I think the most efficient solution would be to have a series of snorkel lines come down from the ceiling that terminate outside of the building, 6 feet apart.
This would allow the occupants inside to pack as closely as possible without worrying about breathing any hazardous air.
Sphere packing is a genuinely interesting math problem. But it really should not be thought of as any kind of solution to a serious public health threat. Six feet apart is only a recommendation. The virus can transfer based on many other factors such as airflow, shared surface contamination, and spending time in proximity to asymptomatic carriers.
Leave such math topics for home school study questions.
The rule is supposed to encourage being apart from each other, not how many can you pack into an arbitrary measurement from Some Agency.
In the UK it's 2m which is MORE than 6ft.
The next release of our PerfectTablePlan seating software (https://www.perfecttableplan.com) will have a social distancing feature. This will allow you to check social distancing at seated events (such as wedding receiptions and galas). It wasn't too difficult to add as we know the x,y position of each seat and the group each occupant belongs to.
This is a pretty bad premise. If you're packing people in cubicles in a building/floor that most likely uses central air and air conditioning, you already lost the game -- sphere packing or not.