Experimental Mathematics

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

Alexander Schwartz and Günter M. Ziegler

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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.

Article information

Experiment. Math., Volume 13, Issue 4 (2004), 385-413.

First available in Project Euclid: 22 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52B12: Special polytopes (linear programming, centrally symmetric, etc.) 52B11: $n$-dimensional polytopes 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]
Secondary: 57Q05: General topology of complexes

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


Schwartz, Alexander; Ziegler, Günter M. Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds. Experiment. Math. 13 (2004), no. 4, 385--413. https://projecteuclid.org/euclid.em/1109106431

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