Number Of Unit Squares Research Articles

Ortaokul Öğrencilerinin Alan Ölçme Bilgi ve Becerilerinin İncelenmesi

In this study, it is aimed to examine the area measurement knowledge and skills of secondary school students. In the study, in which the descriptive method was used, the data were collected in written form with 8 open-ended questions from 92 seventh grade students. Students' answers to open-ended questions were analyzed with a scoring scale. The research shows that the majority of students who correctly estimate the size of the area measurement units are in the majority, but they have difficulty in transforming the standard area measurement units. Students who can calculate the area as the number of unit squares covering a region are in the majority. It was observed that the students who were successful in measuring the area of the rectangle had low success in measuring the area of the parallelogram, triangle and trapezoid, and it was observed that the students tended to multiply the two sides of the polygon while measuring the area. In addition, it was determined that some students confused the area with the environment.

Concept Formation Processes of Area and its Formula with GeoGebra: Case of Rectangle

This research aims to investigate the construction of the area of rectangle with GeoGebra and how students can formulate it. It focuses on describing the cognitive process involved in constructing this concept and how technology mediates such a process. The research was conducted in a classroom with 5th grade students in a secondary school and was designed qualitatively as a case study. A readiness test was prepared to determine the preliminary knowledge of the students about the concept of area. The teaching process was carried out with the activities based on GeoGebra. Considering the test results, researcher observations, and teacher views, five voluntary participants were selected, and we had three interviews with each of them. The obtained data from the teaching process and interviews were analyzed according to APOS theoretical framework. The results of the research demonstrate that one of the participants could not conceptualize the area as an object as the number of unit squares that cover the area without any space, while two of the participants could not conceptualize the formula of area of rectangle as an object. The students generally had difficulty in encapsulating the concept of the area and its formula. Area conservation and unit concept are important for conceptualizing the concept of area by realizing the covering feature of it, and understanding the dimension relationship is significantly effective in conceptualizing its formula.

Online circle and sphere packing

In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receives an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares, isosceles right triangles of leg length one, and unit cubes, respectively. For Online Circle Packing in Squares, we improve the previous best-known competitive ratio for the bounded space version, when at most a constant number of bins can be open at any given time, from 2.439 to 2.3536. For Online Circle Packing in Isosceles Right Triangles and Online Sphere Packing in Cubes we show bounded space algorithms of asymptotic competitive ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and 2.7707 on the competitive ratio of any online bounded space algorithm for these two problems. We also considered the online unbounded space right variant of these three problems which admits a small reorganization of the items inside some of the bins after their packing, and we present algorithms of competitive ratios 2.3105, 2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes, respectively. Throughout the text, we also discuss how our algorithms can be extended to other problems.

On optimal piercing of a square

We treat the following problem: given an n×n square ABCD, determine the minimum number of points that need to be chosen inside the square ABCD such that there does not exist a unit square inside the square ABCD containing none of the chosen points in its interior. In other words, we are interested to know how to most efficiently “destroy” a square-shaped object of side length n, where “destroying” is achieved by piercing as few as possible small holes, and the square is considered “destroyed” if no unpierced square piece of unit side length can be salvaged. This problem actually belongs to the family of problems centered about the so-called piercing number: indeed, if Un denotes the collection of all open unit squares that can be fitted inside a given n×n square, the value that we are looking for is the piercing number of the collection Un, denoted by π(Un). We show that π(Un)=n2 when n⩽7, and give an upper bound for π(Un) that is asymptotically equal to 23n2, which we believe is asymptotically tight. We then generalize our reasoning in order to obtain a similar upper bound when ABCD is a rectangle, as well as an upper bound for π(Ux) when x is not necessarily an integer. Finally, we show that our results have an application to the problem of packing a given number of unit squares in the smallest possible square; it turns out that our results present a general “framework” based on which we are able to reprove many results on the mentioned problem (originally obtained independently of each other) and also obtain a new result on packing 61 unit squares.

A polynomial algorithm for packing unit squares in a hypograph of a piecewise linear function

AbstractWe consider the problem of packing the maximal number of unit squares in a hypograph of a function. A polynomial time algorithm is described to solve this problem for a piecewise linear function.

Proof Without Words: Sums of Odd Integers

SummaryThe number of unit-squares in the left figure equals the sum of the area of the rectangle (of length n), plus the area of the big triangle (of height 2n – 2), plus the area of the n – 1 small 2 × 1 triangles.

Irregular polyomino tiling via integer programming with application in phased array antenna design

A polyomino is a generalization of the domino and is created by connecting a fixed number of unit squares along edges. Tiling a region with a given set of polyominoes is a hard combinatorial optimization problem. This paper is motivated by a recent application of irregular polyomino tilings in the design of phased array antennas. Specifically, we formulate the irregular polyomino tiling problem as a nonlinear exact set covering model, where irregularity is incorporated into the objective function using the information-theoretic entropy concept. An exact solution method based on a branch-and-price framework along with novel branching and lower-bounding schemes is proposed. The developed method is shown to be effective for small- and medium-size instances of the problem. For large-size instances, efficient heuristics and approximation algorithms are provided. Encouraging computational results including phased array antenna simulations are reported.

The Estimation of the Number of Lattice Tilings of a Plane by a Given Area Polyomino

We study a problem of a number of lattice plane tilings by given area polyominoes. A polyomino is a connected plane geometric figure formed by joining edge to edge a finite number of unit squares. A tiling is a lattice tiling if each tile can be mapped to any other tile by translation which maps the whole tiling to itself. Let T(n) be a number of lattice plane tilings by given area polyominoes such that its translation lattice is a sublattice of Z². It is proved that 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2.7)n+1. In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyomino and on the theory of self-avoiding walk. Also, it is proved that almost all polyominoes that give lattice plane tilings have sufficiently large perimeters.

Points Covered an Odd Number of Times by Translates

Given an odd number of axis-aligned unit squares in the plane, it is known that the area of the set whose points in the plane that belong to an odd number of unit squares cannot exceed the area of one unit square, that is, 1. In this paper, we consider the same problem for other shapes. Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of T. This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.

Design of a conductive fabric network by the sheet resistance method

Combining textiles with electronics has become a promising realm in the future development of textiles. A number of researches have been conducted to exploit the potential applications, such as sports, healthcare, etc. However, the growth of wearable electronic textiles in the market is being retarded. The reasons may refer to the poor appearance of current products in that consumers may consider the products to be ‘technical’ rather than ‘garments’, since aesthetics have been ignored as a result of focusing on the functionality of the textiles. Conductive fabrics have become one of the fundamental elements of wearable electronic textiles, which form an electrically conductive network to interconnect various electronic function units, such as textile electrodes, power sources, etc. However, it is necessary to solve the high-dimensional matrix in order to compute the resistance of the conductive network, which hinders the application of conductive fabrics in industry. Therefore, it is essential to develop a direct and fast method to aid the fashion designer without an electronic background to compute the resistance of the conductive fabric so as to meet the high demand for rapid prototype development of wearable electronic garments. This paper describes a systematic approach to calculate the resistance based on the sheet resistance and number of unit squares from a macroscopic view. The experimental results demonstrated that this method could readily calculate the resistance of conductive fabrics as long as the sheet resistance and the number of unit squares are known. In addition, the proposed method can be applied to other types of conductive fabrics.

Packing equal squares into a large square

Let s ( x ) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W ( x ) = x 2 − s ( x ) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that W ( x ) = O ( x ( 3 + 2 ) / 7 log x ) . This improves earlier results of Erdős–Graham and Montgomery in which the upper bounds of W ( x ) = O ( x 7 / 11 ) and W ( x ) = O ( x ( 3 − 3 ) / 2 log x ) , respectively, were obtained. A complementary problem is to determine s ′ ( x ) the minimum number of unit squares needed to cover a large square of side length x. We show that s ′ ( x ) = x 2 + O ( x ( 3 + 2 ) / 7 log x ) , improving an earlier bound of x 2 + O ( x 7 / 11 ) .

LPA4/p2y9/GPR23 Mediates Rho-dependent Morphological Changes in a Rat Neuronal Cell Line

Lysophosphatidic acid (LPA) is a potent lipid mediator that evokes a variety of biological responses in many cell types via its specific G protein-coupled receptors. In particular, LPA affects cell morphology, cell survival, and cell cycle progression in neuronal cells. Recently, we identified p2y(9)/GPR23 as a novel fourth LPA receptor, LPA(4) (Noguchi, K., Ishii, S., and Shimizu, T. (2003) J. Biol. Chem. 278, 25600-25606). To assess the functions of LPA(4) in neuronal cells, we used rat neuroblastoma B103 cells that lack endogenous responses to LPA. In B103 cells stably expressing LPA(4), we observed G(q/11)-dependent calcium mobilization, but LPA did not affect adenylyl cyclase activity. In LPA(4) transfectants, LPA induced dramatic morphological changes, i.e. neurite retraction, cell aggregation, and cadherin-dependent cell adhesion, which involved Rho-mediated signaling pathways. Thus, our results demonstrated that LPA(4) as well as LPA(1) couple to G(q/11) and G(12/13), whereas LPA(4) differs from LPA(1) in that it does not couple to G(i/o). Through neurite retraction and cell aggregation, LPA(4) may play a role in neuronal development such as neurogenesis and neuronal migration.

Resource augmentation in two-dimensional packing with orthogonal rotations

We consider the problem of packing two-dimensional rectangles into the minimum number of unit squares, when 90 ∘ rotations are allowed. Our main contribution is a polynomial-time algorithm for packing rectangles into at most OPT bins whose sides have length ( 1 + ε ) , for any positive ε . Additionally, we show near-optimal packing results for a number of related packing problems.

Near-Optimal Solutions to Two-Dimensional Bin Packing With 90 Degree Rotations

We consider a multidimensional generalization of the bin packing problem, namely, packing 2−dimensional rectangles into the minimum number of unit squares, where 90 degree rotations are allowed. Our main contribution is a polynomial time algorithm for packing rectangles into at most OPT bins whose sides have length (1+ɛ), for any ɛ>0. Additionally, we show how this result can be used to obtain near optimal packing results for a variety of two and three dimensional packing problems in which 90 degree rotations are allowed. These include minimum rectangle packing, two dimensional strip packing, and the z-oriented 3-dimensional packing problem.

On the Pentomino Exclusion Problem

In this paper we are interested in the Pentomino Exclusion Problem due to Golomb: Find the minimum number of unit squares to be placed on a k\times n chessboard so as to exclude all pentominoes . Using an appropriate definition of density of a tiling, we obtain the asymptotic value of this number, and we establish this number for the k\times n chessboard when k≤ 4 .

On a problem connected with a theorem of Jarnik

Divide the plane into unit squares and plot a rectifiable closed jordan curve of lengthl. We show that the error created when approximating the number of unit squares contained inside this curve by the area enclosed by it is less than αl, where α is a constant satisfying $$\frac{{4 + \pi }}{{2\pi }} \leqslant \alpha \leqslant 3 + \frac{2}{{6\pi (1 + \sqrt {1 + 2/9\pi } )}}.$$

An Investigation

Situation: To find the number of unit squares the diagonal of a rectangle passes through when the rectangle has integer sides and is divided into unit squares.