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 ...
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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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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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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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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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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 ...
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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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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 ...
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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 ...
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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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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 $...
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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 $\...
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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
...
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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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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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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 ...
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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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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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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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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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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 $n$...
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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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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 ...
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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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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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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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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 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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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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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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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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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,...
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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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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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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 ...
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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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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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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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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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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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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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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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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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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 ...