# 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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### 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 ...
101 views

### 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 ...
145 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 ...
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 ...
447 views

### 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 ...
1 vote
111 views

### 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 ...
137 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 ...
206 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 ...
145 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 ...
220 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 ...
307 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 ...
83 views

### 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 ...
43 views

### 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 ...
360 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 ...
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 ...
1 vote
80 views

### 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": ...
363 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 ...
127 views

### 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?