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Free arrangements and rhombic tilings. (English) Zbl 0853.52013
Discrete Comput. Geom. 15, No. 3, 307-340 (1996); erratum ibid. 17, 357 (1997).
Let \(Z\), which can also be denoted \(Z(r_1,r_2,\dots,r_l)\), be a centrally symmetric polygon with integer side lengths such that a clockwise consecutive sequence of half the sides of the polygon \(Z\) have lengths \(r_1, r_2, \dots, r_l\) in this order. The authors prove the following:
Theorem: All tilings of \(Z\) by unit rhombi are coherent in the sense of Billera and Sturmfels [L. Billera and B. Sturmfels, Ann. Math., II. Ser. 135, No. 3, 527-549 (1992; Zbl 0762.52003)] if and only if the following four cases hold:
1. \(Z\) is a \(Z(r,s)\) parallelogram;
2. \(Z\) is a \(Z(r,s,t)\) hexagon in which at least one of \(r\), \(s\), \(t\) is at most 2;
3. \(Z\) is projectively equivalent to a \(Z(r,s,1,1)\) octagon;
4. \(Z\) is a \(Z(1,1,1,1,1)\) decagon.
To \(Z\) one can associate a discriminantal hyperplane arrangement. The authors also prove a theorem which classifies the arrangements which are free in the sense of Saito and Terao [P. Orlik and H. Terao, ‘Arrangements of hyperplanes’, Springer Verlag, New York (1992; Zbl 0757.55001)]. This last theorem is similar in structure to the preceding theorem and is used in its proof.
Thus the notions of \(Z\) having the associated hyperplane arrangement free, and \(Z\) having all tilings coherent, are related, but neither implies the other as the authors show in an example.
Finally, considering the discriminantal arrangements for some particular octagons the authors can produce a counterexample to the conjecture by Saito that the complexified complement of a real free arrangement is a \(K(\pi,1)\) space.

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
Full Text: DOI
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