# 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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### Hexagon tiling and affine Weyl group $\widetilde{A}_2$

Let $H$ be a regular hexagonal room centered at the origin. Let $W$ be the group generated by reflections about the six sides of $H$. It's well known that $W$ is the affine Weyl group of type $\...

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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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### Billiard circuits in pentagons

A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently 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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### 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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### 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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### 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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### 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 ...

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### Question about the inverse operator on PSL(2,R) with respect to Liouville measure

In GTM 259 chapter 9 and Katok Hasselbaltt Introduction to Modern Theory of Dynamical System chapter 5 (using the Iwasawa KAN decomposition)
we see the Unit Tangent bundle of Hyperbolic half plane is ...

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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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### Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)

I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the ...

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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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### Maximal length of trajectories in billiard

Consider discrete rectangular billard on lattice with integer dimensions a*b and n balls with radius $\frac{\sqrt 2}{2}$ and ...

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### Finding particular closed paths in geometric plane regions

Let $X_m$ denote a set of $m\geq 3$ lines in $\mathbb{R}^2$ that are not all parallel. Consider the problem of determining a closed path of $kn$ points in $X_m$ $k, n \in \mathbb{Z}^+$, such that the ...

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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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### Why are we interested in operators that share a basis of eigenfunctions?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I ...

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### Why is the billiard problem for obtuse triangles so hard?

This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking if every obtuse triangle admits a periodic billiard path, which has been open ...

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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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### Types of triangles admitting periodic billiard orbits

It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and obtuse ...

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### 6-periodic billiards trajectory in acute triangle

We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?

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### Under which conditions do ellipsoids have a focal property?

Given an ellipsoid $E$, we consider the trajectories of light inside $E$ assuming that $\partial E$ would be a mirror. In other words, let a light trajectory be piecewise linear path $\gamma:[0,\infty)...

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### Can one "hear" the shape of a polygon via external reflections?

This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?"
A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-...

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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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### 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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### A Really Simple Stochastic Dynamic Billiard

Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$, $c>0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t),z(t))$ be the position at time $t$ of a point which moves ...

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### 3D Billiards problem inside a torus

I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "...

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### A question about billiards

This is a question in a rather well investigated subject of which I know very little and I have a hard time "translating" the general results available. Let me also say that I got interested in this ...

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### Current state of Straus's illumination problem

In George W. Tokarsky's Polygonal Rooms Not Illuminable from Every Point (1995) it is stated that the problem
Is a polygonal region illuminable from at least one point in the region?
was still ...

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### Existence of periodic orbits in rational billiards

Recently I've got interested in dynamical billiards. Some results in this field are obtained by elementary methods. For instance, see George W. Tokarsky's Polygonal Rooms Not Illuminable from Every ...

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### Periodic billiard paths in hyperbolic triangles

It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path,
one obeying the reflection law that the angle of incidence equals the angle of reflection. ...

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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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### Complexity of recognizing equivalent translation surfaces

"A translation surface is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence."
I take that succinct (and not fully precise) definition from a ...

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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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### 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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### Are periodic billiard trajectories stable on a manifold with strictly convex boundary?

Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary.
Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reflects specularly at the boundary).
...

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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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### Reflection of light from function graph

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ y_0>0,\,...

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### Which polygons have *simple* periodic billiard paths?

I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Howard Masur proved in the 1980's that every rational polygon
(vertex angles rational ...

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### What "real life" problems can be solved using billiards?

Recently I gave an interview to local media where I explained some basic open problems in billiard dynamics.
After a 45 min interview the reported asked me what "real life" problems can be ...

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### The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose
internal face surfaces were perfect mirrors.
Let's assume $T$'s height is $3{\times}$ yours, so that your
eye is roughly at the ...

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### Billiard dynamics with angle of reflection a fraction of angle of incidence

Suppose that a billiard ball bouncing in a unit square (or a lightray reflecting
in a mirrored square) has the property that the angle of reflection is a fraction
of the angle of incidence, rather ...

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### Unfoldings of trajectories on the Veech triangle $V_4$

Let $V_4$ be the isosceles triangle with base angle $\pi/8$. $V_4$ is a Veech triangle, so the dynamics of billiards on it are very well understood.
Above is the unfolding of $V_4$, with edge ...

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### Existence of nonergodic polygonal billiard

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one.
A standard conjecture is that a ...

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### Raphael Douady's thesis: Applications du théorème des tores invariants

Raphael Douady's thesis, Applications du théorème des tores invariants, has been cited in numerous papers by many experts.
According Wikipedia, he proves of the equivalence of KAM ...

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### Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...

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### Well-definedness of single-particle smooth billiards flow

Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well known;...

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### A special tessellation

Let $P$ be a convex $n$-gon. Suppose that we have an infinite number of $P$s, and that each of them is colored either red or blue. Here, let us consider the following operations :
Operation 1 : Place ...

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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)...