Related papers: A square from similar rectangles

In the present popular-science paper, we find out which rectangles can be dissected into squares. The proof is based on a physical interpretation in terms of electrical networks. Only a secondary school background is assumed in the paper.

We consider so-called squaring the square-puzzles where a given square (or rectangle) should be dissected into smaller squares. For a specific instance of such problems we demonstrate that a mathematically rigorous solution can be quite…

We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles.

This paper deals with the history of the following problem: "Can an arbitrary rectangle be dissected into 3 non-rectangular congruent regions?" We present a new elementary proof that the answer is indeed no.

A procedure that generates parallelograms from any quadrilateral is presented. If the original quadrilateral is itself a parallelogram, then the procedure gives squares. Hence, when applied two times, this procedure generates squares from…

Given a collection of N rectangles such that the side ratio of each one is a quadratic irrationality, we find all rectangles which can be tiled by rectangles similar to one of the given ones. It means that each possible shape can be used…

We prove that one can cover the $1 \times b$ rectangle by equal squares on both sides in one layer iff $b = p \pm \sqrt{p^2 - r^2} $, where $p \ge r \ge 0$ and $p,q \in \mathbb{Q}$.

This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given…

We prove that almost every triangle can be dissected only into $n^2$ triangles which have to be equal one another. Moreover, such a dissection is unique for every $n$. It turns out that to solve this "simple" problem it is convenient to use…

We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number…

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and…

In this article, the issue of choice cuts made to a rectangular region are considered and explored. Results show that this problem is not trivial. Outcomes for teaching and learning are considered.

In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…

An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into…

A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the…

The theorem on squaring a rectangle from a tiling of a quadrilateral (Schramm and Cannon-Floyd-Parry) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of…

Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the…

This paper addresses the idea of the applicability of mathematics, using, as a case study, a construction and software package that partition the unit sphere into regions of equal area. The paper assesses the applicability of this…

Let a polygon be composed of equal rectangles. We find all quadratic irrationals r for which the polygon can be tiled by similar rectangles with given side ratio r.

Question when rectangle can be tiled with similar copies of rectangles witch quetient of sides quadratic irrationalities. New proof of one part F. Sharov's theorem. Other close result.