Mathieu Dutour Sikiric | Ruder Boskovic Institute (original) (raw)

Books by Mathieu Dutour Sikiric

Research paper thumbnail of Random Sequential Packing of Cubes

In this volume very simplified models are introduced to understand the random sequential packing ... more In this volume very simplified models are introduced to understand the random sequential packing models mathematically. The 1-dimensional model is sometimes called the Parking Problem, which is known by the pioneering works by Flory (1939), Renyi (1958), Dvoretzky and Robbins (1962). To obtain a 1-dimensional packing density, distribution of the minimum of gaps, etc., the classical analysis has to be studied. The packing density of the general multi-dimensional random sequential packing of cubes (hypercubes) makes a well-known unsolved problem. The experimental analysis is usually applied to the problem. This book introduces simplified multi-dimensional models of cubes and torus, which keep the character of the original general model, and introduces a combinatorial analysis for combinatorial modelings.

Research paper thumbnail of Geometry of Chemical Graphs: Polycycles and Two-faced Maps

Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling of molecu... more Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling of molecular structures, such as the Nobel prize winning fullerenes, occurring in chemistry and crystallography. The chemistry has inspired and informed many interesting questions in mathematics and computer science, which in turn have suggested directions for synthesis of molecules. Here the authors give access to new results in the theory of polycycles and two-faced maps together with the relevant background material and mathematical tools for their study. Organised so that, after reading the introductory chapter, each chapter can be read independently from the others, the book should be accessible to researchers and students in graph theory, discrete geometry, and combinatorics, as well as to those in more applied areas such as mathematical chemistry and crystallography. Many of the results in the subject require the use of computer enumeration ; the corresponding programs are available from the author’ s website.

Papers by Mathieu Dutour Sikiric

Research paper thumbnail of Generalized cut and metric polytopes of graphs and simplicial complexes

Optimization Letters, 2018

Research paper thumbnail of On the toplogical computation of K4 of the Gaussian and Eisenstein integers

arXiv: K-Theory and Homology, 2014

In this paper we use topological tools to investigate the structure of the algebraic K-groups K_4... more In this paper we use topological tools to investigate the structure of the algebraic K-groups K_4 (Z[i]) and K_4 (Z[rho]), where i := sqrt{-1} and rho := (1+sqrt{-3})/2. We exploit the close connection between homology groups of GL_n(R) for n = 5.

Research paper thumbnail of Geometry of Chemical Graphs: References

Research paper thumbnail of A New Algorithm in Geometry of Numbers

4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), 2007

Research paper thumbnail of Письмо в редакцию

Успехи математических наук, 2013

Research paper thumbnail of On classifying Minkowskian sublattices

Mathematics of Computation, 2011

Research paper thumbnail of Hyperbolic analogues of fullerenes on orientable surfaces

Discrete Mathematics, 2012

Research paper thumbnail of Applications of elementary decompositions to (<i>r, q</i>)-polycycles

Polycycles and Two-faced Maps, 2008

Research paper thumbnail of Combinatorial cube packings in cube and torus

We consider sequential random packing of cubes z+[0,1]^n with z∈1/N^n into the cube [0,2]^n and t... more We consider sequential random packing of cubes z+[0,1]^n with z∈1/N^n into the cube [0,2]^n and the torus ^n2^n as N→∞. In the cube case [0,2]^n as N→∞ the random cube packings thus obtained are reduced to a single cube with probability 1-O(1/N). In the torus case the situation is different: for n≤ 2, sequential random cube packing yields cube tilings, but for n≥ 3 with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on N depend on some parameters. We use use them to derive an expansion of the packing density in powers of 1/N. The explicit computation is done in the cube case. In the torus case, the situation is more complicate and we restrict ourselves to the case N→∞ of strictly positive probability. We prove the following results for torus combinatorial cube packings: We give a general Cartesian product construction. We prove that the number of parameters is at least n(n...

Research paper thumbnail of (2,3, 6)-SPHERES and Their Generalizations

We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical... more We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6-regular plane graph G0, two integers k and l and returns two 6-regular plane graphs. Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and we prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.

Research paper thumbnail of 4-REGULAR and Self-Dual Analogs of Fullerenes

An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of th... more An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties and explain some new algorithms that allow their efficient enumeration. Using this we give the symmetry groups of all i-hedrites and the minimal representative for each. We also review the link of 4-hedrites with knot theory and the classification of 4-hedrites with simple central circuits. An i-self-hedrite is a self-dual plane graph with faces and vertices of size/degree 2, 3 and 4. We give a new efficient algorithm for enumerating them based on i-hedrites. We give a classification of their possible symmetry groups and a classification of 4-self-hedrites of symmetry T, Td in terms of the Goldberg-Coxeter construction. Then we give a method for enumerating 4-self-hedrites with simple zigzags.

Research paper thumbnail of A generalization of Voronoi's reduction theory and its application

We consider Voronoi's reduction theory of positive definite quadratic forms which is based on... more We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.

Research paper thumbnail of Inhomogeneous extreme forms

G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for latti... more G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension 6 and prove their existence in all dimensions beyond. New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast to the packing problem.

Research paper thumbnail of The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices

In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay su... more In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. We obtain a complete list of 110244 affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra. Using a refinement of corresponding secondary cones, we obtain 181394 contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check.

Research paper thumbnail of The seven dimensional perfect Delaunay polytopes and Delaunay simplices

For a lattice L of R^n, a sphere S(c,r) of center c and radius r is called empty if for any v∈ L ... more For a lattice L of R^n, a sphere S(c,r) of center c and radius r is called empty if for any v∈ L we have v - c≥ r. Then the set S(c,r)∩ L is the vertex set of a Delaunay polytope P=conv(S(c,r)∩ L). A Delaunay polytope is called perfect if any affine transformation ϕ such that ϕ(P) is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if n=1 or n≥ 6 and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension 7 which allow us to find that there are only two perfect Delaunay polytopes: 3_21 which is a Delaunay polytope in the root lattice E_7 and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension 7 and found 11 types.

Research paper thumbnail of Delaunay polytopes derived from the Leech lattice

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its verti... more Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S∩ L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S∩ L is exactly S. For a vector v of the Leech lattice Λ_24 we define Λ_24(v) to be the lattice of vectors of Λ_24 orthogonal to v. We studied Delaunay polytopes of L=Λ_24(v) for |v|^2<=22. We found some remarkable examples of Delaunay polytopes in such lattices and disproved a number of long standing conjectures. In particular, we discovered: --Perfect Delaunay polytopes of lattice width 4; previously, the largest known width was 2. --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay polytopes in superlattices of L of the same dimension. --Polytopes that are perfect Delaunay with respect to two lattices L⊂ L' of the same dimension. --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously known examples ...

Research paper thumbnail of Computing symmetry groups of polyhedra

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as we... more Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used for instance in integer linear programming.

Research paper thumbnail of The decomposition of the hypermetric cone into L-domains

The hypermetric cone _n+1 is the parameter space of basic Delaunay polytopes in n-dimensional lat... more The hypermetric cone _n+1 is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone _n+1 is polyhedral; one way of seeing this is that modulo image by the covariance map _n+1 is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone _n+1 of hypermetrics on n+1 points contains exactly 1/2n! principal L-domains. We give a detailed description of the decomposition of _n+1 for n=2,3,4 and a computer result for n=5 (see Table <ref>). Remarkable properties of the root system D_4 are key for the decomposition of _5.

Research paper thumbnail of Random Sequential Packing of Cubes

In this volume very simplified models are introduced to understand the random sequential packing ... more In this volume very simplified models are introduced to understand the random sequential packing models mathematically. The 1-dimensional model is sometimes called the Parking Problem, which is known by the pioneering works by Flory (1939), Renyi (1958), Dvoretzky and Robbins (1962). To obtain a 1-dimensional packing density, distribution of the minimum of gaps, etc., the classical analysis has to be studied. The packing density of the general multi-dimensional random sequential packing of cubes (hypercubes) makes a well-known unsolved problem. The experimental analysis is usually applied to the problem. This book introduces simplified multi-dimensional models of cubes and torus, which keep the character of the original general model, and introduces a combinatorial analysis for combinatorial modelings.

Research paper thumbnail of Geometry of Chemical Graphs: Polycycles and Two-faced Maps

Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling of molecu... more Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling of molecular structures, such as the Nobel prize winning fullerenes, occurring in chemistry and crystallography. The chemistry has inspired and informed many interesting questions in mathematics and computer science, which in turn have suggested directions for synthesis of molecules. Here the authors give access to new results in the theory of polycycles and two-faced maps together with the relevant background material and mathematical tools for their study. Organised so that, after reading the introductory chapter, each chapter can be read independently from the others, the book should be accessible to researchers and students in graph theory, discrete geometry, and combinatorics, as well as to those in more applied areas such as mathematical chemistry and crystallography. Many of the results in the subject require the use of computer enumeration ; the corresponding programs are available from the author’ s website.

Research paper thumbnail of Generalized cut and metric polytopes of graphs and simplicial complexes

Optimization Letters, 2018

Research paper thumbnail of On the toplogical computation of K4 of the Gaussian and Eisenstein integers

arXiv: K-Theory and Homology, 2014

In this paper we use topological tools to investigate the structure of the algebraic K-groups K_4... more In this paper we use topological tools to investigate the structure of the algebraic K-groups K_4 (Z[i]) and K_4 (Z[rho]), where i := sqrt{-1} and rho := (1+sqrt{-3})/2. We exploit the close connection between homology groups of GL_n(R) for n = 5.

Research paper thumbnail of Geometry of Chemical Graphs: References

Research paper thumbnail of A New Algorithm in Geometry of Numbers

4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), 2007

Research paper thumbnail of Письмо в редакцию

Успехи математических наук, 2013

Research paper thumbnail of On classifying Minkowskian sublattices

Mathematics of Computation, 2011

Research paper thumbnail of Hyperbolic analogues of fullerenes on orientable surfaces

Discrete Mathematics, 2012

Research paper thumbnail of Applications of elementary decompositions to (<i>r, q</i>)-polycycles

Polycycles and Two-faced Maps, 2008

Research paper thumbnail of Combinatorial cube packings in cube and torus

We consider sequential random packing of cubes z+[0,1]^n with z∈1/N^n into the cube [0,2]^n and t... more We consider sequential random packing of cubes z+[0,1]^n with z∈1/N^n into the cube [0,2]^n and the torus ^n2^n as N→∞. In the cube case [0,2]^n as N→∞ the random cube packings thus obtained are reduced to a single cube with probability 1-O(1/N). In the torus case the situation is different: for n≤ 2, sequential random cube packing yields cube tilings, but for n≥ 3 with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on N depend on some parameters. We use use them to derive an expansion of the packing density in powers of 1/N. The explicit computation is done in the cube case. In the torus case, the situation is more complicate and we restrict ourselves to the case N→∞ of strictly positive probability. We prove the following results for torus combinatorial cube packings: We give a general Cartesian product construction. We prove that the number of parameters is at least n(n...

Research paper thumbnail of (2,3, 6)-SPHERES and Their Generalizations

We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical... more We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6-regular plane graph G0, two integers k and l and returns two 6-regular plane graphs. Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and we prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.

Research paper thumbnail of 4-REGULAR and Self-Dual Analogs of Fullerenes

An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of th... more An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties and explain some new algorithms that allow their efficient enumeration. Using this we give the symmetry groups of all i-hedrites and the minimal representative for each. We also review the link of 4-hedrites with knot theory and the classification of 4-hedrites with simple central circuits. An i-self-hedrite is a self-dual plane graph with faces and vertices of size/degree 2, 3 and 4. We give a new efficient algorithm for enumerating them based on i-hedrites. We give a classification of their possible symmetry groups and a classification of 4-self-hedrites of symmetry T, Td in terms of the Goldberg-Coxeter construction. Then we give a method for enumerating 4-self-hedrites with simple zigzags.

Research paper thumbnail of A generalization of Voronoi's reduction theory and its application

We consider Voronoi's reduction theory of positive definite quadratic forms which is based on... more We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.

Research paper thumbnail of Inhomogeneous extreme forms

G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for latti... more G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension 6 and prove their existence in all dimensions beyond. New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast to the packing problem.

Research paper thumbnail of The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices

In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay su... more In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. We obtain a complete list of 110244 affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra. Using a refinement of corresponding secondary cones, we obtain 181394 contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check.

Research paper thumbnail of The seven dimensional perfect Delaunay polytopes and Delaunay simplices

For a lattice L of R^n, a sphere S(c,r) of center c and radius r is called empty if for any v∈ L ... more For a lattice L of R^n, a sphere S(c,r) of center c and radius r is called empty if for any v∈ L we have v - c≥ r. Then the set S(c,r)∩ L is the vertex set of a Delaunay polytope P=conv(S(c,r)∩ L). A Delaunay polytope is called perfect if any affine transformation ϕ such that ϕ(P) is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if n=1 or n≥ 6 and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension 7 which allow us to find that there are only two perfect Delaunay polytopes: 3_21 which is a Delaunay polytope in the root lattice E_7 and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension 7 and found 11 types.

Research paper thumbnail of Delaunay polytopes derived from the Leech lattice

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its verti... more Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S∩ L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S∩ L is exactly S. For a vector v of the Leech lattice Λ_24 we define Λ_24(v) to be the lattice of vectors of Λ_24 orthogonal to v. We studied Delaunay polytopes of L=Λ_24(v) for |v|^2<=22. We found some remarkable examples of Delaunay polytopes in such lattices and disproved a number of long standing conjectures. In particular, we discovered: --Perfect Delaunay polytopes of lattice width 4; previously, the largest known width was 2. --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay polytopes in superlattices of L of the same dimension. --Polytopes that are perfect Delaunay with respect to two lattices L⊂ L' of the same dimension. --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously known examples ...

Research paper thumbnail of Computing symmetry groups of polyhedra

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as we... more Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used for instance in integer linear programming.

Research paper thumbnail of The decomposition of the hypermetric cone into L-domains

The hypermetric cone _n+1 is the parameter space of basic Delaunay polytopes in n-dimensional lat... more The hypermetric cone _n+1 is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone _n+1 is polyhedral; one way of seeing this is that modulo image by the covariance map _n+1 is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone _n+1 of hypermetrics on n+1 points contains exactly 1/2n! principal L-domains. We give a detailed description of the decomposition of _n+1 for n=2,3,4 and a computer result for n=5 (see Table <ref>). Remarkable properties of the root system D_4 are key for the decomposition of _5.

Research paper thumbnail of Wythoff polytopes and low-dimensional homology of Mathieu groups

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple... more We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as H_5(M_23,)=_7 and H_3(M_24,)=_12. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free M_n-resolution. Both methods apply in principle to arbitrary finite groups.

Research paper thumbnail of Complexity and algorithms for computing Voronoi cells of lattices

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean latti... more In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.