Polyominoes on twisted cylinders

Barequet, Gill; Shalah, Mira

doi:10.1145/2493132.2462357

Cited by 5 publications

(6 citation statements)

References 8 publications

(10 reference statements)

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“…A slight variation of this automaton can be seen in action in a video animation. 7 See the online appendix for more on automata for modeling polyominoes on twisted cylinders. See also the second sidebar "Representing Boundaries as Motzkin Paths.…”

Section: Methodsmentioning

confidence: 99%

“…We are interested in the growth rate of the sequence of vectors y old that is determined by the dominant eigenvalue λ W of Q. It is not difficult to show 5 . (1) By the Perron-Frobenius Theorem about the eigenvalues of nonnegative matrices, the two bounds converge to λ W , and y old converges to a corresponding eigenvector.…”

Section: Cause Any Problem Because Succ0[s]mentioning

confidence: 99%

“…The number of polyominoes of size n is usually denoted as A(n). No formula is known for this number, but researchers have suggested efficient backtracking 26,27 and transfer-matrix 9,18 algorithms for computing A(n) for a given value of n. The latter algorithm was adapted in Barequet et al 5 and also in this work for polyominoes on so-called twisted cylinders. To date, the sequence A(n) has been determined up to n = 56 by a parallel computation on a Hewlett Packard server cluster using 64 processors.…”

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confidence: 99%

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λ > 4

2016

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Section: Methodsmentioning

confidence: 99%

Section: Cause Any Problem Because Succ0[s]mentioning

confidence: 99%

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confidence: 99%

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λ > 4

2016

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“…Currently, the best known bound is 3.9856 n < t(n) < 4.6496 n [19]. This counting considers fixed polyominoes, known as lattice animals in statistical physics.…”

Section: Polyominoesmentioning

confidence: 99%

Hinged-Tetro: A self-reconfigurable module for nested reconfiguration

Kee

Rojas

Elara

et al. 2014

2014 IEEE/ASME International Conference on Advanced Intelligent Mechatronics

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Nested reconfiguration is an emerging research area in modular robotics. Such a novel design concept utilizes individual robots with distinctive reconfiguration characteristics (intra-reconfigurability) capable of combining with other homogeneous/heterogeneous robots (inter-reconfigurability). The objective of this approach is to generate more complex morphologies for performing specific tasks that are far from the capabilities of a single module or to respond to programmable assembly requirements. The two-level reconfiguration process in nested reconfigurable robotic system implies several technical challenges in hardware design, planning algorithms, and control strategies. In this paper, we discuss the theory, concept, and initial mechanical design of Hinged-Tetro, a self-reconfigurable module conceived for the study of nested reconfiguration. Hinged-Tetro is a mobile robot that uses the principle of hinged dissection of polyominoes to transform itself into any of the seven one-sided tetrominoes, the Tetris pieces, in a straightforward way. The robot can also combine with other modules for shaping complex structures or giving rise to a robot with new capabilities. Some preliminary experiments of intrareconfigurability with an implemented prototype are presented.

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“…Its exact value has remained elusive till these days. The currently best lower and upper bounds known on λ 2 are roughly 3.9801 [1] and 4.6496 [6], respectively.…”

Section: Introductionmentioning

confidence: 99%

Proper n-Cell Polycubes in n − 3 Dimensions

Asinowski

Barequet

et al. 2011

Lecture Notes in Computer Science

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A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d − 1)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in combinatorics and discrete geometry. This is also an important tool in statistical physics for computations and analysis of percolation processes and collapse of branched polymers. A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. In this paper we prove that the number of polycubes of size n that are proper in n − 3 dimensions is 2 n−6 n n−7 (n − 3)(12n 5 − 104n 4 + 360n 3 − 679n 2 + 1122n − 1560)/3.

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Polyominoes on twisted cylinders

Abstract: In this video we show how to enumerate polyominoes on twisted cylinders, and explain how to use them for setting lower bounds on the asymptotic growth rate of polyominoes in the plane.

Cited by 5 publications

References 8 publications

λ > 4

λ > 4

Hinged-Tetro: A self-reconfigurable module for nested reconfiguration

Proper n-Cell Polycubes in n − 3 Dimensions

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