Klein Project Blog

Originating authors are Dirce Uesu Pesco and Humberto José Bortolossi.
The images you see on internet pages and the photos you take with your mobile phone are examples of digital images. It is possible to represent this kind of image using matrices. For example, the small image of Felix the Cat (on the left) can be represented by a matrix whose elements are the numbers and . These numbers specify the color of each pixel (a pixel is the smallest graphical element of a matricial image, which can take only one color at a time): the number indicates black, and the number indicates white. Digital images using only two colors are called binary images or boolean images.

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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 and outside mathematics. In Section 3 we will discuss the fascinating application to image compression.

But, let us start with our game and look at the famous lid of a box of The Laughing Cow.

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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 Congress of Mathematicians in 1900, David Hilbert gave a very famous lecture in which he stated 23 problems that would have deep significance for the advance of mathematical science in the 20-th century. The problem of the densest packing of spheres, also called Kepler’s conjecture, is part of Hilbert’s 18-th problem. Kepler’s conjecture was only proved in 1998 by Thomas Hales, and the details of the proof were published in 2006.

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Figure 1: Illustration of a Calabi-Yau-manifold (Important for the description of higher dimensional models in superstring-theory).

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 possible to get a perception of these objects or do they withdraw any representation? The Theory of Relativity uses four dimensions to explain the concept of space-time, six dimensions are necessary to describe the bending of space-time and different string theories even use representations in up to dimensions (e.g. L. Botelho, R. Botelho, 1999). Another current domain of application for higher dimensional objects and their three-dimensional representations is the study of non-periodic structures in modern crystallography. Within the concept of quasicrystals projections of higher dimensional point-sets (such as the integer-lattice in dimension ) to three dimensional space are supposed to be good models for non-periodic crystalline structures (see section 5 below).

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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 Benford’s law, or law of the first significant digit. If one forgets to follow the rule, then the numbers will fail some statistical tests and are likely to be scrutinized with care. Benford’s law claims that if you collect numbers at random and calculate the frequencies of their first significant digits, the numbers with first significant digit should appear around % of the time, while the numbers with first significant digit appear only % of the time. This rule is observed in many other sets of numbers, like powers of or Fibonacci numbers.

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This picture is a property of mathscareers.org.uk, who kindly granted permission to use it in this work.

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 three colours are sufficient for a particular map.

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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 on.
At the end of the 19th century, the Russian mathematician and mineralogist Yevgraf Fyodorov established that there are types of symmetry for patterns in the plan [WPG]. That is, we can have exactly different wallpapers in terms of replications of symmetry, and no more! Notably, all these types of symmetries can be found in decorative arts in antiquity.

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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 octagon ? The answer is : « No » and it can be proved, for the octagon, as follows (Payan, 1994) :

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