Related papers: Bridging Classical and Modern Approaches to Thales…

An elementary geometric construction known as Napoleon's theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward,…

We prove the following result on the timelike spherical Hilbert geometry of simplices: Let $\Delta_2$ be a simplex on the 2-sphere and $\tilde{\Delta}_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated…

Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by…

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph…

In the present chapter, we obtain the reconstruction of Theaetetus' theory of ratios of magnitudes based, according to Aristotle's Topics 158b, on the definition of proportion in terms of equal anthyphairesis. Our reconstruction is built on…

Unlike some other formal systems, the proof system Metamath has no built-in concept of "decimal number" in the sense that arbitrary digit strings are not recognized by the system without prior definition. We present a system of theorems and…

We present the first example of an interacting Carroll supersymmetric field theory with both temporal and spatial derivatives, belonging to the Galileon class, where the non-linear field equation remains second-order in derivative. To…

According to Aristotle, a philosopher in Ancient Greece, "the whole is greater than the sum of its parts". This observation was adopted to explain human perception by the Gestalt psychology school of thought in the twentieth century. Here,…

After having investigated the geodesic and translation triangles and their angle sums in $\SOL$ and $\SLR$ geometries we consider the analogous problem in $\NIL$ space that is one of the eight 3-dimensional Thurston geometries. We analyze…

D. Hensley showed in 1994 that the number of steps taken by the Euclidean algorithm to find the greatest common divisor of two natural numbers less than or equal to n follows a normal distribution in the limit as n tends to infinity. V.…

We report on a pedagogical experiment to make mathematics easy by changing its philosophy. The Western philosophy of math originated in religious beliefs about mathesis, cursed by the church. Later, mathematics was "reinterpreted", in a…

To improve our understanding of connected systems, different tools derived from statistics, signal processing, information theory and statistical physics have been developed in the last decade. Here, we will focus on the graph comparison…

Using concepts and techniques of bilinear algebra, we construct hyperbolic planes over a euclidean ordered field that satisfy all the Hilbert axioms of incidence, order and congruence for a basic plane geometry, but for which the hyperbolic…

The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three…

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make…

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…

Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using…

Classical mathematical statistics deals with models that are parametrized by a Euclidean, i.e. finite dimensional, parameter. Quite often such models have been and still are chosen in practical situations for their mathematical simplicity…

The purpose of this paper is to give a twisted version of the Eynard-Orantin topological recursion by a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic definition for a topological recursion to…

Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a…