almost simple cubical (d — 1)-spheres.  ...  (D-STGT-B; Stuttgart) The almost simple cubical polytopes. (English summary) Discrete Math. 184 (1998), no. 1-3, 25-48.  ...

### Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds [article]

Alexander Schwartz, Guenter M. Ziegler
2004 arXiv   pre-print
We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation).  ...  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  , via javaview of Polthier et. al.  . (A small number of figures was drawn with xfig.)  ...

### Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds

Alexander Schwartz, Günter M. Ziegler
2004 Experimental Mathematics
The illustrations of polytopes and balls were produced in the polymake system of Gawrilow and Joswig [Gawrilow and Joswig 03], via javaview of Polthier et. al. [Polthier et al. 02].  ...  The second author was partially supported by the Deutsche Forschungs-Gemeinschaft, via the DFG Research Center "Mathematics in the Key Technologies" (FZT86), the Research Group "Algorithms, Structure,  ...  Almost Cubical Polytopes All proper faces of a cubical d-polytope have to be combinatorial cubes.  ...

### Torus actions, combinatorial topology and homological algebra [article]

Victor M. Buchstaber, Taras E. Panov
2000 arXiv   pre-print
The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements.  ...  Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres.  ...  Each face of a simple polytope is a simple polytope. The product P 1 × P 2 of two simple polytopes P 1 and P 2 is a simple polytope as well.  ...

### Isocanted alcoved polytopes [article]

María Jesús de la Puente
2020 arXiv   pre-print
Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type.  ...  They are realizations of d–elementary cubical polytopes. The f–vector of a d–dimensional isocanted alcoved polytope attains its maximum at the integer ⌊ d/3⌋.  ...  Further properties are proved, showing that isocanted alcoved polytopes are d-elementary cubical, almost simple and zonotopes.  ...

### Combinatorial Space Tiling [article]

Egon Schulte
2010 arXiv   pre-print
The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling;  ...  The tiling properties of simple convex polytopes are not well understood. Recall here that a convex d-polytope is called simple if all its vertices are d-valent.  ...  It may be conjectured that every simple convex 3-polytope is the prototile of a monotypic face-to-face tiling of E 3 by topological polytopes.  ...

### Site percolation and random walks ond-dimensional Kagomé lattices

Steven C van der Marck
1998 Journal of Physics A: Mathematical and General
These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d.  ...  The site percolation problem is studied on d-dimensional generalisations of the Kagome' lattice.  ...  Acknowledgments I would like to thank Ed Stephens for critically reading the manuscript and Shell International Exploration & Production for permission to publish this paper.  ...

### Page 6816 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews
a nonabelian simple group such that L < G < Aut(L)—from the fact that G is an almost simple group—an2¢d it is identical with the socle of G).  ...  It is also proved that AutI’ is an almost simple group.  ...

### Normal crossing immersions, cobordisms and flips [article]

2020 arXiv   pre-print
We study various analogues of theorems from PL topology for cubical complexes.  ...  In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between generic immersions of  ...  In cubical complexes, popularized for their connection to low-dimensional topology and geometric group theory, the situation is not as simple.  ...

### POLYTOPES AND PROJECTION METHOD : AN APPROACH TO COMPLEX STRUCTURES

1986 Le Journal de Physique Colloques
Abstract: Decoration with atomic motifs in high dimension space is considered in simple cases, and related to the theory of curved spaces in non-periodic structures.  ...  Great -Britain ~ksurn6: La mise en place de motifs atomlques dans les espaces de grandes dimensions est envisagde dans des cas simples en rapport avec la thgorie des espaces courbes pour les structures  ...  The simple plane surface in a cubic lattice (1,0,0) leads to twinning defects. (Fig. 9 ). 2.  ...

### Page 4194 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews
(CZ-KARL-AM; Prague) Almost-tiling the plane by ellipses. (English summary) Discrete Comput. Geom. 22 (1999), no. 3, 367-375.  ...  Summary: “The universal polytope is the polytope defined as the convex hull of the characteristic vectors of all triangulations for a given point configuration.  ...

### Non-constructible Complexes and the Bridge Index

Richard Ehrenborg, Masahiro Hachimori
2001 European journal of combinatorics (Print)
We show that if a three-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible  ...  As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres.  ...  The second author was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.  ...
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