Questions tagged [billiards]
Billiards are a class of dynamical systems in which a point particle moves uniformly in a domain $D\subset \mathbb{R}^d$ except for mirror-like reflections from the boundary. Varying $D$ leads to examples satisfying many ergodic properties. Billiards enhance visual explanations of dynamical concepts to students and the general public. There are many applications in physics and image processing. The free motion and/or reflection rule may be generalized.
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Trapping lightrays with segment mirrors
Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors?
I posed this question in several forums before (e.g., here
and in an ...
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Minimum reflection paths in a mirror polygon
Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles,
and is non-self-intersecting;
also known as a rectilinear polygon.
Treat every edge of $P$ as a ...
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Is there an inventory of closed billiard paths in a regular tetrahedron?
Conway found a closed billiard-ball trajectory in a regular tetrahedron:
Image: Izidor Hafner
Since then Bedaride and Rao
Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic ...
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Billiards with incompatible regions
An existing question asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong counter-examples:...
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square-tiled surfaces and the Euler phi function
In Billiards in Rectangles with Barriers, Eskin-Masur-Schmoll count the number of primitive square-tiled surfaces with two cone points with angle $4\pi$ for one cone point with angle $6\pi$: (See also)...
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How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls?
Consider the following scenario. Let $\alpha>1$. Suppose whenever two superelastic balls collide at speed $\gamma$ they bounce off each other at speed $\gamma\cdot\alpha$ (i.e. $\alpha$ is the ...
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Differential equation of line tangent to caustics
This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...
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Pocket billiards with balls in general position
There were at least two earlier MO questions about ideal pocket billiards.
(Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does ...
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Illuminating a just-barely irrational polygon
As has been discussed earlier on MO,1,2
recently an impressive advance was proved concerning
internally illuminating a mirrored polygon.
Here is the result:
Let $P$ be a rational polygon.
Then for ...
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Do identical orbit tiles imply identical combinatorial types?
Given a periodic trajectory on a triangle, we can associate to this trajectory a sequence of integers $1,2$ and $3$ by labeling the edges of the triangle and taking the sequence of edges the ...
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Can a laser hit all the mirrors out of order?
For this question, a "cycle" is a sequence of distinct points
$X = (x_1,x_2,\cdots,x_k)\in\mathbb{R}^3$ which defines a piecewise linear path starting at $x_1$ and visiting the points in ...
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2-ball billiards in a circle
Consider a 2D circular billiards table with diameter 1m containing two
balls with diameter 0.25m. Let each ball start with a speed of 1m/s.
In general, this speed could change after the balls hit ...
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Trapped Billiard trajectories on non-convex billiard tables
Let $\Omega$ be a domain in $\mathbb{R}^2$ with smooth boundary. A billiard trajectory is a continuous curve $c: \mathbb{R}\supseteq I \longrightarrow \overline{\Omega}$ such that
$c(t) \in \partial ...
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Reference for standard lemmas in polygonal billiards
I'm trying to track down references for a couple of well-known results in the theory of polygonal billiards for a paper I'm working on.
Recall that periodic trajectories on a polygonal table occur in ...
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Polygon illumination with perturbed reflections
Here is a variation on the classical polygon illumination problem. For $c \geq 0$ we say that a mirror has reflection index $c$ if whenever a ray hits the mirror with angle of incidence $\alpha$ then ...
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Proving light escapes mirrors via ergodic theory of billiards
There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
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Mathematical Billiards: Set of starting points and velocities hitting boundary at time t
In the simplest setting $\Omega$ smooth compact and convex in $R^n$ with linear constant speed trajectories that is ($q_t=q_0+t\cdot v$ until the collision point). What is known about the structure ...
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Mixing for a gas of hard spheres
The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
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Periodic orbits in planar smooth billiard table with large periods
Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period.
Formulation of my question: We are considering ...
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Trapping lightrays under nonstandard reflections and/or paths
Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open:
"It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
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Boundedness of partial products for a divergent trig product
I am looking at a discrete dynamical system and I wish to show that it is bounded. I know that the displacement after $n$ iterations is given by the product
$$\Delta_n=\prod_{k=0}^n \left(1+\frac{2\...
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Is there a reversible fully polynomial-time approximation scheme for polygonal billiards?
Let $P \subset \mathbb{R}^2$ be a polygon with rational coordinates, and consider discrete billiards inside $P$, where a ball (of zero radius) moves by steps of fixed length on each step, in a ...
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Infinite number of points reflecting on the circle, must some two (or more) ever meet?
I just created a following problem.
Suppose that we have an infinite number of points on the circle and that they at the same time start to travel (all with the same speed) from the point where they ...
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Arithmetic billiards, prime numbers and the Goldbach conjecture
I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post.
On ...
