48 Altered States


I saw this intriguing map produced by Brian Brettschneider.

He made it on and for Twitter, as best I can determine. I found it from a stray post in Usenet newsgroup soc.history.what-if, dedicated to ways history could have gone otherwise. It also covers ways that it could not possibly have gone otherwise but would be interesting to see happen. Very different United States state boundaries are part of the latter set of things.

The location of these boundaries is described in English and so comes out a little confusing. It’s hard to make concise. Every point in, say, this alternate Missouri is closer to Missouri’s capital of … uhm … Missouri City than it is to any other state’s capital. And the same for all the other states. All you kind readers who made it through my recent A To Z know a technical term for this. This is a Voronoi Diagram. It uses as its basis points the capitals of the (contiguous) United States.

It’s an amusing map. I mean amusing to people who can attach concepts like amusement to maps. It’d probably be a good one to use if someone needed to make a Risk-style grand strategy game map and didn’t want to be to beholden to the actual map.

No state comes out unchanged, although a few don’t come out too bad. Maine is nearly unchanged. Michigan isn’t changed beyond recognition. Florida gets a little weirder but if you showed someone this alternate shape they’d recognize the original. No such luck with alternate Tennessee or alternate Wyoming.

The connectivity between states changes a little. California and Arizona lose their border. Washington and Montana gain one; similarly, Vermont and Maine suddenly become neighbors. The “Four Corners” spot where Utah, Colorado, New Mexico, and Arizona converge is gone. Two new ones look like they appear, between New Hampshire, Massachusetts, Rhode Island, and Connecticut; and between Pennsylvania, Maryland, Virginia, and West Virginia. I would be stunned if that weren’t just because we can’t zoom far enough in on the map to see they’re actually a pair of nearby three-way junctions.

I’m impressed by the number of borders that are nearly intact, like those of Missouri or Washington. After all, many actual state boundaries are geographic features like rivers that a Voronoi Diagram doesn’t notice. How could Ohio come out looking anything like Ohio?

The reason comes to historical subtleties. At least once you get past the original 13 states, basically the east coast of the United States. The boundaries of those states were set by colonial charters, with boundaries set based on little or ambiguous information about what the local terrain was actually like, and drawn to reward or punish court factions and favorites. Never mind the original thirteen (plus Maine and Vermont, which we might as well consider part of the original thirteen).

After that, though, the United States started drawing state boundaries and had some method to it all. Generally a chunk of territory would be split into territories and later states that would be roughly rectangular, so far as practical, and roughly similar in size to the other states carved of the same area. So for example Missouri and Alabama are roughly similar to Georgia in size and even shape. Louisiana, Arkansas, and Missouri are about equal in north-south span and loosely similar east-to-west. Kansas, Nebraska, South Dakota, and North Dakota aren’t too different in their north-to-south or east-to-west spans.

There’s exceptions, for reasons tied to the complexities of history. California and Texas get peculiar shapes because they could. Michigan has an upper peninsula for quirky reasons that some friend of mine on Twitter discovers every three weeks or so. But the rough guide is that states look a lot more similar to one another than you’d think from a quick look. Mark Stein’s How The States Got Their Shapes is an endlessly fascinating text explaining this all.

If there is a loose logic to state boundaries, though, what about state capitals? Those are more quirky. One starts to see the patterns when considering questions like “why put California’s capital in Sacramento instead of, like, San Francisco?” or “Why Saint Joseph instead Saint Louis or Kansas City?” There is no universal guide, but there are some trends. Generally states end up putting their capitals in a city that’s relatively central, at least to the major population centers around the time of statehood. And, generally, not in one of the state’s big commercial or industrial centers. The desire to be geographically central is easy to understand. No fair making citizens trudge that far if they have business in the capital. Avoiding the (pardon) first tier of cities has subtler politics to it; it’s an attempt to get the government somewhere at least a little inconvenient to the money powers.

There’s exceptions, of course. Boston is the obviously important city in Massachusetts, Salt Lake City the place of interest for Utah, Denver the equivalent for Colorado. Capitals relocated; Atlanta is Georgia’s eighth(?) I think since statehood. Sometimes they were weirder. Until 1854 Rhode Island rotated between five cities, to the surprise of people trying to name a third city in Rhode Island. New Jersey settled on Trenton as compromise between the East and West Jersey capitals of Perth Amboy and Burlington. But if you look for a city that’s fairly central but not the biggest in the state you get to the capital pretty often.

So these are historical and cultural factors which combine to make a Voronoi Diagram map of the United States strange, but not impossibly strange, compared to what has really happened. Things are rarely so arbitrary as they seem at first.

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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.

My Mathematics Blog, As March 2015 Would Have It


And now for my monthly review of publication statistics. This is a good month to do it with, since it was a record month: I had 1,022 pages viewed around these parts, the first time (according to WordPress) that I’ve had more than a thousand in a month. In January I’d had 944, and in February a mere 859, which I was willing to blame on the shortness of that month. March’s is a clean record, though, more views per day than either of those months.

The total number of visitors was up, too, to 468. That’s compared to 438 in January and 407 in short February, although it happens it’s not a record; that’s still held by January 2013 and its 473 visitors. The number of views per visitor keeps holding about steady: from 2.16 in January to 2.11 in February to 2.18 in March. It appears that I’m getting a little better at finding people who like to read what I like to write, but haven’t caught that thrilling transition from linear to exponential growth.

The new WordPress statistics tell me I had a record 265 likes in March, up from January’s 196 and February’s 179. The number of comments rose from January’s 51 and February’s 56 to a full 93 for March. I take all this as supporting evidence that I’m better at reaching people lately. (Although I do wonder if it counts backlinks from one of my articles to another as a comment.)

The mathematics blog starts the month at 22,837 total views, and with 454 WordPress followers.

The most popular articles in March, though, were the set you might have guessed without actually reading things around here:

I admit I thought the “how interesting is a basketball tournament?” thing would be more popular, but it’s hampered by having started out in the middle of the month. I might want to start looking at the most popular articles of the past 30 days in the middle of the month too.

The countries sending me the greatest number of readers were the usual set: the United States at 658 in first place, and Canada in second at 66. The United Kingdom was a strong third at 57, and Austria in fourth place at 30.

Sending me a single reader each were Belgium, Ecuador, Israel, Japan, Lebanon, Mexico, Nepal, Norway, Portugal, Romania, Samoa, Saudi Arabia, Slovakia, Thailand, the United Arab Emirates, Uruguay, and Venezuela. The repeats from February were Japan, Mexico, Romania, and Venezuela. Japan is on a three-month streak, while Mexico has sent me a solitary reader four months in a row. India’s declined slightly in reading me, from 6 to 5. Ah well.

Among the interesting search terms were:

  • right trapezoid 5 (I loved this anime as a kid)
  • a short comic strip on reminding people on how to order decimals correctly (I hope they found what they were looking for)
  • are there other ways to draw a trapezoid (try with food dye on the back of your pet rabbit!)
  • motto of ideal gas (veni vidi v = nRT/P ?)
  • rectangular states (the majority of United States states are pretty rectangular, when you get down to it)
  • what is the definition of rerun (I don’t think this has come up before)
  • what are the chances of consecutive friday the 13th’s in a year (I make it out at 3/28, or a touch under 11 percent; anyone have another opinion?)

Well, with luck, I should have a fresh comic strips post soon and some more writing in the curious mix between information theory and college basketball.

Something I Didn’t Know About Trapezoids


I have a little iPad app for keeping track of how this blog is doing, and I’m even able to use it to compose new entries and make comments. (The entry about the lottery was one of them.) Mostly it provides a way for me to watch the count of unique visits per day, so I can grow neurotic wondering why it’s not higher. But it also provides supplementary data, such as, what search queries have brought people to the site. The “Trapezoid Week” flurry of posts has proved to be very good at bringing in search referrals, with topics like “picture of a trapezoid” or “how do I draw a trapezoid” or “similar triangles trapezoid” bringing literally several people right to me.

Continue reading “Something I Didn’t Know About Trapezoids”