Tiling polygons with parallelograms

Kannan, Sampath; Soroker, Danny

doi:10.1007/bf02187834

Cited by 10 publications

(14 citation statements)

References 9 publications

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“…It follows from this symmetry that, in each direction a, the boundary contains equally many unit vectors in directions a and a. This is the balance condition in [6].…”

Section: General Casementioning

confidence: 96%

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Sub Rosa, A System of Quasiperiodic Rhombic Substitution Tilings with n-Fold Rotational Symmetry

Kari

Rissanen²

2016

Discrete Comput Geom

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In this paper we prove the existence of quasiperiodic rhombic substitution tilings with 2n-fold rotational symmetry, for any n. The tilings are edge-to-edge and use ⌊ n 2 ⌋ rhombic prototiles with unit length sides. We explicitly describe the substitution rule for the edges of the rhombuses, and prove the existence of the corresponding tile substitutions by proving that the interior can be tiled consistently with the given edge substitutions.

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“…It follows from this symmetry that, in each direction a, the boundary contains equally many unit vectors in directions a and a. This is the balance condition in [6].…”

Section: General Casementioning

confidence: 96%

“…Only in the case of the (1, n−1) super-rhombus for odd n the roses at the opposite n−1 corners touch in the middle, but without crossing each other. In the terminology of [6] the boundary is simple.…”

Section: General Casementioning

confidence: 99%

Sub Rosa, A System of Quasiperiodic Rhombic Substitution Tilings with n-Fold Rotational Symmetry

Kari

Rissanen²

2016

Discrete Comput Geom

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“…23 ' 28 Polygon decomposition is also studied in conjunction with the theory of packings, coverings, and tilings, which concerns arranging a set of polygons, each usually a copy of a single polygon or one of a small set of polygons, in such a way as to occupy as much of a given region as possible with minimal or no overlap of the polygons. 14,18,20 Another interesting variation on the polygon decomposition problem comes from the mathematics literature concerned with dissection theory. 5 ' 13 There it is shown that, given any two polygons V\ and V2 of the same area, either can be cut into a set of pieces that can be rearranged to produce the other.…”

Section: Introductionmentioning

confidence: 99%

Polygon Area Decomposition for Multiple-Robot Workspace Division

Hert

Lumelsky

1998

Int. J. Comput. Geom. Appl.

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A new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem is presented. This problem concerns dividing a given polygon V into n polygonal pieces, each of a specified area and each containing a certain point (site) on its boundary or in its interior. First the algorithm for the case when V is convex and contains no holes is presented. Then the generalized version that handles nonconvex and nonsimply connected polygons is presented. The algorithm uses sweep-line and divide-and-conquer techniques to construct the polygon partition. The input polygon V is assumed to have been divided into a set of p convex pieces (p = 1 when V is convex), which can be done in 0(vp log log up) time, where vp is the number of vertices of V and p = 0(vp), using algorithms presented elsewhere in the literature. Assuming this convex decomposition, the running time of the algorithm presented here is 0(pn 2 -run), where v is the sum of the number of vertices of the convex pieces.

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“…Kannan and Soroker [8] characterize all simple polygons that can be tiled by finitely many parallelograms. In particular, a convex polygon P admits such a tiling if and only if it is a zonotope [8,Sect.…”

Section: Proof Of Proposition 1 We Consider a Convex Quadranglementioning

confidence: 99%

“…In particular, a convex polygon P admits such a tiling if and only if it is a zonotope [8,Sect. 3], which means that P is centrally symmetric [15, Theorem 3.5.1].…”

Section: Proof Of Proposition 1 We Consider a Convex Quadranglementioning

confidence: 99%