The Tilings of Deficient Squares by Ribbon &amp;lt;i&amp;gt;L&amp;lt;/i&amp;gt;-Tetrominoes Are Diagonally Cracked

Niţică, Viorel

doi:10.4236/ojdm.2017.73015

Cited by 5 publications

(8 citation statements)

References 7 publications

(8 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then the strips can be repeated on the top or at the bottom of the deficient strip to fill a deficient half plane of type HP(U) or HP(D). A similar argument is valid for 8).We show in[6] that the tile set T 4 consisting of four ribbon L-tetrominoes, with only translations allowed, tile the opposite deficient quadrants DQ(II) and DQ(IV), allowing for any missing cell, but does not tile any of the other two deficient quadrants DQ(I) and DQ(III), no matter what cell is missing. We also show that T 4 does not tile any deficient half-strip, deficient bent-strip or deficient strip, while it tiles all deficient half-planes and deficient planes.…”

supporting

confidence: 54%

“…As the tile set T 4 already tiles rectangles, we study implications of tiling capabilities for deficient regions of tile sets that tile rectangles and consist of simply connected tiles. It is shown in [6] that a tile set that tiles rectangles also tiles deficient planes. We asked in [6] if it is possible for such a tile set to not tile any other deficient region.…”

Section: Hs(l) and Hs(u) But Does Not Tile Dhs(d)mentioning

confidence: 99%

“…It is shown in [6] that a tile set that tiles rectangles also tiles deficient planes. We asked in [6] if it is possible for such a tile set to not tile any other deficient region. The following result answers this question.…”

Section: Hs(l) and Hs(u) But Does Not Tile Dhs(d)mentioning

confidence: 99%

See 2 more Smart Citations

Revisiting a Tiling Hierarchy (II)

Niţică¹

2018

OJDM

Self Cite

View full text Add to dashboard Cite

In a recent paper, we revisited Golomb's hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles. In this classification, for several levels in Golomb's hierarchy, more types appear. We showed that there is no general relationship among tiling capabilities for types corresponding to same level. Then we found the relationships from Golomb's hierarchy that remain valid in this setup and found those that fail. As a consequence we discovered two alternative tiling hierarchies. The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones. We show that almost all of them fail. If one refines the hierarchy for tile sets that tile rectangles and for deficient regions then most of the implications of tiling capabilities can be recovered.

show abstract

supporting

confidence: 54%

Section: Hs(l) and Hs(u) But Does Not Tile Dhs(d)mentioning

confidence: 99%

See 1 more Smart Citation

Revisiting a Tiling Hierarchy (II)

Niţică¹

2018

OJDM

Self Cite

View full text Add to dashboard Cite

show abstract

“…This paper, in conjunction with [14], which studies tilings of deficient rectangles by the tile set 4 T ; points out an interesting dichotomy between the behavior of n T , n even and that of n T , n odd. We show in [14] that the obstructions to tiling of deficient rectangles by 4 T are much more severe and we claim with high confidence that this is the case for all n T , n even. Other infinite family related to Conway-Lagarias tile set is the family of ribbon tiles of length n introduced by Pak [4].…”

Section: Resultsmentioning

confidence: 99%

Tiling Rectangles with Gaps by Ribbon Right Trominoes

Junius¹,

Niţică²

2017

OJDM

Self Cite

View full text Add to dashboard Cite

We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.

show abstract

“…-A modified rectangle is simply an a × b rectangle with both the upper-left and lower-right corner cells removed [29]. -A deficient rectangle means a rectangle in which some of its square cells are ignored or occupied by obstacles [30].…”

Section: Elaborating On the Applied Tiling Theoremsmentioning

confidence: 99%

Multi-Criteria Decision Making for Efficient Tiling Path Planning in a Tetris-Inspired Self-Reconfigurable Cleaning Robot

et al. 2018

View full text Add to dashboard Cite

In this study, we aim to optimize and improve the efficiency of a Tetris-inspired reconfigurable cleaning robot. Multi-criteria decision making (MCDM) is utilized as a powerful tool to target this aim by introducing the best solution among others in terms of lower energy consumption and greater area coverage. Regarding the Tetris-inspired structure, polyomino tiling theory is utilized to generate tiling path-planning maps which are evaluated via MCDM to seek a solution that can deliver the best balance between the two mentioned key issues; energy and area coverage. In order to obtain a tiling area that better meets the requirements of polyomino tiling theorems, first, the whole area is decomposed into five smaller sub-areas based on furniture layout. Afterward, four tetromino tiling theorems are applied to each sub-area to give the tiling sets that govern the robot navigation strategy in terms of shape-shifting tiles. Then, the area coverage and energy consumption are calculated and eventually, these key values are considered as the decision criteria in a MCDM process to select the best tiling set in each sub-area, and following the aggregation of best tiling path-plannings, the robot navigation is oriented towards efficiency and improved optimality. Also, for each sub-area, a preference order for the tiling sets is put forward. Based on simulation results, the tiling theorem that can best serve all sub-areas turns out to be the same. Moreover, a comparison between a fixed-morphology mechanism with the current approach further advocates the proposed technique.

show abstract

The Tilings of Deficient Squares by Ribbon <i>L</i>-Tetrominoes Are Diagonally Cracked

Abstract: We consider tilings of deficient rectangles by the set 4 T of ribbon L-tetro-

Cited by 5 publications

References 7 publications

Revisiting a Tiling Hierarchy (II)

Revisiting a Tiling Hierarchy (II)

Tiling Rectangles with Gaps by Ribbon Right Trominoes

Multi-Criteria Decision Making for Efficient Tiling Path Planning in a Tetris-Inspired Self-Reconfigurable Cleaning Robot

Contact Info

Product

Resources

About