## Counting Integer Lattice Points inside a Plane Figure

In this short article we will present some problems related to the calculation of points with integer coordinates, contained inside or on the boundary of some plane figures. This type of problem has very interesting aspects and finds important applications Read more…

## Bertrand Hypothesis and Ramanujan Prime Numbers

The French mathematician Bertrand (1822-1900) formulated the conjecture that for every positive integer $$n$$ there is always at least one prime number $$p$$ such that $n \lt p \le 2n$ This conjecture was proved by the Russian mathematician Chebyshev (1821-1894). In Read more…

## Splines and Bézier Curves and their application in Video Games

The geometry of curves and surfaces is of fundamental importance in computer graphics and in video game programming. In this article we will describe the mathematics of interpolation curves, in particular splines and Bézier curves, with examples of use in Read more…

## Sprite Animation in Unity 3D and Finite State Machines

Animation has an essential role in video games. Most video games try to simulate real world physics; in order to achieve this objective it is essential to be able to represent, in the most realistic way, the movement of people, Read more…

## Ordinary Generating Functions and Recurrence Equations

1) Ordinary generating functions of a variable Generating functions are an important tool for solving combinatorial problems of various types. A typical problem is the counting of the number of objects as a function of the size $$n$$, which Read more…

## Lambert Series, the Arithmetic Function $$r(n)$$ and Gauss’s Probability Integral

In this article we will study some properties of Lambert series. Then, using Lambert series relative to the representation of integers as sum of two squares, we will compute the value of Gauss’s classical probability integral. 1) Dirichlet generating functions Let’s Read more…

## Motion in a Plane and Unity’s 2D Physics Engine

This article gives a brief illustration of Newton’s laws of motion in the plane and analyzes the tools that the Unity 2D engine makes available to programmers to simulate the movement of bodies in the 2D environment.For the study of Read more…

## Iterated Function Systems, Fractals and Sierpinski Triangle

In a previous article we have introduced some examples of fractals, illustrating their main characteristics, both qualitative and quantitative: self-similarity, geometric irregularity, fractional dimension. To continue the study of fractal science, it’s first necessary to give a more rigorous definition Read more…

## Dirichlet’s Box Principle and Ramsey Numbers

The box principle is attributed to the German mathematician Dirichlet (1805-1859). It is also called the pigeonhole principle. This article illustrates Dirichlet’s principle and gives a brief introduction to Ramsey’s theory, with some examples of computation of Ramsey numbers. For Read more…

## Euler and Möbius Arithmetic Functions and RSA Cryptography

This article illustrates the properties of the Euler and Möbius functions, which have great importance in Number Theory and in other fields. As an example of application we describe the RSA algorithm for public key cryptography. 1) Arithmetic functions An Read more…