Originating authors are David Mumford and Christiane Rousseau.
Figure 1: The shock wave caused by a supersonic jet.
Foreword: This vignette is more difficult than others. However, in a few pages, it tells you how to explain in simple terms one of the most difficult open problems at the beginning of the 21st century. The vignette contains enrichment material, that you can choose to read or skip. The editors of the Klein blog hesitated for a while posting this vignette. After testing it with teachers during two Klein workshops, who expressed that they enjoy being challenged by more difficult vignettes, they decided to test it on the blog. They are eager to hear your comments, and if some of you were motivated by this topic.
You have probably heard some planes breaking the sound barrier. What does that mean? It means that a shock wave in the atmosphere is created as in Figure 1. But what is a shock wave? Imagine heavy traffic on highways as a wave. A shock wave corresponds to collisions. To explain this, we develop our intuition with a 1D model: traffic on a one lane road at different speeds. You know that collisions could occur if drivers do not adjust their speed. The atmosphere is a fluid, and traffic is a rough model of 1D fluid which is convenient to develop our intuition. Under which conditions do shock waves or other singularities occur in fluids? A million dollar prize is offered for answering this question. This is what we are going to explain you.
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) :
Originating authors are Graeme L. Cohen (University of Technology, Sydney), Steven Galbraith (University of Auckland) and Edoardo Persichetti (University of Auckland).
How can we safely send our credit card details over the internet, or using a mobile phone, when others can intercept our messages? How can we trust software updates, when we know that computer viruses are common? Cryptography (the study of techniques for secure communication in the presence of adversaries) provides answers to these questions, and mathematics provides its foundations.
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.
Originating author is Marcelo Escudeiro Hernandes.
This picture is a property of mathscareers.org.uk, who kindly granted permission to use it in this work.
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.
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).
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.
Originating authors are Michèle Artigue, Ferdinando Arzarello and Susanna Epp.
Studying the evolution of a natural phenomenon often leads to studying numerical sequences , especially their long-term behavior and whether they eventually converge. Polynomial, exponential, and logarithmic sequences are frequently encountered in secondary school, but certain other sequences with very simple definitions exhibit much more complex behavior. Examples include the chaotic sequences that arise in the study of dynamical systems (see ) and the Syracuse sequence (or sequence), introduced by Luther Collatz in 1937. The Syracuse sequence has challenged mathematicians for decades. Despite the huge number of values that have been computed, it is currently unknown whether the sequence is infinite or is finite and always ends in (see ).