Related papers: Billiard Ordered Games and Books
The billiard problem concerns a point particle moving freely in a region of the horizontal plane bounded by a closed curve $\Gamma$, and reflected at each impact with $\Gamma$. The region is called a `billiard', and the reflections are…
Inspired by the theory of desirable gambles that is used to model uncertainty in the field of imprecise probabilities, I present a theory of desirable things. Its aim is to model a subject's beliefs about which things are desirable. What…
Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a…
Domineering is a combinatorial game played on a subset of a rectangular grid between two players. Each board position can be put into one of four outcome classes based on who the winner will be if both players play optimally. In this note,…
The traditional mathematical model for an impartial combinatorial game is defined recursively as a set of the options of the game, where the options are games themselves. We propose a model called gamegraph, together with its generalization…
The illumination problem is a popular topic in recreational mathematics: In a mirrored room, is every region illuminable from every point in the region? So-called \enquote{unilluminable rooms} are related to \enquote{trapped sets} in…
Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The…
We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two…
We investigate the semiclassical energy spectrum of quantum elliptic billiard. The nearest neighbor spacing distribution, level number variance and spectral rigidity support the notion that the elliptic billiard is a generic integrable…
In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer…
Random-matrix theory is used to show that the proximity to a superconductor opens a gap in the excitation spectrum of an electron gas confined to a billiard with a chaotic classical dynamics. In contrast, a gapless spectrum is obtained for…
We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these…
We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of…
We introduce a class of billiards with chaotic unidirectional flows (or non-chaotic unidirectional flows with "vortices") which go around a chaotic or non-chaotic "core", where orbits can change their orientation. Moreover, the…
We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the…
We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb R}^d$ until it hits the boundary and bounces randomly inside according to some reflection…
Billiard models of single particles moving freely in two-dimensional regions enclosed by hard walls, have long provided ideal toy models for the investigation of dynamical systems and chaos. Recently, billiards with (semi-)permeable walls…
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and…
We discuss four famous card games that can help learn linear algebra. The games are: SET, Socks, Spot it!, and EvenQuads. We describe the game in the language of vector, affine, and projective spaces. We also show how these games are…
Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian…