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For two years, we have been working on constant diameter sets. They are flat geometrical figures with a distance between two points which is always inferior or equal to the diameter. The circle is an example of constant diameter set but there are many others. After drawing and building some models of these sets, we tried to work out properties that they could have. Then, we did a study in 3D. For a few months we have been thinking about applications of the theory. Cams and British coins are the most famous applications. But the application to tubular boilers is an innovation. Moreover it’s a good illustration of the properties of constant diameter sets and could result in a patent.

Understanding Distance (Invariants of Finite Metric Spaces): Metric spaces are one of the key notions in mathematics. A metric space is a set (whose elements are ``points’’) equipped with a notion of distance, that is to every pair of points we assign a nonnegative real number, the distance between them. This geometric notion has found many application, not only in geometry itself but also in algebra and number theory, where we can use the relation of division by a given prime number to define distance between integers. The reason for usefulness of metric spaces lies within the definition, which is simple enough to capture a broad class of examples, while expressive enough to build a rich theory. My paper approaches the study of (finite) metric spaces by assigning to them various invariants which are less complicated than the spaces themselves, but allow us to distinguish between spaces in a simple way. These invariants are twofold: algebraic and combinatorial. The algebraic invariant is the ``isometry group’’ of a space, that is a set with a binary operation (we can think of it as ``multiplication’’); the combinatorial one is a graph (a set of vertices connected by edges) which decodes information about distance in a simpler way that is easier to implement. The main theorem shows how to compare these invariants. Professor Peter Clark (University of Georgia, USA) suggested on a mathematical forum, Math Overflow, that these invariants contain the same information about a given metric space. The paper shows that this in fact is not the case and studies the relation between them in detail.

“Islamic artists were 500 years ahead of Western scientists” –headlines like this one appeared all over the world when an acclaimed Science paper investigated Islamic mosaics in 2007. These so-called girih patterns were suggested to have been conceived as tilings, the jigsaw-like geometrical concept of putting shapes together to cover the plane. The Science authors argued that some of these Islamic tilings have a quasi-periodic structure – whereas in the Western world, such complex structures were not invented until the 1970s. However, my analysis of the Science patterns and six additional mosaics from Iran suggests that this sensational claim should be revised: The Science proof of the quasiperiodicity is not convincing. In addition, the mosaics were not constructed in the 15th but rather in the 18th or 19th century. My discovery of Ammann bars – a feature of quasi-periodic tilings – in girih patterns might serve as an alternative proof of their quasi-periodicity.

The four-color theorem states that every planar map can be colored with four colors such that every adjacent region is colored with a different color. However nothing is said about the number of possible colorings. This research inspects the impact of the features of planar maps on the number of ways the regions can be colored through graph theory and computer-assisted simulation. We present a relation between the features and colorings applicable to e.g. time-complexity analysis and optimization of algorithms. We created a program which calculates the possible colorings and inspected a total of 633 maps used to prove the four-color theorem. We present a formula y = 3338754,0 ∙1, 7997866 ∙ z -17;705562, which ties together the colorings y, the vertices (regions) x and the vertices’ proportion to edges (adjacent regions) z. The formula’s coefficient of determination in our cases was 99.98%, which is enough for analysing computational complexity.

Consider the following deterministic game (a percolation game) suggested by the mathematician Itai Benjamini in 2002. Two species compete on a graph: the red and the blue. Start with one red vertex and one blue vertex. Each second the blue colour blue all its neighbours which are not yet coloured. The red is lazy and grow only at even times. If both species aim at the same vertex, it becomes blue. A question arises: will the blue surround the red after some finite time? If the red will be able to grow infinitely many times, we say that the blue lose, otherwise the blue win. In our work, we investigate the game on Cayley graphs of finitely generated abelian groups and Cayley graphs of the integer Heisenberg group.

A perfect cuboids (PC) is a rectangular parallelepiped with rational sides: a, b, c whose face diagonals d_ab, d_bc, d_ac and space (body) diagonal d_s are rationales. The existence or otherwise of PC is a problem known since at least the time of Leonard Euler (1707-1783).

The perfect cuboids problem is equivalent to the following system[1], [2]: {█(2mn(1-k^2 )=(1-m^(2 ) )(1-n^2 )k@2mn(1-p^(2 ) )=(1-m^2 )(1+n^2 )p)┤

Our research addresses the solution of the second equation. In our project we find all rational solutions of the second equation of the given system. n=√XZ/N

m=√((X-N)/(X+N) (Z+N)/(Z-N))

p=√(Z/X (X^2-N^2)/(Z^2-N^2 ))

These solutions are obtained by two arbitrary nontrivial different rational solution of arbitrary congruent number equation. Ten solutions in case of congruent number N=34 are found. Finding Perfect Cuboids, doesn’t only mean finding an answer for a three-century problem, but it can also be used for science for environment. Mainly it will be used in architecture, for constructing music halls with the best acoustic.

Zobrazené:

**Mathematics**(total**6**)### Constant diameter sets and their applications

*France, Mathematics*

**Fabian Rainouard, Margaux Le Brun**

For two years, we have been working on constant diameter sets. They are flat geometrical figures with a distance between two points which is always inferior or equal to the diameter. The circle is an example of constant diameter set but there are many others. After drawing and building some models of these sets, we tried to work out properties that they could have. Then, we did a study in 3D. For a few months we have been thinking about applications of the theory. Cams and British coins are the most famous applications. But the application to tubular boilers is an innovation. Moreover it’s a good illustration of the properties of constant diameter sets and could result in a patent.

Actualized: 18 Sep 2013

### Invariants of Finite Metric Spaces

*Poland, Mathematics*

**Aleksander Horawa**

Understanding Distance (Invariants of Finite Metric Spaces): Metric spaces are one of the key notions in mathematics. A metric space is a set (whose elements are ``points’’) equipped with a notion of distance, that is to every pair of points we assign a nonnegative real number, the distance between them. This geometric notion has found many application, not only in geometry itself but also in algebra and number theory, where we can use the relation of division by a given prime number to define distance between integers. The reason for usefulness of metric spaces lies within the definition, which is simple enough to capture a broad class of examples, while expressive enough to build a rich theory. My paper approaches the study of (finite) metric spaces by assigning to them various invariants which are less complicated than the spaces themselves, but allow us to distinguish between spaces in a simple way. These invariants are twofold: algebraic and combinatorial. The algebraic invariant is the ``isometry group’’ of a space, that is a set with a binary operation (we can think of it as ``multiplication’’); the combinatorial one is a graph (a set of vertices connected by edges) which decodes information about distance in a simpler way that is easier to implement. The main theorem shows how to compare these invariants. Professor Peter Clark (University of Georgia, USA) suggested on a mathematical forum, Math Overflow, that these invariants contain the same information about a given metric space. The paper shows that this in fact is not the case and studies the relation between them in detail.

Actualized: 18 Sep 2013

### LSLLSLSLLSLLSLS – Modern Mathematics in Islamic Mosaics

*Switzerland, Mathematics*

**Jasmin Allenspach**

“Islamic artists were 500 years ahead of Western scientists” –headlines like this one appeared all over the world when an acclaimed Science paper investigated Islamic mosaics in 2007. These so-called girih patterns were suggested to have been conceived as tilings, the jigsaw-like geometrical concept of putting shapes together to cover the plane. The Science authors argued that some of these Islamic tilings have a quasi-periodic structure – whereas in the Western world, such complex structures were not invented until the 1970s. However, my analysis of the Science patterns and six additional mosaics from Iran suggests that this sensational claim should be revised: The Science proof of the quasiperiodicity is not convincing. In addition, the mosaics were not constructed in the 15th but rather in the 18th or 19th century. My discovery of Ammann bars – a feature of quasi-periodic tilings – in girih patterns might serve as an alternative proof of their quasi-periodicity.

Actualized: 18 Sep 2013

### Map Coloring - Finding the Number of Colorings of Maps Colorable with Four Colors

*Finland, Mathematics*

**Eero-Pekka Räty, Samuli Thomasson**

The four-color theorem states that every planar map can be colored with four colors such that every adjacent region is colored with a different color. However nothing is said about the number of possible colorings. This research inspects the impact of the features of planar maps on the number of ways the regions can be colored through graph theory and computer-assisted simulation. We present a relation between the features and colorings applicable to e.g. time-complexity analysis and optimization of algorithms. We created a program which calculates the possible colorings and inspected a total of 633 maps used to prove the four-color theorem. We present a formula y = 3338754,0 ∙1, 7997866 ∙ z -17;705562, which ties together the colorings y, the vertices (regions) x and the vertices’ proportion to edges (adjacent regions) z. The formula’s coefficient of determination in our cases was 99.98%, which is enough for analysing computational complexity.

Actualized: 18 Sep 2013

### Percolation games on Cayley graphs of groups

*Belarus, Mathematics*

**Maksim Bezrukov, Aliaksandr Stadolnik**

Consider the following deterministic game (a percolation game) suggested by the mathematician Itai Benjamini in 2002. Two species compete on a graph: the red and the blue. Start with one red vertex and one blue vertex. Each second the blue colour blue all its neighbours which are not yet coloured. The red is lazy and grow only at even times. If both species aim at the same vertex, it becomes blue. A question arises: will the blue surround the red after some finite time? If the red will be able to grow infinitely many times, we say that the blue lose, otherwise the blue win. In our work, we investigate the game on Cayley graphs of finitely generated abelian groups and Cayley graphs of the integer Heisenberg group.

Actualized: 18 Sep 2013

### Space diagonal equation of perfect cuboids

*Georgia, Mathematics*

**Ketevan Topchishvili**

A perfect cuboids (PC) is a rectangular parallelepiped with rational sides: a, b, c whose face diagonals d_ab, d_bc, d_ac and space (body) diagonal d_s are rationales. The existence or otherwise of PC is a problem known since at least the time of Leonard Euler (1707-1783).

The perfect cuboids problem is equivalent to the following system[1], [2]: {█(2mn(1-k^2 )=(1-m^(2 ) )(1-n^2 )k@2mn(1-p^(2 ) )=(1-m^2 )(1+n^2 )p)┤

Our research addresses the solution of the second equation. In our project we find all rational solutions of the second equation of the given system. n=√XZ/N

m=√((X-N)/(X+N) (Z+N)/(Z-N))

p=√(Z/X (X^2-N^2)/(Z^2-N^2 ))

These solutions are obtained by two arbitrary nontrivial different rational solution of arbitrary congruent number equation. Ten solutions in case of congruent number N=34 are found. Finding Perfect Cuboids, doesn’t only mean finding an answer for a three-century problem, but it can also be used for science for environment. Mainly it will be used in architecture, for constructing music halls with the best acoustic.

Actualized: 18 Sep 2013