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Results tagged with billiards
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user 6094
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.
9
votes
1
answer
2k
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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 tha …
- 151k
25
votes
1
answer
493
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 billiard …
- 151k
2
votes
1
answer
101
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 fro …
- 151k
100
votes
6
answers
5k
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 …
- 151k
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 b …
- 151k
10
votes
3
answers
1k
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 mult …
- 151k
10
votes
1
answer
570
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. …
- 151k
18
votes
0
answers
480
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 ea …
- 151k
42
votes
2
answers
3k
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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-bou …
- 151k
1
vote
0
answers
84
views
Trapping lightrays under nonstandard reflections and/or paths
Geometry and Billiards, p. 116.
Can we trap light in a polygonal room? … I think for billiards under gravity the answer is likely Yes,
but I have little intuition for reflections at a fraction of angle of incidence.
…
- 151k
10
votes
0
answers
167
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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 perfec …
- 151k
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. … For single particle billiards, it is well known
that (1) a trajectory of rational slope that avoids the corners
is periodic, and (2) a trajectory of irrational slope that avoids
the corners will be "uniformly …
- 151k
37
votes
6
answers
3k
views
Billiard dynamics under gravity
Has the dynamics of billiards in a polygon subject to gravity been
studied? … I am wondering if such a system can somehow be converted into one
without gravity, so that our understanding of, e.g., the dynamics of
billiards in a square may be applied. …
- 151k
4
votes
1
answer
324
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 comple …
- 151k
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 balls …
- 151k