The United States flag has as many stars as the country has states. For a long while star arrangement was up to the flag-maker, with no specific rule in place. This is where the occasional weird and ugly 19th century flag comes from. But the arrangement has got codified. It’s to be stars in rows, or at least staggered rows.
It’s easy to understand how to arrange 48 stars, which the flag had for a while. Or 49 stars, which it had almost long enough to get a new flag made. 50 stars, which it’s had for longer than 48 now, are familiar from experience. But a natural question is how to arrange an arbitrary number of stars? And courtesy the MTBos Blogbog, linking to essays about mathematics, I don’t have to answer it myself.
Experience First Math reviewed the problem recently. You can find a pattern by playing around, of course. It’s not very efficient, but we don’t need new flags very often. We don’t need to save time on this.
And uniformly spacing stuff can be a hard problem. For example, no one knows what is the most uniform way to put thirteen spots on the surface of a sphere. We’re certain that we’re close, though.
This is a simpler problem. We have to fit stars in a rectangle. The stars have to be arranged in rows, or in staggered rows. Each row can’t be too much bigger or smaller than its neighbors. And with that, a little bit of factoring and geometric reasoning and counting produces a lovely result: how to generally arrange stars.
Well, almost generally. There are some numbers that don’t work with alternating rows. We’ve seen this before. There were some ugly compromises necessary to have a 44-star flag, in the 1890s, or the 36-star flag in 1865. But with this alternating-rows example, we’ve got a hint to working out other nearly-staggered and nearly-alternating row patterns.
nebusresearch 
New Zealand’s flag has four stars, representing the Southern Cross, and we might be losing those if a flag change referendum under way just now votes for an alternative.
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Both the current one and the possible alternate both have the four star pattern on them.
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Yes, however I have been trying to ignore the new one.
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I am intrigued … hm.
Well, if Wikipedia’s not leading me wrong and I’m looking correctly at what it calls the designs by Mike Archer … they’re not bad, I suppose. I feel old-fashioned enough to not be perfectly comfortable with curves in flags, except those that surround Great Seals and the like. I’m from New Jersey, which just plants its seal on a buff background. I imagine it didn’t take a lot of design imagination to do, but it looks thick with symbolism and dignity and all.
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The designer of the chosen alternative to the current flag is Kyle Lockwood. Not too bad a look in some ways, but the whole process of finding a new flag has been rather badly mishandled, and the fact that everybody and their dog had alternatives to the ones offered indicates there wasn’t a public buy-in. Some of the proposals looked like branding logos, not national flags. It’s a hot-button issue in NZ for sure.
http://www.scoop.co.nz/stories/PO1603/S00077/flawed-referendum-process-on-changing-new-zealands-flag.htm
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Ah, thank you. I understand more of this now. … New Zealand flag redesigns haven’t been a major item in the United States news lately, which I can’t really say is provincial on our part. It also strikes me the dynamics of this are very close to when my grad school attempted to impose a new mascot on the student body. The consent manufacture didn’t go very well, not at least the years I was there.
I did belatedly think of a flag with curves that I still find attractive. Singapore’s crescent-moon and pentagon of stars I think works very nicely even if, as Professor Helmer Aslaksen points out, the moon is pointing in the wrong direction as viewed from Singapore.
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