'Twas the night before Christmas and under the tree
Was a heap of new balls, stacked tight as can be.
The balls so gleaming, they reflect all light rays,
Which bounce in the stack every which way.
When, what to my wondering mind does occur:
A question of interest; I hope you concur!
From each point outside, I wondered if light
Could reach deep inside through gaps so tight?
More precisely,
let $\cal{B}$ be a finite collection of congruent, perfect mirror balls
arranged, say, in a cubic close-packing cannonball stack.
Let $H$ be the set of points inside the closed convex hull of $\cal{B}$,
$H^+ = \mathbb{R}^3 \setminus H$ the points outside,
and $H^- = H \setminus \cal{B}$ the points in the crevasses inside.
Q1. Is it true that every point $a \in H^+$ can illuminate every point $b \in H^-$ in the sense that there is a light ray from $a$ that reaches $b$ after a finite number of reflections?
I believe the answer to Q1 is 'No': If $a$ is sufficiently close to a point of contact between a ball of $\cal{B}$ and $H$, then all rays from $a$ deflect into $H^+$. If this is correct, the question becomes: which pair of points $(a,b)$ can illuminate one another, for a given collection $\cal{B}$? Specifically:
Q2. Is there some finite radius $R$ of a sphere $S$ enclosing a collection $\cal{B}$ such that every point $a$ outside $S$ can illuminate every point $b \in H^-$? More precisely, are there conditions on $\cal{B}$ that ensure such a claim holds?
If the centers of the balls in $\cal{B}$ are collinear, then points in the bounding cylinder do not fully illuminate. If the centers of the balls are coplanar, then points on that plane do not fully illuminate. So some configurations must be excluded. Perhaps a precondition analogous to this might suffice: If the hull $H$ of $\cal{B}$ encloses a sphere of more than twice the common radius of the balls, then ... ? Failing a general result, can it be established for stackings as illustrated above?
The answers (especially Bill Thurston's) in response to the earlier MO question on lightrays bouncing between convex bodies may be relevant. Even speculative 'answers' are welcomed!
Edit (23Dec). Although I remain optimistic that there is a nice theorem lurking here, fedja's observation that points near the boundary of the hull remain dark makes it a challenge to formulate a precise statement of a possible theorem. Something like this:
If $\cal{B}$ is sufficiently "fat," then every point $a$ sufficiently far from $\cal{B}$ illuminates every point $b$ in $H^-$ that is not too close to the boundary of $H$.
Edit (24Dec). There is an associated computational question, interesting even in two dimensions:
Given $a$ and $b$, what is the complexity of deciding if $a$ can illuminate $b$?
Is it even decidable?
