The four colour problem

In 1852, while colouring a map representing the english counties, the british mathematician Francis Guthrie realized that only four colours where necessary to satisfy the criterion that neighbouring counties should have different colours. It turns out that this is true for any (real or imaginary) map. The poster shows such a colouring for the countries of Europe and, in small, for three other imaginary maps.  It has taken more than 100 years before a correct proof for the four colour theorem has been found. The proof by Appel and Haken in 1976 has solicited much of a controversial discussion since it heavily relies on computer calculations. These are so extensive that humans cannot verify them "by hand".  Consequently,  proving the four colour theorem correct also means proving that the program is implemented correctly and that the computer works correctly.

Instead of taking Europe and its countries, we could illustrate the four colour theorem with Germany and its regions. This can be found at MathePrisma (in German). There you can also colour some other maps on your own and develop strategies to get the right colouring fast.

This poster is one within a series of three illustrating famous mathematical problems and the way mathematical thinking works. The general idea is to give the broader public an idea about what mathematics is really about (fighting the common misunderstanding that mathematics is calculation and numbers in the first place) and to trace some of the history of mathematics. The two other posters are: "the bridges of Königsberg" and "primes".

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The Bridges of Königsberg


Königsberg  (today: Kaliningrad) lies close to the baltic sea. Formerly a city of Eastern Prussia, it is now part of Russia. The 18th century philosopher Immanuel Kant is certainly the most prominent citizen of Königsberg. The mathematical problem known as the "bridges of Königsberg" is said to have been a popular riddle in 18th century Königsberg.

The challenge is to find out whether or not there is a way to walk over all seven bridges exactly once. By itself, this problem is indeed easy to understand and it could be solved, in principle, by systematically trying all possibilities. However, mathematics is more interested in understanding how the structure of the underlying problem determines whether such a way exists or not. We are therefore interested in an easy way to answer the question for all imaginable town maps with an arbitrary arrangement of rivers, land parts and connecting bridges. The text of the poster gives this general answer. It was first formulated by the mathematician Leonhard Euler (1707-1783). His approach is regarded as giving birth to modern graph theory, a branch of mathematics which has become increasingly important as a means of modelling networks, dependencies in production processes, logistic processes etc.

An animated, interactive treatment of the bridges problem with more details and a biography of Euler can be found at  MathePrisma (in German).

This poster is one within a series of three illustrating famous mathematical problems and the way mathematical thinking works. The general idea is to give the broader public an idea about what mathematics is really about (fighting the common misunderstanding that mathematics is calculation and numbers in the first place) and to trace some of the history of mathematics. The two other posters are: "the 4 colour problem" and "primes".

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Primes

The concept of a prime number is very simple and easy to understand, even for the mathematical lay person. The definition fits onto the poster without problems. One of the fascinating aspects of prime numbers is that easy sounding questions often have very involved answers - if there is an answer at all. Primes have also made their way into current technology: very large primes are the major ingredients of today's encryption systems used in e-commerce, e.g.

The proof of the fact that there are infinitely many primes is attributed to Euclid (3th century BC). Many other conjectures about prime numbers are not yet proven today: the prime twin conjecture (see poster), the Goldbach conjecture (every even number is a sum of two primes), the square numbers conjecture (there is at least one prime between two consecutive square numbers) and many more. Consult  MathePrisma (in German) for other nice conjectures and Eratosthenes' sieve as a strategie for getting all primes up to some threshold.

This poster is one within a series of three illustrating famous mathematical problems and the way mathematical thinking works. The general idea is to give the broader public an idea about what mathematics is really about (fighting the common misunderstanding that mathematics is calculation and numbers in the first place) and to trace some of the history of mathematics. The two other posters are: "the four colour problem" and "the bridges of Königsberg".

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Crash Test


Car manufacturers are ardent users of supercomputers.  "Virtual" computer simulations more and more replace  costly and time consuming "real" tests. To this purpose, engineers develop virtual prototypes of cars and then simulate the dynamics and statics of the cars on a computer. These simulations require the formulation of a complete mathematical model of the virtual car and its interaction with the road, the air etc. as a system of partial differential equations. These equations must then be solved on high performance computers using appropriate numerical algorithms.

Virtual crash tests are certainly amongst the most spectacular applications of this simulation technology. 

This poster is one within a series of two advertising Mathematics pretty much the same way industrial companies work out their public relations campaigns. Their goal is to foster the public understanding that Mathematics plays a key role in current high tech. The other poster is "computerized tomography".

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Computerized Tomography


Computerized tomography has revolutionarized medical diagnosis. As opposed to "old fashioned" X-raying, which is just taking photographs using X-rays instead of light, computerized tomography relies crucially on Mathematics in order to reconstruct images from the data taken up by the sensors moving around the body in the scanning process. Computerized tomography images are much neater and deliver more detailed information than X-ray photographs. Moreover, computerized tomography also allows to reconstruct 3-dimensional (instead of flat) images. 

This poster is one within a series of two advertising Mathematics pretty much the same way industrial companies work out their public relations campaigns. Their goal is to foster the public understanding that Mathematics plays a key role in current high tech. The other poster is "crash test".

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