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2004 Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds
Alexander Schwartz, Günter M. Ziegler
Experiment. Math. 13(4): 385-413 (2004).


We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical $4$-polytope that has a nonorientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995).

More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into {\small $\R^3$} is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubification of Boy's surface as a dual manifold immersion, and with an odd number of facets. Our explicit example has 17,718 vertices and 16,533 facets. Thus we get a parity-changing operation for three-dimensional cubical complexes (hex meshes); this solves problems of Eppstein, Thurston, and others.


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Alexander Schwartz. Günter M. Ziegler. "Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds." Experiment. Math. 13 (4) 385 - 413, 2004.


Published: 2004
First available in Project Euclid: 22 February 2005

zbMATH: 1110.52015
MathSciNet: MR2118264

Primary: 52B05 , 52B11 , 52B12
Secondary: 57Q05

Keywords: Boy's surface , Cubical polytopes , hex meshes , normal crossing immersions , regular subdivisions

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 4 • 2004
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