The Very Mathematical U. S. Star Flag

Regular visitors here might recall that the good people at Firefly Books sent me a review copy of Ivan Moscovitch's beautiful puzzle book at top just before Christmas. If you missed my first recommendations, let me urge you to check it out.


I mention it again because one of the puzzles came up in a nice illustration of one of the mathematical concepts in the US flag's star field.
So first, the problem.

So there is the set up, a tightly packed square with 100 circles in a 10 by 10 array. And the challenge: Is it possible to squeeze one more circle into the same space? If you need to take some time, just stop reading and grab a pencil or whatever and work on it.

W
A
I
T
I
N
G

OK, and now for the upgraded challenge, how many more can you squeeze into the same space.

Could you see a way to squeeze a total of 105 into the same space? (actualy I think it is very slightly less space.)
Here is a way:

Ok, and now the tie-in question, what does this puzzle have to do with the US flag star field.

If you've been around since the sixties, you probably know that the 48 star flag, the one we had from 1912 to 1959 was a rectangle of 6 rows of eight stars. Within a little less than two years, we had added a 49th, and then a 50th star. If you look at the two star fields, you may notice a similarity.

Although we've had some unusual shaped flags, usually the star field is in a rectangle with the stars displaying some kind of (generally rectangular) similarity. Some have strayed greatly from the rectangle form however. This one with 38 stars from 1877 until 1890 is an example.

But the preference for a rectangular field over a square field was clearly demonstrated in the flags with 25,  which seems to be one of the least symmetric flags ever, the 36 star flag, and the briefly flown 49 star flag with 7 rows of 7, but not in a square. .

So what do you think the 51 star flag might look like?