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Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds Alexander Schwartz and Günter M

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Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds Alexander Schwartz and Günter M Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds Alexander Schwartz and Günter M. Ziegler CONTENTS We provide a number of new construction techniques for cubi- 1. Introduction cal complexes and cubical polytopes, and thus for cubifications 2. Basics (hexahedral mesh generation). As an application we obtain an 3. Lifting Polytopal Subdivisions instance of a cubical 4-polytope that has a nonorientable dual 4. Basic Construction Techniques manifold (a Klein bottle). This confirms an existence conjecture 5. A Small Cubical 4-Polytope with a Dual Klein Bottle of Hetyei (1995). 6. Constructing Cubifications More systematically, we prove that every normal crossing R3 7. Cubical 4-Polytopes with Prescribed Dual Manifold codimension one immersion of a compact 2-manifold into Immersions is PL-equivalent to a dual manifold immersion of a cubical 4- polytope. As an instance we obtain a cubical 4-polytope with a 8. An Odd Cubical 4-Polytope with a Dual Boy’s Surface cubification of Boy’s surface as a dual manifold immersion, and 9. Consequences with an odd number of facets. Our explicit example has 17,718 10. Applications to Hex Meshing vertices and 16,533 facets. Thus we get a parity-changing op- 11. The Next Step eration for three-dimensional cubical complexes (hex meshes); Acknowledgments this solves problems of Eppstein, Thurston, and others. References 1. INTRODUCTION A d-polytope is cubical if all its proper faces are com- binatorial cubes, that is, if each k-face of the polytope, k ∈{0,...,d− 1} is combinatorially equivalent to the k-dimensional standard cube. It has been observed by Stanley, MacPherson, and others (see [Babson and Chan 00, Jockusch 93]) that ev- ery cubical d-polytope P determines a PL immersion of an abstract cubical (d − 2)-manifold into the polytope ∼ −1 boundary ∂P = Sd . The immersed manifold is ori- entable if and only if the 2-skeleton of the cubical d- polytope (d ≥ 3) is “edge-orientable” in the sense of Het- yei, who conjectured that there are cubical 4-polytopes that are not edge-orientable [Hetyei 95, Conjecture 2]. In the more general setting of cubical PL (d − 1)- 2000 AMS Subject Classification: Primary 52B12, 52B11, 52B05; Secondary 57Q05 spheres, Babson and Chan [Babson and Chan 00] have observed that every type of normal crossing PL immer- Keywords: Cubical polytopes, regular subdivisions, normal crossing immersions, hex meshes, Boy’s surface sion of a (d − 2)-manifold into a (d − 1)-sphere appears c A K Peters, Ltd. 1058-6458/2004$ 0.50 per page Experimental Mathematics 13:4, page 385 386 Experimental Mathematics, Vol. 13 (2004), No. 4 among the dual manifolds of some cubical PL (d − 1)- opposite parity (Section 10). This answers questions sphere. by Bern, Eppstein, Erickson, and Thurston [Bern et No similarly general result is available for cubical poly- al. 02, Eppstein 99, Thurston 93]. topes. The reason for this may be traced/blamed to a lack of flexible construction techniques for cubical poly- Electronic geometry models of the instances con- topes, and more generally, for cubical complexes (such structed in Sections 5 and 8 are available at the as the “hexahedral meshes” that are of great interest in second author’s home page, http://www.math.tu- CAD and in numerical analysis). berlin.de/˜schwartz/c4p/. In this paper, we develop a number of new and im- proved construction techniques for cubical polytopes. We try to demonstrate that it always pays off to carry along convex lifting functions of high symmetry. The most 2. BASICS complicated and subtle element of our constructions is For the following we assume that the readers are fa- the “generalized regular Hexhoop” of Section 6.4, which miliar with the basic combinatorics and geometry of yields a cubification of a d-polytope with a hyperplane convex polytopes. In particular, we will be dealing of symmetry, where a (suitable) lifting function may be with cubical polytopes (see [Gr¨unbaum 03, Section 4.6]), specified on the boundary. Our work is extended by the polytopal (e.g., cubical) complexes, regular subdivisions first author in [Schwartz 03], where additional construc- (see [Ziegler 98, Section 5.1]), and Schlegel diagrams tion techniques for cubifications (i.e., cubical subdivisions [Gr¨unbaum 03, Section 3.3], [Ziegler 98, Section 5.2]. For of d-polytopes with prescribed boundary subdivisions) cell complexes, barycentric subdivision and related no- are discussed. tions we refer to [Munkres 84]. Suitable references for Using the constructions developed here, we achieve the the basic concepts about PL manifolds, embeddings, and following constructions and results: (normal crossing) immersions include [Hudson 69] and • a rather simple construction yields a cubical 4- [Rourke and Sanderson 82]. polytope (with 72 vertices and 62 facets) for which the immersed dual 2-manifold is not orientable: one 2.1 Almost Cubical Polytopes of its components is a Klein bottle. Apparently this All proper faces of a cubical d-polytope have to be com- is the first example of a cubical polytope with a binatorial cubes. We define an almost cubical d-polytope nonorientable dual manifold. Its existence confirms as a pair (P, F), where F is a specified facet of P such a conjecture of Hetyei (Section 5). that all facets of P other than F are required to be com- • more generally, all PL-types of normal crossing im- binatorial cubes. Thus, F need not be a cube, but it will mersions of 2-manifolds appear as dual manifolds in be cubical. C the boundary complexes of cubical 4-polytopes (Sec- By (P ) we denote the polytopal complex given by C tion 7). In the case of nonorientable 2-manifolds of a polytope P and all its faces. By (∂P) we denote odd genus, this yields cubical 4-polytopes with an the boundary complex of P , consisting of all proper faces C odd number of facets. From this, we also obtain a of P .IfP is a cubical polytope, then (∂P) is a cu- complete characterization of the lattice of f-vectors bical complex. If (P, F) is almost cubical, then the C \{ } of cubical 4-polytopes (Section 9). Schlegel complex (∂P) F is a cubical complex that is combinatorially isomorphic to the Schlegel diagram • in particular, we construct an explicit example with Schlegel(P, F)ofP based on F . 17, 718 vertices and 16, 533 facets of a cubical 4- polytope which has a cubification of Boy’s surface 2.2 Cubifications (projective plane with exactly one triple point) as a A cubification of a cubical PL (d − 1)-sphere Sd−1 is a dual manifold immersion (Section 8). cubical d-ball Bd with boundary Sd−1. A double count- • via Schlegel diagrams, this implies that every 3- ing argument shows that every cubical (d − 1)-sphere cube has a cubical subdivision into an even num- that admits a cubification has an even number of facets. ber of cubes that does not subdivide the boundary Whether this condition is sufficient is a challenging open complex. Thus for every cubification of a three- problem, even for d = 3 (compare [Bern et al. 02, Epp- dimensional domain there is also a cubification of the stein 99]). Schwartz and Ziegler: Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds 387 (a) C (b) C and D(C) (c) D(C) FIGURE 1. The derivative complex of a cubical 2-complex C. 2.3 Dual Manifolds Restricted to a dual manifold, this immersion may be an embedding or not. For every cubical (pure) d-dimensional complex C, d>1, the derivative complex is an abstract cubical (d − 1)- dimensional cell complex D(C) whose vertices may be identified with the edge midpoints of C, while the facets “separate the opposite facets of a facet of C,” that is, they correspond to pairs (F, [e]), where F is a facet of C and [e] denotes a “parallel class” of edges of F .Thisisa cell complex with f1(C) vertices and df d(C) cubical facets of dimension d − 1, d of them for each facet of C. Hence the derivative complex D(C) is pure (d − 1)-dimensional. FIGURE 2. The cubical octahedron O8 (the only combi- See Babson and Chan [Babson and Chan 00, Section 4]. natorial type of a cubical 3-polytope with eight facets), In the case of cubical PL manifolds, such as spheres and its single immersed dual manifold. (for instance, boundary complexes of cubical polytopes), or cubical PL balls, the derivative complex is a (not nec- In the case of cubical 3-polytopes, the derivative com- essarily connected) manifold, and we call each connected plex may consist of one or many 1-spheres. For example, component of the derivative complex D(P ) of a cubical for the 3-cube it consists of three 1-spheres, while for C C the “cubical octahedron” O8 displayed in Figure 2 the complex a dual manifold of . If the cubical com- 1 plex C is a sphere, then the dual manifolds of C are dual manifold is a single immersed S (with eight double manifolds without boundary. If C is a ball, then some points). (possibly all) dual manifolds have nonempty boundary In the case of 4-polytopes, the dual manifolds are components, namely the dual manifolds of ∂C. surfaces (compact 2-manifolds without boundary). As The derivative complex, and thus each dual mani- an example, we display here a Schlegel diagram of a fold, comes with a canonical immersion into the bound- “neighborly cubical” 4-polytope (that is, of a 4-polytope ary of P . More precisely, the barycentric subdivision whose 1-skeleton (graph) is equivalent to that of a higher- sd (D(P )) of D(P ) has a simplicial map to the barycen- dimensional cube), with f-vector (32, 80, 96, 48).
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