Rational Triangles Research Articles

Intermediate spectral statistics of rational triangular quantum billiards.

Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle, which permits one to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion, and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short-range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

On certain Diophantine equations concerning the areas and perimeters of rational triangles

On certain Diophantine equations concerning the areas and perimeters of rational triangles

Integer triangles with integer circumradii

When studying properties of the circumcircle of an integer triangle, it quickly becomes evident that the radius of such a circle (circumradius) need not itself be an integer. When it is not an integer, the circumradius can still be rational but it can also be irrational, as exemplified in the following examples. It is left to the reader to verify that the triangle with sides 10, 24, 26 has circumradius 13, and that the corresponding values for the triangles with sides 13, 14, 15 and 1, 1, 1 are 65/8 and respectively. It is shown in Theorem 1 below that a necessary condition for the circumradius to be an integer is that the area of the triangle is itself an integer (Heronian triangle) but this condition is not in itself sufficient. A simple counterexample is given by the 13, 14, 15 triangle above which has area 84. However, as a consequence of Theorem 1, we can restrict ourselves to considering Heronian triangles, and relevant properties of such triangles, proved in [1], are given in Theorems 2 and 3 below. We also need to quote some well-known results involving the sums of two squares (see e.g. [2]) and these are listed in Lemma 3. In all that follows, we will use the convention that if T is a triangle with sides a, b, c and z > 0, then zT will denote the triangle, similar to T, with sides za, zb, zc.

Periodic points on the regular and double n-gon surfaces

Using the transfer principle of Lie theory, we classify the periodic points on the regular n-gon and double n-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.

On the class of an integer triangle

Any mathematics student who has ever used the cosine rule to investigate simple properties of an integer triangle will immediately have realised that the cosine of each angle of the triangle must be a rational number. It is clear, however, that the same is not in general true for the sines. In [1], it is shown how to use a property of the sines of the angles of an integer triangle to categorise the triangle as being of a particular class. In this article, we develop some of the concepts and results of [1] to derive a method for generating integer triangles of a given class. Finally, we apply our results to find all primitive integer triangles in the particular case of Heronian triangles.

Strongly obtuse rational lattice triangles

We classify rational triangles which unfold to Veech surfaces when the largest angle is at least $\frac{3\pi}{4}$. When the largest angle is greater than $\frac{2\pi}{3}$, we show that the unfolding is not Veech except possibly if it belongs to one of six infinite families. Our methods include a criterion of Mirzakhani and Wright that built on work of Moller and McMullen, and in most cases show that the orbit closure of the unfolding cannot have rank 1.

On the Diophantine equations z^2 = f(x)^2 ± f(x)f(y) + f(y)^2

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations $z^2=f(x)^2\pm f(x)f(y)+f(y)^2$ for certain polynomials f(x), which mean to construct integral triangles with two sides given by the values of polynomials f(x) and f(y) with the intersection angle $120^\circ$ or $60^\circ$.

An Affine Model of a Riemann Surface Associated to a Schwarz–Christoffel Mapping

In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated n-gon in the complex plane.

On Ceva points of (almost) equilateral triangles

A Ceva point of a rational-sided triangle is any internal or external point such that the lengths of the three cevians through this point are rational. Buchholz [Buc89] studied Ceva points and showed a method to construct new Ceva points from a known one. We prove that almost-equilateral and equilateral rational triangles have infinitely many Ceva points by establishing a correspondence to points in certain elliptic surfaces of positive rank.

Pairs of rational triangles with equal symmetric invariants

The semiperimeter s, inradius r and circumradius R are the symmetric invariants of a triangle. In this short note, we complement previous work of Hirakawa and Matsumura by determining all pairs (up to similitude) consisting of a rational right angled triangle and a rational isosceles triangle having two corresponding symmetric invariants equal. In the proof of our main theorems, we determine the set of rational points on two hyperelliptic curves of genus 3. The first of these can be settled via a classical descent argument, whereas the second one is proved using ‘‘elliptic curve Chabauty’’.

The $${\theta }$$-Congruent Number Elliptic Curves via Fermat-type Algorithms

A positive integer N is called a $$\theta $$ -congruent number if there is a $${\theta }$$ -triangle (a, b, c) with rational sides for which the angle between a and b is equal to $$\theta $$ and its area is $$N \sqrt{r^2-s^2}$$ , where $$\theta \in (0, \pi )$$ , $$\cos (\theta )=s/r$$ , and $$0 \le |s|<r$$ are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a $${\theta }$$ -congruent number if and only if the elliptic curve $$E_N^{\theta }: y^2=x (x+(r+s)N)(x-(r-s)N)$$ has a point of order greater than 2 in its group of rational points. Moreover, a natural number $$N\ne 1,2,3,6$$ is a $${\theta }$$ -congruent number if and only if rank of $$E_N^{\theta }({{\mathbb {Q}}})$$ is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational $${\theta }$$ -triangle for a $${\theta }$$ -congruent number N from given ones by generalizing the Fermat’s algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle $${\theta }$$ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in $$E_N^{\theta }({{\mathbb {Q}}})$$ . Then, based on the addition of two distinct points in $$E_N^{\theta }({{\mathbb {Q}}})$$ , we provide a way to find new rational $${\theta }$$ -triangles for the $${\theta }$$ -congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara’s Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $$E_N^{\theta }({{\mathbb {Q}}})$$ with corresponding rational $${\theta }$$ -triangles.

Rational triangle pairs and cyclic quadrilateral pairs with areas and perimeters in certain proportions

The problem of finding a pair of distinct rational triangles (resp. cyclic quadrilaterals) with the same area and the same perimeter has been solved. Now we generalize these results and get several parametric solutions to find infinitely many pairs of rational triangles (resp. cyclic quadrilaterals) with areas and perimeters in fixed proportions $(\alpha,\beta)$ respectively, where $\alpha$ and $\beta$ are positive rational numbers.

Integral triangles and perpendicular quadrilateral pairs with a common area and a common perimeter

By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\geq 4$, and show the existence of infinitely many elliptic curves of rank $\geq 5$, parameterized by the points of an elliptic curve of positive rank.

Fast tetrahedral meshing in the wild

We propose a new tetrahedral meshing method, fTetWild, to convert triangle soups into high-quality tetrahedral meshes. Our method builds on the TetWild algorithm, replacing the rational triangle insertion with a new incremental approach to construct and optimize the output mesh, interleaving triangle insertion and mesh optimization. Our approach makes it possible to maintain a valid floating-point tetrahedral mesh at all algorithmic stages, eliminating the need for costly constructions with rational numbers used by TetWild, while maintaining full robustness and similar output quality. This allows us to improve on TetWild in two ways. First, our algorithm is significantly faster, with running time comparable to less robust Delaunay-based tetrahedralization algorithms. Second, our algorithm is guaranteed to produce a valid tetrahedral mesh with floating-point vertex coordinates, while TetWild produces a valid mesh with rational coordinates which is not guaranteed to be valid after floating-point conversion. As a trade-off, our algorithm no longer guarantees that all input triangles are present in the output mesh, but in practice, as confirmed by our tests on the Thingi10k dataset, the algorithm always succeeds in inserting all input triangles.

Complete and computable orbit invariants in the geometry of the affine group over the integers

The subject matter of this paper is the geometry of the affine group over the integers, $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ . Turing-computable complete $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit invariants are constructed for rational affine spaces, angles, segments, triangles and ellipses. In rational affine $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -geometry, ellipses are classified by the Clifford–Hasse–Witt invariant, via the Hasse–Minkowski theorem. We classify ellipses in $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch–Jung continued fraction algorithm. We then consider rational polyhedra, i.e., finite unions of simplexes in $${\mathbb {R}}^n$$ with rational vertices. Markov’s unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra P and $$P'$$ are continuously $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -equidissectable. The same problem for the continuous $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -equidissectability of P and $$P'$$ is open. We prove the decidability of the problem whether two rational polyhedra P and $$P'$$ in $${\mathbb {R}}^n$$ have the same $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit.

Class of Neumann-Type Problems for the Polyharmonic Equation in a Ball

A set of necessary solvability conditions for the class $${{\mathcal{N}}_{k}}$$ of Neumann-type problems for the polyharmonic equation with a polynomial right-hand side in the unit ball is obtained. These conditions have the form of the orthogonality of homogeneous harmonic polynomials to linear combinations of boundary functions with coefficients from the Neumann integer triangle perturbed by certain derivatives of the right-hand side of the equation.

Rational cuboids and Heron triangles II

We study the connection of Heronian triangles with the problem of the existence of rational cuboids. It is proved that the existence of a rational cuboid is equivalent to the existence of a rectangular tetrahedron, which all sides are rational and the base is a Heronian triangle. Examples of rectangular tetrahedra are given, in which all sides are integer numbers, but the area of the base is irrational. The example of the rectangular tetrahedron is also given, which has lengths of one side irrational and the other integer, but the area of the base is integer.&#x0D;

Rational cuboids and Geron triangles

The problem of existence of rational cuboids is considered in the article. Equivalent statements ofthis problem are obtained which lead to the existence of Heron’s triangles whose lengths are integralsquares and satisfy other additional conditions. Also, the connections between Euler bricks and theHeron triangles are obtained, which correspond to additional conditions.

A unique pair of triangles

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of p-adic abelian integrals.

Participation in Fraud/Cheat in the Buying and Selling of Meats Without Legal Metrology: A Theoretical and Empirical Investigations

ABSTRACTFraud/cheat in the buying and selling of commodities is part and parcel of modern businesses worldwide, and a common phenomenon in Nigeria. Personal purchases of meats using a measurement scale and others without measurement scale were made in Abakiliki and Calabar meat markets. Empirically, holding the quality of meats constant, a paired t-test analytical tool shows significant difference in the weight of meats bought and sold by non-measurement scale and those with measurement scale. Theoretically, perspectives from rational choice, fraud triangle, economic, social learning/contagious, anomie theories and political economy are drawn on to suggest an integrated multiple reinforcing-social-forces theory of fraud in the sale of meats. Results confirm that consumers receive less value for their money when they purchase meats without metrology. Cheat is conceived as a product of individual, structural variables and process that interact at different levels to produce negative consequences. An EpiCrim approach which, include strategies to promote consumers’ rights awareness, impose sanctions with certainty, celerity and pains, and other measures to assist helpless victims of cheat is advocated.