31 July 2007

how to break up time and space

Squaring the hexagon, from Strange Maps. This map illustrates a proposal for dividing France into perfectly rectangular (well, as rectangular has something on a sphere can be -- but France is small enough that one needn't worry too much about this) regions, of which there were 80 or so. (As you may note, France is not a rectangle.)

This proposal was made around the time of the French Revolution, and it has the ring of proposals from that period; these are the same people that invented the French Republican calendar and the metric system. The metric system has turned out to be a good idea (although some people will argue that many of its units don't correspond to anything on a human scale). The Republican calendar, for those who aren't familiar with it, broke up the year into twelve thirty-day months, of three ten-day "decades" each (these are analogous to weeks); there seems to be no a priori reason why this doesn't work, except that we're used to seven-day weeks. It's rather inconvenient that seven is prime, in fact; it would be useful to have a unit of the "half-week".

If you look at a Major League Baseball schedule some time, you'll see that that's actually the fundamental unit of baseball scheduling; teams generally play two series a week, one of them running from Monday to Wednesday, Tuesday to Thursday, or Monday to Thursday, and the other from Thursday or Friday to Sunday. You might notice that Thursdays are kind of ambiguous in this scheme; sometimes Thursday is the end of a series, sometimes it's the beginning, and sometimes it's neither. Teams often have off on Mondays or Thursdays. The weirdness of Thursday in baseball scheduling is a direct consequence of the fact that seven is odd.) In fact, I'd argue that the fact that the week has seven days is a lot more inconvenient than a thirteen-month year would be; the fact that thirteen is prime, and that therefore no simple fraction of a hypothetical 13-month year is a whole number of months, is pretty much irrelevant. When was the last time you ever did something that took exactly three months (one quarter) or six months (half a year)?

Returning to geography, drawing straight lines as boundaries often seems to be a bad idea; lines that are drawn without regard for population centers inevitably, if you draw enough of them, pass through some population center and divide it up. This causes the people in charge on either side of the line to ignore the other side in government, making the region less able to function as a whole. Surprisingly, it's hard to think of population centers in the U.S. that lie along a state border that's just a line on the map; most of the big population centers near state borders are along rivers, such as Philadelphia, New York, or Washington (note that even if the District of Columbia didn't exist, Washington would be on the Maryland-Virginia border). This isn't surprising; rivers are natural dividing lines. But I suspect that the situation would be different in more densely populated France. And drawing lines on a map without regard for settlement patterns is a cause of the chaos in the Middle East.

The difference between the U. S. and Europe is that in Europe the lines have evolved along with the historical population patterns (which haven't changed that much even in centuries), whereas in the U. S. a lot of the lines between states were drawn before the states in question were settled. One can't expect the people making the maps to forecast in advance where people are going to choose to live, especially in modern times when the population centers grow up along roads (which people can build) and not rivers (which are where they are).

A lot more thoughts on how territory is often divided up -- countries into states, states into counties, etc. -- can be found in Ed Stephan's book The Division of Territory in Society, text available online. His hypothesis is that there's a relationship between the size of a state, county, etc. and its population density; smaller states within a country, counties within a state, and so on tend to be denser. This can be derived from the assumption that "social structures evolve in such a way as to minimize the time expended in their operation", although it only gives a relation between density and county area. It doesn't make forecasts about shape, and it seeems to assume that densities are locally uniform when this is very far from the case.


Matt Noonan said...

I grew up in Kansas City, Missouri which is in both Kansas and (mostly) Missouri. Kansas City is split by the Missouri River, which is why the city exists in the first place. But while the northwest border of Missouri is the Missouri River, the rest of the western border is a straight north-south line. The intersection of the river and this line is right in the middle of Kansas City, making the city split by an artificial boundary rather than the natural boundary already present. And if you've ever driven through the midwest, you know that population density surely isn't what forced such a strange division!

Here is the Kansas City area, for the curious.

Anonymous said...

The Toledo War was fought between Michigan and Ohio over just such a "perfect" border. The original survey and law would have given Toledo to the Michigan Territory. However, this left Northwest Ohio without a port on Lake Erie (the Maumee river). Michigan's contention was that the straight lines of the (mis) surveyed Great Lakes and the letter of the law should prevail, Ohio's position was that the natural geographic features should be distributed to the greatest benefit.

During Michigan's bid for statehood, this cause quite a row. Skirmishes, arrests, an aborted invasion of Ohio, and gunfire followed.

The "War" was settled by giving the (468 mi^2) Toledo Strip back to Ohio, and Michigan received the bulk of the Upper Peninsula from the Wisconsin Territory.

Thus, geography won over straight lines. Michigan, Indiana, Ohio, and Illinois adjoining borders aren't straight for just this reason.

Isabel said...


I'd heard of the Toledo War before. It always struck me as a bit strange that the 500-square-mile Toledo Strip was viewed as being roughly equivalent to the entire Upper Peninsula.

cwitty said...

When was the last time you ever did something that took exactly three months (one quarter) or six months (half a year)?

Note that a lot of people care very much about fiscal quarters (Google gives about 2 million results for "fiscal quarter").

Isabel said...


you make a good point about fiscal quarters. I didn't think of that, probably because I have never worked in the sort of place where people care about those things. When I hear "quarter" I think of an academic term of approximately ten weeks; although these come four to a year and roughly divide the year into quarters at most institutions that use them (think ten weeks of classes, one week of exams, two-week break before the next term starts), they rarely line up exactly with the month boundaries.

Michael Cassidy said...

The seven day week: those of us who take pills spend time trying to figure out is Monday:Friday or Monday:Thursday better for taking a medication twice a week.

I guess I could set my watch's alarm to go off during Thursday afternoon but its easier to do Monday:Friday.

Isabel said...


they're probably equally good, I'd think. Either way one of the intervals is three days and the other is four days. If the difference between three and four were relevant your doctor would probably tell you.

that being said, I doubt there's something magical about "twice a week", and I suspect that they could either make the pills slightly smaller and say "every three days" or slightly larger and say "every four days". But then people would probably have trouble remembering which days were which, since there's no three- or four-day cycle we habitually keep track of.

Michael Cassidy said...

I wonder if the growth of New York City: the absorption of Brooklyn, Queens, Staten Island and The Bronx; and Los Angeles absorption of its surrounding counties runs counter to his arguement.