# «Abstract. We discuss two problems in combinatorial geometry. First, given a geometric polyhedral complex in R3 (a family of 3-polytopes attached ...»

simply perturb all the vertices. In particular, this explains why we must use non-simplicial polytopes in the proof of Theorem 1.1.

It is perhaps less obvious that all geometric realizations produced in Theorem 6.1 are rational. Although the resulting polyhedral complex must have simplicial interior faces, the boundary faces can be arbitrary. Here rationality is a corollary resulting from the nature of the proof: all steps, in particular all projective transformations can be done over Q.

7.6. Recently, two new explicit examples of simplicial balls with further properties were announced in [BL]. They have 12 and 15 vertices, respectively. This can be contrasted with the 9 vertices of the topological polyhedral ball X ′ we construct in the proof of Theorem 1.3.

Acknowledgements The authors are grateful to Karim Adiprasito, Bruno Benedetti, Jes´s De Loera, J´nos Pach, Rom Pinchasi, Carsten Thomassen, Russ Woodroofe and u a G¨nter Ziegler for helpful comments and conversations. The second named author was u partially supported by the BSF and NSF.

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26 IGOR PAK, STEDMAN WILSON Appendix A. Belt construction in figures and numbers It follows from Theorem 1.1 that the belts cannot be given by explicit (integer) coordinates. We give the explicit description of the belts by specifying the arcs on which the triangular facets of the prisms lie, as well as a function describing their lateral lengths (see Figure 12). Note that the prism lengths are made small except at the boundary to ensure that the belts do not intersect. Furthermore, we must ensure that the arcs bend suﬃciently to avoid each other at the top of the core. To create the arcs we start with a family of circles, and then apply a parametrized family of rotations to stretch them. The Mathematica code describing the explicit details of the construction, and used to generate the complete irrational complex and the 3D graphics in this paper, can be found at http://www.math.ucla.edu/~stedmanw/research/.

1.0 2.0 0.8 1.5 0.6 1.0 0.4 0.5 0.2 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0

In each belt, our construction uses 318 triangular prisms, exactly 2(80 − 1) + 1 = 159 prisms per semi-belt. The core consists of 5 triangular prisms and 1 pentagonal pyramid.

The complete irrational complex thus consists of a total of 4 · 318 + 5 = 1277 triangular prisms and 1 pentagonal pyramid, as in the theorem.

## GEOMETRIC REALIZATIONS OF POLYHEDRAL COMPLEXES 27

Since the belts come close to intersecting near the boundary of the core, some checking is necessary. In Figure 13 we show how the belts near-miss each other due to their shape.We conclude with a rotated view of the irrational polyhedral complex.

Department of Mathematics, UCLA Los Angeles, CA 90095, USA Email: {pak, stedmanw}@math.ucla.edu