Related papers: Rationality of certain triangle tilings
We give a simple proof of T. Stehling's result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.
This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible…
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with…
Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…
In this paper, we give some sufficient conditions for a $n$-dimensional rectangle to be tiled with a set of bricks. These conditions are obtained by using the so-called Frobenius number.
Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. He pointed out that a certain ratio corresponding to two such regions has a nice product formula. In…
The conditions determining that two triangles are congruent play a basic role in planimetry. By comparing not congruent triangles with respect to given sets of corresponding elements it is important to discover if they have any common…
We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and…
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
A positive integer $n$ is called a $\theta$-congruent number if there is a triangle with sides $a,b$ and $c$ for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $n\sqrt{r^2 - s^2}$, where $0 < \theta < \pi$, $\cos…
We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled…
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and…
The problem that we consider is the following: given an $n \times n$ array $A$ of positive numbers, find a tiling using at most $p$ rectangles (which means that each array element must be covered by some rectangle and no two rectangles must…
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.
The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…
A method is developed to compute analytically fully symmetric cubature rules on the triangle by using symmetric polynomials to express the two kinds of invariance inherent in these rules. Rules of degree up to 15, some of them new and of…
We define a new family of non-periodic tilings with square tiles that is mutually locally derivable with some family of tilings with isosceles right triangles. Both families are defined by simple local rules, and the proof of their…
Proctor proved a formula for the number of lozenge tilings of a hexagon with side-lengths $a,b,c,a,b,c$ after removing a "maximal staircase." Ciucu then presented a weighted version of Proctor's result. Here we present weighted and…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
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…