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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Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I ...
Joseph O'Rourke's user avatar
95 votes
2 answers
114k views

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 ...
Phedg1's user avatar
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54 votes
3 answers
3k views

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 ...
Joseph O'Rourke's user avatar
42 votes
2 answers
3k views

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-...
Joseph O'Rourke's user avatar
39 votes
2 answers
3k views

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 "...
ShnitzelKiller's user avatar
37 votes
6 answers
3k views

Billiard dynamics under gravity

Has the dynamics of billiards in a polygon subject to gravity been studied? What I have in mind is something like this:            Still Snell's Law ...
Joseph O'Rourke's user avatar
36 votes
11 answers
7k views

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 ...
Ferran V.'s user avatar
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33 votes
4 answers
3k views

Does there exist a shot in ideal pocket billiards?

Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with the game idealized in that no spin is placed on the cue ball in the initial shot, all collisions between billiard ...
Joseph O'Rourke's user avatar
32 votes
5 answers
1k views

Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$, except leave the origin $(0,0)$ unoccupied by a disk. Q. Is it the case that every disk can be hit ...
Joseph O'Rourke's user avatar
32 votes
3 answers
2k views

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 ...
user918212's user avatar
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31 votes
3 answers
3k views

Optic fibers after Joseph O'Rourke

Let $\gamma\colon[a,b]\to \mathbb R^3$ be a smooth curve with curvature $< 1$. Consider a tube, formed by one parameter family of unit circles with center at $\gamma(t)$ in the plane orthogonal to $...
Anton Petrunin's user avatar
25 votes
4 answers
1k views

Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\...
Joseph O'Rourke's user avatar
24 votes
2 answers
1k views

Billiard dynamics for multiple balls

I am interested to learn to what extent results on billiards in polygons have been extended to multiple balls. Assume the balls have equal radii and the same mass, the same initial speed, and all ...
Joseph O'Rourke's user avatar
21 votes
3 answers
1k views

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,\,...
user64494's user avatar
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18 votes
0 answers
473 views

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 ...
Joseph O'Rourke's user avatar
14 votes
2 answers
1k views

Polygonal billards programs

I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure. (source) It was a good exercise, but at this point I ...
john mangual's user avatar
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12 votes
3 answers
1k views

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 ...
mathlove's user avatar
  • 4,727
10 votes
1 answer
317 views

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 ...
Liviu Nicolaescu's user avatar
10 votes
3 answers
981 views

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 ...
Joseph O'Rourke's user avatar
10 votes
1 answer
551 views

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. ...
Joseph O'Rourke's user avatar
10 votes
1 answer
445 views

Does the $n$-gonal billiards conjecture follow from the $m$-gonal conjecture when $m>n$?

For $n\ge 3$, define the $n$-gonal billiards conjecture as the statement All convex $n$-gons admit periodic billiard trajectories. To the best of my knowledge this question remains open for all $n$...
Alex Becker's user avatar
10 votes
0 answers
157 views

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 ...
Joseph O'Rourke's user avatar
9 votes
4 answers
2k views

Birkhoff conjecture about integrable billiards

There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse. Integrability here might be ...
DamienC's user avatar
  • 8,103
9 votes
3 answers
581 views

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 ...
AlterTim's user avatar
  • 315
9 votes
1 answer
2k views

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 ...
Joseph O'Rourke's user avatar
8 votes
1 answer
665 views

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 ...
Pengfei's user avatar
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8 votes
2 answers
2k views

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 ...
AlterTim's user avatar
  • 315
8 votes
4 answers
655 views

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 ...
Selim G's user avatar
  • 2,626
8 votes
1 answer
298 views

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)...
Josué Tonelli-Cueto's user avatar
8 votes
0 answers
105 views

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 ...
Joseph O'Rourke's user avatar
8 votes
0 answers
243 views

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:...
mjqxxxx's user avatar
  • 131
7 votes
2 answers
266 views

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;...
James Propp's user avatar
  • 19.4k
7 votes
2 answers
592 views

Does the random Lorenz gas have a non-trivial diffusion coefficient?

For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If not,...
Paul Tupper's user avatar
7 votes
1 answer
379 views

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 ...
Quoka's user avatar
  • 133
7 votes
0 answers
239 views

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)...
john mangual's user avatar
  • 22.6k
6 votes
0 answers
179 views

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 ...
Joseph Van Name's user avatar
6 votes
0 answers
447 views

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 ...
A B's user avatar
  • 281
5 votes
3 answers
2k views

Dense orbits in billiards

This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ...
Zatrapilla's user avatar
5 votes
2 answers
313 views

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). ...
Joonas Ilmavirta's user avatar
5 votes
1 answer
182 views

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 ...
Alex Becker's user avatar
5 votes
2 answers
234 views

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 ...
Victor Galitski's user avatar
5 votes
0 answers
154 views

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 ...
Joseph O'Rourke's user avatar
4 votes
1 answer
266 views

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 ...
Joseph O'Rourke's user avatar
4 votes
1 answer
313 views

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 ...
DSblizzard's user avatar
4 votes
1 answer
481 views

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 ...
user918212's user avatar
  • 1,005
4 votes
0 answers
228 views

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 ...
Joseph O'Rourke's user avatar
4 votes
0 answers
80 views

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 ...
Alex Becker's user avatar
3 votes
2 answers
188 views

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 ...
Maurizio Barbato's user avatar
3 votes
0 answers
111 views

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 ...
felipeh's user avatar
  • 337
3 votes
0 answers
143 views

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 ...
bobuhito's user avatar
  • 1,537