Stars On The Flag

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.

Tessellation Using Equilateral Triangles, Isosceles Triangles, Squares, Regular Pentagons, and Equilateral, Non-Convex Octakaitetracontagons

I’m afraid I lack the time to talk about this in more detail today, but, Robert Loves Pi, a geometry-oriented blog, has a lovely tessellation that you might like to see. Tessellations are ways to cover a surface, usually a plane, with an, ideally, small set of a couple pieces infinitely repeated. As a field of mathematics it’s more closely related to kitchen floors than the usual, but it’s also wonderfully artistic, and the study of these patterns brings one into abstract algebra.

In abstract algebra you look at things that work, in some ways, like arithmetic does — you can add and multiply things — without necessarily being arithmetic. The things that you can do to a pattern without changing it — sliding it in some direction, rotating it some angle, maybe reflecting it across some dividing line — can often be added together and multiplied in ways that look strikingly like what you do with regular old numbers, which is part of why this is a field that’s fascinating both when you first look at it and when you get deeply into its study.

Tessellation Using Equilateral Triangles, Isosceles Triangles, Squares, Regular Pentagons, and Equilateral, Non-Convex Octakaitetracontagons

In this tessellation, regular polygons have been given the brighter colors, while the two non-regular polygons have pastel colors.

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