On 31/10/2005, I posted this problem to sci.optics and sci.physics. The problem is described in more detail below.

Consider the following optical structures:

1. A void sphere S with totally reflecting inner walls.
2. A void torus T with totally reflecting inner walls.
3. A (point or extended) light source L inside S and T.
4. An observer O inside S and T.

Here are some interesting questions for you to ponder about:

1. What would O see when O tries to look at the walls of S or T at various angles, while L is illuminating the inside of S or T?
2. What will O see when O tries to look at the walls of S or T at various angles, when either O or L move relative to the walls of S or T, while O is observing?
3. What happens to a single light ray which emanates from L inside S and T?
4. Will any light ray emanating from L, eventually reach O in S and T?
5. Will O be able to see L's reflection anywhere in S or T? Everywhere? Nowhere?
6. Will O be able to see O's reflection anywhere in S or T? Everywhere? Nowhere?
7. Will O be able to see L's reflection, even when L is hidden from O in T?
8. What will O see in S or T, if L is a strobo flash which lights exactly once?
9. Can S and T's of reasonable sizes (with radii 10-20 m) be built?
10. Do you think they would be "safe" for human observers?
11. Justify your answer about their "safety".
12. How might S and T be related to "cavity resonators"?
13. Assume S or T is built, without O/L in it. What will happen if one opens a little hole on the wall of S or T for some light to get in?
14. Same question as above with O already in S or T.
15. Can you perhaps think of a useful application of S and T in Physics?
16. Would you volunteer to be O in experiments with S and T?
17. Why/Why not?

A Short Analysis

The central problem above is the general case of what happens in two dimensions, which was a question posed by Costas Vlachos on sci.math:

Suppose a user shines a beam of light (say a laser) inside a perfectly reflective unit circle. Which points will be visited by the light ray and which will not?

Let us define the problem geometrically: Consider the unit circle and a user A who "shines" a laser at an angle θ from the tangent at A. Clearly we have:

ρ+2*φ=π,
θ+φ=π/2, Þ
ρ=2*θ

User A shining a laser at angle θ inside the unit circle

Immediately then, when θ=π/n, n Î N, ρ=2*π/n, and the resulting path will be a regular polygon with n sides. Therefore the visited points are exactly the n-th roots of unity: zⁿ=1, or explicitly the points e^2*k*π/n*i, kÎ {0,1,2,...,n-1}.

What about the points which will cannot be visited? These are the points z on the unit circle which do not satisfy zⁿ=1, or the points e^2*ρ*π*i, with ρÎR\Q.

Because ρ is also the central angle, these points are the points whose central angle ρ is not a rational multiple of 2*π. In other words, they are the points whose central angle ρ is an irrational multiple of 2*π.

The n-th roots of unity are dense on the unit circle, hence for any non-visited point subtending an (irrational) angle ρ, there exists n₁, n₂ Î N, and consequently θ₁, θ₂, such that for any ε>0, we have |θ₁-θ₂|=π*|1/n₁-1/n₂|<ε, and that there exist points p₁ on the path of shooting angle θ₁ and p₂ on the path of shooting angle θ₂, such that: angle(p₁) < ρ < angle(p₂).

In other words, we can come as close as we want to the non-visited (irrational) points, by choosing appropriate angles θ₁ and θ₂.

Are the angles θ=π/n the only angles which give n-th roots of unity? Peter Webb on sci.math observed that the angle θ can be of the form θ=k/n*π, kÎ{0,1,2,...,n-1} and made the following conjecture:

For any ρ=m/n*2*π where m and n are relatively prime the light ray will trace out the endpoints of a regular n sided polygon, though not necessarily in the correct order.

Peter's conjecture can be reformulated in terms of θ as:

For any θ=k/n*π the light ray will trace out the endpoints of a regular n/GCD(k,n) sided polygon, though not necessarily in the correct order.

Indeed, the n-th roots of unity form a cyclic group of order n and the elements of this group of order k with GCD(k,n)=1 are generators. The situation is depicted on the following figure for n=6:

The paths traced according to the shooting angle θ are:

1. n=6, k=1, θ=π/6, period: 6/GCD(1,6)=6, orbit: O-p1-p2-p3-p4-p5-O, hexagon
2. n=6, k=2, θ=2*π/6, period: 6/GCD(2,6)=3, orbit: O-p2-p4-O, triangle
3. n=6, k=3, θ=3*π/6, period: 6/GCD(3,6)=2, orbit: O-p3-O, line
4. n=6, k=4, θ=4*π/6, period: 6/GCD(4,6)=3, orbit: O-p4-p2-O, triangle (in reverse)
5. n=6, k=5, θ=5*π/6, period: 6/GCD(5,6)=6, orbit: O-p5-p4-p3-p2-p1-O, hexagon (in reverse)

The cases 2,3 and 4 exhibit subgroups of the cyclic group of order 6 formed by the second, third and fourth elements.

An Unexpected Connection With Infinite Exponentials

When considering the orbit of the sequence {z,c^z,c^cz,...} for various c Î C, it turns out that the unit circle is exactly the boundary which separates the convergent sequences from the divergent ones.

Consider the map: φ:C®C, defined by: φ(t)=e^t/et. φ is invertible. Specifically, φ^-1(c)=-W(-log(c)), where W is Lambert's function.

When t=φ^-1(c) and |t|<1, i.e., when t falls inside the unit circle, the sequence {z,c^z,c^cz,...} converges. When |t|>1, i.e., when t falls outside the unit circle, the sequence diverges.

When t is ON the unit circle, things are very interesting.

Specifically, when t_k=φ^-1(c) is an n-th root of unity (i.e., when t_k=e^2*k*π/n*i, kÎ{0,1,2,...,n-1}), the sequence {z,c^z,c^cz,...} converges when z=e^t_k or falls into periodic cycling of period p=n/GCD(n,k) when z¹e^t_k.

When the sequence falls into periodic cycling, the orbit trajectory traces various (irregular) polygons, similar to the polygons on the light-mirror problem. Here is the orbit of the above sequence for t_k=e^2*k*π/6*i, i.e., for n=6 and kÎ{1,2,...,5}, with z perturbed -1/2 away from e^t_k.

n=6, k=1, period: 6/GCD(1,6)=6, orbit: hexagon

n=6, k=2, period: 6/GCD(2,6)=3, orbit: triangle

n=6, k=3, period: 6/GCD(3,6)=2, orbit: line

n=6, k=4, period: 6/GCD(4,6)=3, orbit: triangle (in reverse)

n=6, k=5, period: 6/GCD(5,6)=6, orbit: hexagon (in reverse)

What happens to points for which |t|=1, but t=φ^-1(c) is NOT an n-th root of unity? These points correspond to the points which subtend a central angle ρ which is an irrational multiple of 2*π on the problem with the light-mirror trajectory. In this case, the sequence {z,c^z,c^cz,...} diverges (chaotically!).

For more info on this problem and on iterated exponential powers, consult my pages on infinite exponentials, here and here.

A POVRay Illustration of the General Case

Brian Vanderkolk investigates the general case optically using POVRay (a ray-tracing program), on this page.

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