Related papers: A different perspective on teaching Geometry at hi…

Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…

We show how Cartesian method can be used in the proof of fundamental planimetric topics of the school course, such as introduction of trigonometric functions, equation of a line and similarity of triangles. This work also can be considered…

One of the goals that physicists have been pursuing is to get the same explanation from different angles for the same phenomenon, so as to realize the unity of basic physical laws. Geometry, classical mechanics and classical thermodynamics…

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods,…

This paper frames calculus as a global, centuries-long development rather than a subject that began only with Newton and Leibniz. Drawing on ideas from Greek, Indian, Islamic, and later European mathematics, it highlights how concepts like…

An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…

We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…

This is an essay that considering the knowledge structure and language of a different nature, attempts to build on an explanation of the object of study and characteristics of the mathematical science. We end up with a learning cycle of…

An approach is shown that proves various theorems of plane geometry in an algorithmic manner. The approach affords transparent proofs of a generalization of the Theorem of Morley and other well known results by casting them in terms of…

A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…

This is an elementary geometrical proof of Birkhoff theorem. It is hardly important, but the pictures behind are quite nice.

Inspired by a didactic experience in an academic environment, and following the idea given by M. Villa in \cite{Villa}, we illustrate two different proofs of an important result in Euclidean geometry studied in the first two years of…

Objects are a centerpiece of the mathematical realm and our interaction with and reasoning about it, just as they are of the physical one (if not more). And humans' mathematical reasoning must ultimately be grounded in our general…

Poincar\'e held the view that geometry is a convention and cannot be tested experimentally. This position was apparently refuted by the general theory of relativity and the successful confirmation of its predictions; unfortunately,…

Since the subject of noncommutative geometry is now entering maturity, we felt there is need for presentation of the material at an undergraduate course level. Our review is a zero order approximation to this project. Thus, the present…

The initial techniques developed in Euclid's Elements, well before the use of the parallel postulate, are reexamined in order to clarify even the most obscure details, particularly those related to equality, superposition and angle…

The reasons of the crisis in the contemporary (Riemannian) geometry are discussed. The conventional method of the generalized geometries construction, based on a use of the topology, leads to a overdetermination of the Riemannian geometry.…

In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same…

The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties. In this paper, we present algorithms for…