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

Abstract

We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). Thus we obtain the first instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms the existence conjecture of Hetyei [17, Conj. 2, p. 325]. More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into 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 cubation of Boy's surface as a dual manifold immersion, and with an odd number of facets. This solves problems of Eppstein, Thurston and others. Our explicit example has 19 520 vertices and 18 333 facets.

A 44-element mesh of Schneiders’ pyramid: Bounding the difficulty of hex-meshing problems

“…The maximum number of hexahedra H max of the solution is set to a smaller value than |C |. Changing the parity of a hexahedral mesh is known to be a difficult operation [14], so we set H max to |C | − 2. We also set the limit to the number of interior vertices V max to one less than the number of interior vertices in C to accelerate the search.…”

Efficiently Hex-Meshing Things with Topology

“…Babson and Chan call the natural cubical structure on the dual immersion the derivative complex. Using a generalization of the Hexhoop template developed by Yamakawa and Shimada for constructing hexahedral meshes [74], Schwartz [63] and Schwartz and Ziegler [64] proved that every self-transverse piecewise-linear immersion of a (d − 1)-manifold into the d-sphere can be refined into the dual immersion of a cubical d-polytope. In particular, they construct the first cubical 4-polytope with an odd number of facets, by refining Boy's classical immersion of the projective plane [13] (specifically, an orthogonal version of Boy's surface popularized by Petit [57]).…”

“…For any hex mesh X , the union of the interior 2-dimensional cells of the dual complex X * is the image of a topological surface immersion. At the risk of confusing the reader, we use the same notation X * to denote this dual immersion, which is variously called the spatial twist continuum [51], the derivative complex of X [3,33,39,64], and the canonical surface of X [1,2]. The duality between cube complexes and immersed surfaces was already observed in the late 1800s, at least in preliminary form, in Fedorov's seminal study of zonotopes [29,30,65].…”

Nonrational configurations, polytopes, and surfaces

“…But how about cubical polytopes of dimension 4? The boundary of such a polytope consists of combinatorial 3-cubes; its combinatorics is closely related with that of immersed cubical surfaces [26]. On the other hand, if we impose the condition that the cubes in the boundary have to be affine cubesso all 2-faces are centrally symmetric -then there are easy non-rational examples, namely the zonotopes associated to non-rational configurations [33,Lect.…”