arXiv subjects
Publications and source records attributed to Maurice Margenstern.
In this paper we define an infinite family of triangular tilings of the hyperbolic plane defined by two parameters ranging in the natural nummbers and we give a uniform way to define coordinates for locating the triangles of the tiling.
In this paper we give three examples of expending patterns defined by hyperbolic cellular automata whose growth seems to be very similar to the growth of colonies of bacteria.
In this paper, following the way opened by a previous paper deposited on arXiv, we give an upper bound to the number of states for a hyperbolic cellular automaton in the pentagrid. Indeed, we prove that there is a hyperbolic cellular automaton which is rotation invariant and whose halting problem is undecidable and which has 9~states.
In this paper, we show a construction of a weakly universal cellular automaton in the 3D hyperbolic space with two states. The cellular automaton is rotation invariant and, moreover, based on a new implementation of a railway circuit in the dodecagrid,the construction is a truly 3D-one.
In this paper, we look at two ways to implement determinisitic one dimensional cellular automata into hyperbolic cellular automata in three contexts: the pentagrid, the heptagrid and the dodecagrid, these tilings being classically denoted by $\{5,4\}$, $\{7,3\}$ and $\{5,3,4\}$ respectively.
This paper introduces a method of navigation in a large family of tilings of the hyperbolic plane and looks at the question of possible applications in the light of the few ones which were already obtained.
In this paper, we give an application of iterated pushdown automata to contour words of balls and two other domains in infinitely many tilings of the hyperbolic plane. We also give a similar application for the tiling {5,3,4} of the hyperbolic 3D space and for the tiling {5,3,3,4} of the hyperbolic 4D space as well.
In this paper, we look at the following question. We consider cellular automata in the hyperbolic plane, (see Margenstern, 2000, 2007 and Margenstern, Morita, 2001) and we consider the global function defined on all possible configurations. Is the injectivity of this function undecidable? The problem was answered positively in the case of the Euclidean plane by Jarkko Kari, in 1994. In the present paper, we show that the answer is also positive for the hyperbolic plane: the problem is undecidable.
In this paper, we construct a new weakly universal cellular automaton on the ternary heptagrid. The previous result, obtained by the same author and Y. Song required six states only. This time, the number of states is four. This is the best result up to date for cellular automata in the hyperbolic plane.
In this paper, we distinguish two levels for the plane-filling property. We consider a simple and a strong one. In this paper, we give the construction which proves that the simple plane-filling property also holds for the hyperbolic plane. The plane-filling property was established for the Euclidean plane by J. Kari, in 1994, in the strong version.
In this paper, we prove that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is undecidable for the Euclidean plane, initially proved by Robert Berger in 1966.
In this paper, we complete the construction of paper arXiv:cs.CG/0701096v2. Together with the proof contained in arXiv:cs.CG/0701096v2, this paper definitely proves that the general problem of tiling the hyperbolic plane with {\it à la} Wang tiles is undecidable.
In this paper, we consider the periodic tiling problem which was proved undecidable in the Euclidean plane by Yu. Gurevich and I. Koriakov in 1972. Here, we prove that the same problem for the hyperbolic plane is also undecidable.
In this paper, we consider the finite tiling problem which was proved undecidable in the Euclidean plane by Jarkko Kari in 1994. Here, we prove that the same problem for the hyperbolic plane is also undecidable.
In this paper, we look at the extention of Hedlund's characterization of cellular automata to the case of cellular automata in the hyperbolic plane. This requires an additionnal condition. The new theorem is proved with full details in the case of the pentagrid and in the case of the ternary heptagrid and enough indications to show that it holds also on the grids $\{p,q\}$ of the hyperbolic plane.
In this paper we consider tiling $\{p, q \}$ of the Euclidean space and of the hyperbolic space, and its dual graph $Γ_{q, p}$ from a combinatorial point of view. A substitution $σ_{q, p}$ on an appropriate finite alphabet is constructed. The homogeneity of graph $Γ_{q, p}$ and its generation function are the basic tools for the construction. The tree associated with substitution $σ_{q, p}$ is a spanning tree of graph $Γ_{q, p}$. Let $u_n$ be the number of tiles of tiling $\{p, q \}$ of generation $n$. The characteristic polynomial of the transition matrix of substitution $σ_{q, p}$ is a characteristic polynomial of a linear recurrence. The sequence $(u_n)_{n \geq 0}$ is a solution of this recurrence. The growth of sequence $(u_n)_{n \geq 0}$ is given by the dominant root of the characteristic polynomial.
This contribution belongs to a combinatorial approach to hyperbolic geometry and it is aimed at possible applications to computer simulations. It is based on the splitting method which was introduced by the author and which is reminded in the second section of the paper. Then we sketchily remind the application to the classical case of the pentagrid, i.e. the tiling of the hyperbolic plane which is generated by reflections of the regular rectangular pentagon in its sides and, recursively, of its images in their sides. From this application, we derived a system of coordinates to locate the tiles, allowing an implementation of cellular automata. At the software level, cells exchange messages thanks to a new representation which improves the speed of contacts between cells. In the new setting, communications are exchanged along actual geodesics and the contribution of the cellular automaton is also linear in the coordinates of the cells.
In this paper, we prove that the general problem of tiling the hyperbolic plane with à la Wang tiles is undecidable.