**1**

vote

**0**answers

33 views

### 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 ...

**2**

votes

**1**answer

52 views

### 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 ...

**6**

votes

**0**answers

121 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 ...

**2**

votes

**0**answers

40 views

### 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 ...

**5**

votes

**2**answers

208 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).
...

**3**

votes

**0**answers

98 views

### 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 ...

**10**

votes

**3**answers

384 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 Masur proved in the 1980's that every rational polygon
(vertex angles rational multiples ...

**26**

votes

**8**answers

4k 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 solved ...

**43**

votes

**2**answers

2k 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 ...

**8**

votes

**1**answer

475 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 ...

**5**

votes

**1**answer

120 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 ...

**7**

votes

**3**answers

333 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 ...

**8**

votes

**1**answer

472 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 ...

**79**

votes

**2**answers

109k 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 ...

**7**

votes

**2**answers

205 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 ...

**11**

votes

**3**answers

776 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 ...

**7**

votes

**0**answers

197 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 ...

**3**

votes

**0**answers

111 views

### 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 ...

**1**

vote

**1**answer

166 views

### Computing saddle connections in flat structures

Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ which start and end at a ...

**30**

votes

**5**answers

876 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 ...

**3**

votes

**0**answers

68 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 ...

**10**

votes

**1**answer

344 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 ...

**21**

votes

**4**answers

988 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 ...

**26**

votes

**3**answers

2k 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 ...

**3**

votes

**0**answers

238 views

### 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 ...

**32**

votes

**6**answers

2k 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 ...

**78**

votes

**5**answers

3k views

### 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 ...

**6**

votes

**2**answers

478 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 ...

**21**

votes

**2**answers

910 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
...

**8**

votes

**0**answers

201 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 ...

**5**

votes

**3**answers

1k 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 ...

**6**

votes

**0**answers

419 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 ...

**12**

votes

**1**answer

768 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.
It was a good exercise, but at this point I wonder if ...