Category Archives: Mathematics Within the Last 100 Years

Banach’s microscope to find a fixed point

Originating author is Christiane Rousseau. In this vignette, we will show how we start from a small game to discover one of the most powerful theorems of mathematics, namely the Banach fixed point theorem. This theorem has fantastic applications inside … Continue reading

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What is the way of packing oranges? — Kepler’s conjecture on the packing of spheres

Originating author is Christiane Rousseau. What is the densest packing of spheres? Kepler conjectured that it was the one you observe with oranges at the fruit shop, and which is called the face-centered cubic lattice (Figure 1). At the International … Continue reading

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Higher Dimensions

Originating authors are Markus Ruppert and Hans-Georg Weigand. 1. Looking for the next dimension Does our world really have more than three dimensions? If so, do objects in higher dimension have a relation to the world around us? Is it … Continue reading

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Benford’s law: learning to fraud or to detect frauds?

Originating author is Christiane Rousseau. It is very risky to change too many numbers in some fi nancial statements if one does not know some mathematics. Indeed, most often the numbers appearing in fi nancial statements follow some strange mathematical rule, called … Continue reading

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Map colouring and Gröbner Bases

Originating author is Marcelo Escudeiro Hernandes. By the famous “Four Colour Theorem”, only four colours we need to colour a map so that no bordering regions have the same colour. Using polynomial equations and Gröbner bases we can determine if … Continue reading

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Symmetry Step by Step

Originating author is Ana Cannas da Silva. Symmetry has always fascinated and served humankind in architecture, arts, engineering and science. Over thousands of years symmetric patterns have been used to create fabrics, baskets, floors, wallpapers and wrapping papers, and so … Continue reading

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Recurrence and induction

Originating authors are Michèle Artigue and Ferdinando Arzarello. Given a square grid, it is easy to draw squares whose vertices are intersections of the grid lines. But is it possible to do so for other regular polygons, for instance an … Continue reading

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Trying to predict a floating leaf: chaos and predictions

Originating author is César R. de Oliveira, Universidade Federal de São Carlos. What path will a leaf follow floating down a turbulent stream? Is it even possible to make a mathematical model that will predict such motion? Is this the … Continue reading

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How to get rid of quantifiers?

Originating authors are Reinhard Oldenburg and Michele Artigue. How do computer packages do abstract algebraic problems such as proving statements “for all ” or finding whether a Real Number with certain conditions exists? Recent advances draw on theorems in mathematical … Continue reading

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The Revenge of the Infinitesimals

Originating author is Michèle Artigue. Infinitesimals played an essential role in the emergence and development of differential and integral calculus. The evident productivity of this calculus did not prevent recurrent and fierce debates about the nature of these objects and … Continue reading

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