Economical triangle-square dissection

Cohn, M. J.

doi:10.1007/bf00181377

Cited by 12 publications

(8 citation statements)

References 2 publications

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“…For example, any two simple polygons in the plane -i.e., polygons with non-intersecting sides-of equal area can be dissected by straight line cuts into a finite number of congruent polygonal figures which can be rearranged without overlapping to form the other polygon [25], a result known as the Wallace-Bolyai-Gerwien theorem [26]. Some popular geometric dissections include the Tangram -the dissection puzzle invented in ancient China, the Hindu problem -the Greek cross dissection into five pieces to form a square [27], and the Bhaskara's proof of the Pythagorean theorem [28].…”

Section: Hinged Dissections Of 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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Section: Hinged Dissections Of 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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“…Hence, a number of analytic approaches have been developed to find geometric dissections with the fewest number of pieces. However, they mainly focus on 2D primitive shapes such as triangles and squares [Coh75,KKU00,CKU07].…”

Section: Related Workmentioning

confidence: 99%

“…In practice, it is usually required that the dissection uses just a few pieces, such that the puzzle is manageable for playing. Early research works [Coh75, KKUOO, AAC* 12] focus on analytic approaches to dissect 2D primitive shapes such as triangles, squares and other regular polygons into congruent polygonal pieces. Until recently, a few computational methods have been developed to dissect general 2D shapes represented as discrete squares [ZW12] or dissect naturalistic 2D shapes approximately [DYYT17].…”

Section: Introductionmentioning

confidence: 99%

Computational Design of Steady 3D Dissection Puzzles

Tang

Song

Wang

et al. 2019

Computer Graphics Forum

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Dissection puzzles require assembling a common set of pieces into multiple distinct forms. Existing works focus on creating 2D dissection puzzles that form primitive or naturalistic shapes. Unlike 2D dissection puzzles that could be supported on a tabletop surface, 3D dissection puzzles are preferable to be steady by themselves for each assembly form. In this work, we aim at computationally designing steady 3D dissection puzzles. We address this challenging problem with three key contributions. First, we take two voxelized shapes as inputs and dissect them into a common set of puzzle pieces, during which we allow slightly modifying the input shapes, preferably on their internal volume, to preserve the external appearance. Second, we formulate a formal model of generalized interlocking for connecting pieces into a steady assembly using both their geometric arrangements and friction. Third, we modify the geometry of each dissected puzzle piece based on the formal model such that each assembly form is steady accordingly. We demonstrate the effectiveness of our approach on a wide variety of shapes, compare it with the state‐of‐the‐art on 2D and 3D examples, and fabricate some of our designed puzzles to validate their steadiness.

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“…A geometric figure can be dissected into a finite number of smaller pieces that can be rearranged to form another figure [29,30]. For instance, any two simple polygons in the plane (i.e., polygons with non-intersecting sides) of equal area can be dissected by straight line cuts into a finite number of congruent polygonal figures that can be rearranged without overlapping to form the other polygon [31]. This is known as the Wallace-Bolyai-Gerwien theorem [32].…”

Section: Reconfigurability Of Hinged Dissections Of Polyminoesmentioning

confidence: 99%

Nested Reconfigurable Robots: Theory, Design, and Realization

Tan

Rojas

Elara

et al. 2015

International Journal of Advanced Robotic Systems

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Rather than the conventional classification method, we propose to divide modular and reconfigurable robots into intra-, inter-, and nested reconfigurations. We suggest designing the robot with nested reconfigurability, which utilizes individual robots with 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. 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 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. Finally, the validation experiments verify the nested reconfigurability of Hinged-Tetro. Extensive tests and analyses of intra-reconfiguration are provided in terms of energy and time consumptions. Experiments using two robots validate the inter-reconfigur ability of the proposed module

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Economical triangle-square dissection

Cited by 12 publications

References 2 publications

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

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

Computational Design of Steady 3D Dissection Puzzles

Nested Reconfigurable Robots: Theory, Design, and Realization

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