Escher-like Tilings with Weights

Imahori, Shinji; Kawade, Shizuka; Yamakata, Yoko

doi:10.1007/978-3-319-48532-4_12

Cited by 2 publications

(15 citation statements)

References 8 publications

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“…Koizumi and Sugihara originally placed the same number of points on every tiling edge. Later, Imahori et al [3] suggested in the conclusion of their paper that it is also possible to place different numbers of points on each tiling edge. We denote the numbers of points placed on the tiling edges (white circles in the figure) as k 1 , k 2 , .…”

Section: Parameterization Of Isohedral Tilingsmentioning

confidence: 99%

“…The optimization problem (10) is known as the Rayleigh quotient; the optimal value is the maximum eigenvalue of B ⊤ V B and the optimal solution ξ * is the eigenvector associated with the maximum eigenvalue (the length is arbitrary). By utilizing the fact that B ⊤ V B is a symmetric positive semidefinite matrix of maximum rank two, Imahori et al [3] proposed a projection method [11] to compute the maximum eigenvalue and its corresponding eigenvector in O(n 2 ) time.…”

Section: Escherization Problem As An Eigenvalue Problemmentioning

confidence: 99%

“…Then, we select the best tile shape as that with the minimum distance value. As suggested by Imahori et al [3], we need to consider only 28 isohedral types of those originally classified by Heesch and Kienzle [2], because the remaining 65 isohedral types are regarded as special cases of the 28 types; other 65 types are obtained by imposing internal symmetries on the 28 types. For each isohedral type IHi, it takes O(n 3 ) time to compute Eq (12) for j = 1, 2, .…”

Section: Escherization Problem As An Eigenvalue Problemmentioning

confidence: 99%

“…Instead, we propose an O(n) time construction procedure of the matrix B. (ii) We propose an O(n) time algorithm to compute the Procrustes distance, which required O(n 2 ) time in the previous calculation method [3]. (iii) We introduce a mechanism to evaluate a lower bound on the Procrustes distance to omit the calculation of the Procrustes distance that does not lead to an improvement in the best tile shape found so far.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Efficient Algorithm for the Escherization Problem in the Polygon Representation

Nagata,

Imahori

2019

Preprint

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In the Escherization problem, given a closed figure in a plane, the objective is to find a closed figure that is as close as possible to the input figure and tiles the plane. Koizumi and Sugihara's formulation reduces this problem to an eigenvalue problem in which the tile and input figures are represented as n-point polygons. In their formulation, the same number of points are assigned to every tiling edge, which forms a tiling template, to parameterize the tile shape. By considering all possible configurations for the assignment of the n points to the tiling edges, we can achieve much flexibility in terms of the possible tile shapes and the quality of the optimal tile shape improves drastically, at the cost of enormous computational effort. In this paper, we propose an efficient algorithm to find the optimal tile shape for this extended formulation of the Escherization problem.

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Section: Parameterization Of Isohedral Tilingsmentioning

confidence: 99%

Section: Escherization Problem As An Eigenvalue Problemmentioning

confidence: 99%

Section: Escherization Problem As An Eigenvalue Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Efficient Algorithm for the Escherization Problem in the Polygon Representation

Nagata,

Imahori

2019

Preprint

Self Cite

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“…The exhaustive search of the templates was very time-consuming because (i) the order of K i is O(n 3 ) for IH5 and IH6 and O(n 4 ) for IH4 (see Fig. 3), and (ii) for each combination of i and k, it took O(n 3 ) time for computing eval E ikj for all j ∈ J [6,5]. As a compromise, Imahori and Sakai [6] proposed a local search algorithm to search only promising configurations of (k, j) ∈ (K i , J) for each isohedral type i ∈ I.…”

Section: Koizumi and Sugihara's Formulation And Its Extensionsmentioning

confidence: 99%

Escherization with a Distance Function Focusing on the Similarity of Local Structure

Nagata

2018

Parallel Problem Solving From Nature – PPSN XV

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The Escherization problem involves finding a closed figure that tiles the plane that is most similar to a given goal figure. In Koizumi and Sugihara's formulation of the Escherization problem, the tile and goal figures are represented as n-point polygons where the similarity between them is measured based on the difference in the positions between the corresponding points. This paper presents alternative similarity measures (distance functions) suitable for this problem. The proposed distance functions focus on the similarity of local structures in several different manners. The designed distance functions are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. Efficient exhaustive and incomplete search algorithms for the formulated problems are also developed to obtain results within a reasonable computation time. Experimental results showed that the proposed algorithms found satisfactory tile shapes for fairly complicated goal figures in a reasonable computation time.

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Escher-like Tilings with Weights

Cited by 2 publications

References 8 publications

An Efficient Algorithm for the Escherization Problem in the Polygon Representation

An Efficient Algorithm for the Escherization Problem in the Polygon Representation

Escherization with a Distance Function Focusing on the Similarity of Local Structure

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