# Tagged Questions

**2**

votes

**2**answers

136 views

### A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line

I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a ...

**27**

votes

**4**answers

734 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 ...

**2**

votes

**2**answers

486 views

### Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of ...

**10**

votes

**2**answers

439 views

### Periodic lightray paths trapped between two nested mirror circles

I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known?
The behavior of such rays seems chaotically complicated. For example, ...

**69**

votes

**4**answers

2k 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 ...

**2**

votes

**0**answers

126 views

### System dynamic of space euclidean and hyperbolic tilings

Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC)
tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of
$R^{d}$ by translation is on ...

**28**

votes

**4**answers

2k views

### $\exists$ 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 ...

**9**

votes

**2**answers

848 views

### Midpoint geodesic polygon / Birkhoff curve shortening

I would like to know under what conditions the process
of creating a midpoint piecewise geodesic polygon converges
on a surface $S \subset \mathbb{R}^3$.
$S$ may be assumed smooth, closed, and ...

**5**

votes

**2**answers

797 views

### Light rays bouncing around inside a sphere in d-dimensions

Suppose $S=\mathbb{S}^d$ is a unit sphere in $(d-1)$ dimensional space, with $d=3$ of special interest.
The surface of $S$ is a perfect (internal) mirror.
You stand at point $x$ (not the sphere center ...

**3**

votes

**1**answer

336 views

### Connections between a polytope's symmetry group and the existence of periodic orbits

Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...