Related papers: A simple method for generating rational triangles
A triangular field of rational numbers is characterized, with relations to Stirling numbers 2nd, Hyperbolic functions, and centered Binomial distribution. A Generating function is given.
In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we…
A Heronian triangle is a triangle that has integer side lengths and integer area. Praton and Shalqini [1] define amicable Heronian triangles to be two Heronian triangles where the area of one equals the perimeter of the other, and vice…
One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper…
The Pythagorean triples have the structure of a ternary rooted tree; the tree is based on the Cayley graph of a free subgroup of the modular group
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as…
Let X be a scroll over a rational surface. We construct a linear system of surfaces in P^3 yielding a birational map from P^3 to X. We apply this construction to the scrolls of Bordiga and Palatini.
A Heron triangle is a triangle whose side lengths and area are integers. Two Heron triangles are amicable if the perimeter of one is the area of the other. We show, using elementary techniques, that there is only one pair of amicable Heron…
This article discusses two versions of elliptic equations obtained from a system of equations describing a rational cuboid. Analysis of elliptic equations shows that they are equivalent, and that there are rational points on the elliptic…
The paper found a geometric and algebraic interpretation of the parameters m and n from the formulas for obtaining primitive Pythagorean triples, which are solutions of the equation ${x^2+y^2=z^2}$, namely: ${x=m^2-n^2}$, ${y=2mn}$,…
This article proves a Pythagoras-type formula for the sides and diagonals of a polygon inscribed in a semicircle having one of the sides of the polygon as diameter.
This article reports the occurrence of binary quadratic forms in primitive Pythagorean triangles and their geometric interpretation. In addition to the well-known fact that the hypotenuse, z, of a right triangle, with sides of integral…
In 1934 B. Berggren first discovered the surprising result that every Pythagorean triplet is the pre product of the triplet (3, 4, 5) presented as a column by a product of three matrices, that every triplet is obtained in this manner…
A triangular partition is a partition whose Ferrers diagram can be separated from its complement (as a subset of $\mathbb{N}^2$) by a straight line. Having their origins in combinatorial number theory and computer vision, triangular…
Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define…
We introduce a bulging triangle like the generalization of the Reuleaux triangle. We may be able to propose various ways to bulge a triangle, but this paper presents the way so that its vertices are the same as them of the original…
A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $\Delta$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a…
In this article we exploit the fact that the special relativistic formula which relates the energy and the 3-momentum of an elementary particle with its rest mass, resembles the pythagorean theorem for right triangles. Using such triangles,…