```
'Twas the night before Christmas, and throughout the net
Not a question was stirring, at least---not yet.
The tree was all draped with garlands and lights,
With presents beneath for morning delights.
As I enjoyed this warm-hearted scene,
A question arose as if in a dream:
The tree is a cone, the garlands are O's.
I wonder how many can fit of those?
Has this been studied? Can you shed some light?
Merry Christmas to all! And to all a good night!
```

Q1. Have packings of congruent disks on (finite) cones been explored?

*Notation*.
Let $C$ be a (finite) right circular cone.
A *generator* of $C$ is a segment connnecting
the apex $a$ to a point on the rim of the base.
Let $\alpha$ be the angle of the cone at $a$ when cut open
along a generator; so $\alpha$ is the total
surface angle incident to $a$.
Let $L$ be the length of a generator.
The surface area of $C$ is $A=A(L,\alpha)=\pi L^2 (\alpha/(2 \pi))$.

Q2. For a fixed surface area $A$, which cone angle $\alpha$ permits the most unit-radius disks to be packed on $C$?

In the example above, $L= 3 \sqrt{2} \approx 4.14$ and $\alpha = \frac{2}{3} \pi = 120^\circ$; so $A=6 \pi$, which is the area of $6$ unit-radius disks, although I only packed $4$ disks (and so achieved a density of $\delta=\frac{2}{3}$).

As $\alpha \to 0$, no unit-radius disk can fit, and for $\alpha = 2 \pi$, we are packing disks in a circle of radius $L$, a heavily studied problem. In the absence of an answer to Q2, let me suggest:

Q3. Can the behavior of the packing density along the $A(L,\alpha) = \mathrm{const}$ hyperbola-like curves be described qualitatively? E.g., is the density $\delta$ unimodal along those curves, perhaps for sufficiently large $A$?

Ladies' Diary. [‟All mathematical Questions in The Diary were stated in verse until 1729.”] $\endgroup$ – Marius Kempe Dec 24 '14 at 17:20