Related papers: The Domino Problem of the Hyperbolic Plane Is Unde…
The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special…
An algorithm is provided to tile the plane with the aperiodic monotile Tile(1,1) recently discovered by Smith et al. (2023). Their geometric construction guidelines are expanded into a numerical MATLAB algorithm. The intention is to remove…
We construct and study the unique random tiling of the hyperbolic plane into ideal hyperbolic triangles (with the three corners located on the boundary) that is invariant (in law) with respect to Moebius transformations, and possesses a…
We survey the status of decidabilty of the consequence relation in various axiomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski's conjecture on the…
A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane $\mathbb{R}^2$. Recognizing them is known to be $\exists\mathbb{R}$-complete, i.e., as hard as solving a system of polynomial…
We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can be…
A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a $O(n\log^2{n})$-time algorithm for deciding if a…
Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three…
A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of…
This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of…
We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value…
More than two centuries ago Malfatti (see \cite{malfatti}) raised and solved the following problem (the so-called Malfatti's construction problem):Construct three circles into a triangle so that each of them touches the two others from…
The Honeycomb Conjecture states that among tilings with unit area cells in the Euclidean plane, the average perimeter of a cell is minimal for a regular hexagonal tiling. This conjecture was proved by L. Fejes T\'oth for convex tilings, and…
We give a complete classification of edge-to-edge tilings of the sphere by regular polygons under a unified framework. Without assuming convexity of the tiles or polyhedrality of the underlying graph, our proof is independent of the…
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.…
We prove existence and uniqueness of an unstable manifold for a degenerate hyperbolic map of the plane arising in statistics.
In our article in MCU'2013 we state the the Domino problem is undecidable for all Baumslag-Solitar groups $BS(m,n)$, and claim that the proof is a direct adaptation of the construction of a weakly aperiodic subshift of finite type for…
We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra…
Consider the semisimple mod p reduction of the Galois representation associated to a Hilbert newform f by Carayol and Taylor. This paper discusses how, under certain conditions on f, the universal ring for deformations of this residual…
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper…