历史与综述

We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times. This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.

This paper is about the beauty of fractals and the surprising connections between them. We will explain the pioneering role that the Sierpinski triangle plays in the Ulam-Warburton automata and show you a number of pictures along the way.

In a previous article we gave the general foundations of the theory of movement considered from a philosophical and mathematical point of view. Philosophical it meant to understand the opposition of the one and the multiple, mathematically…

This article has his origin in some lectures given at the University of Bologna, inside an interdisciplinary program of mathematics, history of science, physics and philosophy. Since they are at the junction of these fields, movement and…

This article considers the Dissertatio de Arte Combinatoria, published in 1666 and relatively neglected by Leibniz s scholars. However in recent times the tide seems to be changing. Our work presents three main parts (sections II, III and…

In this paper, we study the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of…

To account for the first proof of existence of an irrational magnitude, historians of science as well as commentators of Aristotle refer to the texts on the incommensurability of the diagonal in Prior Analytics, since they are the most…

The integration of digital tools in mathematics education is considered both promising and problematic. To deal with this issue, notions of webbing and instrumental orchestration are developed. However, the two seemed to be disconnected,…

In the paper, we study the situation of the mathematical community in Brno, the maintown of Moravia, between 1900 and 1930. During this time, the First World War and, as one of its consequences, the creation of the new independent…

This is the last of a trilogy of papers on triangle centers. A fairly obscure "conformal center of gravity" is computed for the class of all isosceles triangles. This calculation appears to be new. A byproduct is the logarithmic capacity or…

This primer explains how continuous-time stochastic processes (precisely, Brownian motion and other Ito diffusions) can be defined and studied on manifolds. No knowledge is assumed of either differential geometry or continuous-time…

The purpose of this short paper is to describe a project to manufacture a 3D-print of the big letters in the logo for JDRF (formerly known as the Juvenile Diabetes Research Foundation). The methods described in this paper involve processing…

Rousseau's simple proof of the quadratic reciprocity law, followed by the proof of its equivalence with Hilbert's product formula. The Hilbert symbol is explained in terms of the reciprocity isomorphism, and the places of Q are determined.

At two examples dealt with in methodologically different ways it will be pointed out how the concept of an empirical theory (in the sense of the Structuralists) can be useful to specify contents relevant to maths didactics.

In the middle of the third millennium BC the Sumerians must have noticed that the reciprocal of the number 7,in contrast to the numbers 1,2,3,4,5,and 6,could not be expressed by a finite sexagesimal fraction but it recurred every three…

The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential…

This material is a rewriting and expansion of notes for beginning graduate students in seminars in combinatorics (Department of Mathematics, University of California San Diego). Solid skills in linear and multilinear algebra were required…

Given a dollar, how many ways are there to make change using pennies, nickels, dimes, and quarters? What if you are given a different amount of money? What if you use different coin denominations? This is a well known problem and formulas…

Starting from a short review of the "classical" space problem in the sense of the 19th century (Helmholtz -- Lie -- Klein) it is discussed how the challenges posed by special and general relativity to the classical analysis were taken up by…

The purpose of this short paper is to describe a project to manufacture a regular octohedron on a 3D printer. We assume that the reader is familiar with the basics of 3D printing. In the project, we use fundamental ideas to calculate the…