Related papers: The ancient Greeks present: Rational Trigonometry
To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for…
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.
Pythagorean triples are the positive integer solutions to the Pythagoras equation for right triangles, a2+b2 = c2. They have been studied for many years, many centuries in fact. In this short paper we present a method for computing…
Noticing that all of the 19th, 20th and 21st centuries treatments of trigonometry surveyed in this article are conceptually or logically defective, it is required to seek a conceptually sound and logically correct foundations of the…
Any permutation-invariant function of data points $\vec{r}_i$ can be written in the form $\rho(\sum_i\phi(\vec{r}_i))$ for suitable functions $\rho$ and $\phi$. This form - known in the machine-learning literature as Deep Sets - also…
The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…
We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the…
The formula for the dihedral angle of the simplex of n dimensions, arccos(1/n), is derived using classical geometry.
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's…
The relevance of this paper lies in the fact that it resolves two previously unsolved open problems. In the first part of the paper, a new lemma is proved, from which it follows that if there exists a triangle with integer sides and…
While the contents of Euclid's Elements are well-known these days, some characters of the original text have been overlooked due to interpretation by modern mathematical languages. The lens of modern mathematics once anachronistically…
Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum…
The author proposes a new geometry in this book. The author named this new geometry Intercenter Geometry. Intercenter Geometry is different from traditional Euclidean geometry and analytic geometry (coordinate geometry). The idea of…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
Recollect that Heron's formula for the area of a triangle given its sides has a counterpart given the medians instead, which carries an extra factor of $\frac{4}{3}$. On the one hand, we formulate the pair of these in Linear Algebra terms,…
This paper presents algorithms for computing the length of a sum of squares and a Pythagoras element in a global field $K$ of characteristic different from $2$. In the first part of the paper, we present algorithms for computing the length…
We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles,…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were…
Unquestionably the greatest discovery of the Pythagoreans is the existence of incommensurable magnitudes, most probably the incommensurability of the diameter to the side of a square, but there is no agreement among historians of Greek…