On packing of rectangles in a rectangle

Joós, Antal

doi:10.1016/j.disc.2018.06.007

Cited by 8 publications

(6 citation statements)

References 9 publications

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“…In this paper all rectangles and squares are understood to have sides parallel to the coordinate axes, and to be topologically closed. 2 As pointed out in [6], some of the lemmas in this paper were not proven correctly, but the gaps in this paper were recently repaired in [4].…”

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confidence: 91%

Perfectly packing a square by squares of nearly harmonic sidelength

Tao¹

2022

Preprint

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A well known open problem of Meir and Moser asks if the squares of sidelength 1/n for n ≥ 1 can be packed perfectly into a square of area6 . In this paper we show that for any 1/2 < t < 1, and any n 0 that is sufficiently large depending on t, the squares of sidelength n −t for n ≥ n 0 can be packed perfectly into a square of area ∞ n=n0 1 n 2t . This was previously known for 1/2 < t < 2/3 (in which case one can take n 0 = 1).

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confidence: 91%

Perfectly packing a square by squares of nearly harmonic sidelength

Tao¹

2022

Preprint

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“…According to the regularity of shapes of participating parts, the 2D packing problem can be further divided into 2D regular (such as Figure 3) and irregular packing problems. The 2D regular packing problem includes the layout problem of regular polygons such as rectangles (Martello and Monaci, 2015;Wu et al, 2016;Wu et al, 2017;Joós, 2018;Erzin et al, 2021) and circles (George et al, 1995;Nurmela and Östergård, 1997;Huang and Xu, 1999;Wang et al, 2002;Zhang and Deng, 2005;Liu et al, 2016;Fekete et al, 2019). The 2D irregular layout problem mainly involves the layout of irregular shapes.…”

Section: Packing Problem Classificationmentioning

confidence: 99%

Two-dimensional irregular packing problems: A review

Guo

et al. 2022

Front. Mech. Eng.

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Two-dimensional (2D) irregular packing problems are widespread in manufacturing industries such as shipbuilding, metalworking, automotive production, aerospace, clothing and furniture manufacturing. Research on 2D irregular packing problems is essential for improving material utilization and industrial automation. Much research has been conducted on this problem with significant research results and certain algorithms. The work has made important contributions to solving practical problems. This paper reviews recent advances in the domain of 2D irregular packing problems based on a variety of research papers. We first introduce the basic concept and research background of 2D irregular packing problems and then summarize algorithms and strategies that have been proposed for the problems in recent years. Conclusion summarize development trends and research hotspots of typical 2D irregular shape packing problems. We hope that this review could provide guidance for researchers in the field of 2D irregular packing.

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“…For instance, Paulhus [9] showed that the squares of sidelength for could be packed into a square of area . However, it was pointed out in [4] that the proof contained some errors; these errors were corrected in [1].…”

Section: Introductionmentioning

confidence: 99%

Perfectly packing a cube by cubes of nearly harmonic sidelength

MCCLENAGAN

2023

Can. Math. Bull.

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Let d be an integer greater than 1, and let t be fixed such that 1 d < t < 1 d−1 . We prove that for any n0 chosen sufficiently large depending upon t, the d-dimensional cubes of sidelength n −t for n ≥ n0 can perfectly pack a cube of volume ∞ n=n 0 1 n dt . Our work improves upon a previously known result in the three-dimensional case for when 1/3 < t ≤ 4/11 and n0 = 1 and builds upon recent work of Terence Tao in the two-dimensional case.

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On packing of rectangles in a rectangle

Cited by 8 publications

References 9 publications

Perfectly packing a square by squares of nearly harmonic sidelength

Perfectly packing a square by squares of nearly harmonic sidelength

Two-dimensional irregular packing problems: A review

Perfectly packing a cube by cubes of nearly harmonic sidelength

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