Background
I've been recently been thinking about the packing problem.
I noticed something odd. In the case of tight packing (no jiggling of a particular part of the packages if the container is shook). One can always define a line of tightest packing. Along this line the packages are touching each other and the edges of the container.
Example $1$
To explain what I mean consider:
The yellow line is line of tightest packing. Note: in this case there are $2$ lines of maximum packing. In fact one of the lines is the longest possible edge one can have.
Now in this example the longer tightest line of packing sees to be $3 \sqrt 2 s + s$, where $s$ is a side of a packaged square.
Observation $1$
Here the length of the tightest packing is the integer solution to:
$$ L =\max( a / \underbrace{ \sin(45^o)}_{\text{projection angle}}s + bs) \leq \underbrace{\sqrt S}_{\text{Diagonal of container}} $$
where $a$ and $b$ are integer variables.
Example $2$
Consider another example:
There is $1$ line of tightest packing we focus on. The lines of tightest packing hug the circumference and is of $18 r$ where $r$ is the radius packaged circles.
Observation $2$:
In sphere case the length of lines of tightest packing are the integer solutions to:
$$ L = \max{nr} \leq 2 \pi (\underbrace{R}_{\text{Radius of container}} -r) $$
Where $n$ is an integer variable.
Question
Have these observations been made? Given the solution has no jiggling, is it possible to first guess the lengthiest line of tightest packing and then try solving the packing problem (especially in higher dimension)? Would that approach be more useful?