Last night's Final Jeopardy clue: "With 301 miles, it has the most coastline of current states that were part of the original 13 colonies." (Thanks to the Jeopardy! forum for the wording.)
This agrees with the Wikipedia list which is sourced from official US government data.
But as Mandelbrot told us, coastlines are self-similar . (Link goes to the paper How Long is the Coast of Britain as reproduced on Mandelbrot's web page, which unfortunately doesn't have pictures. I'm not sure if the original version of this paper in Science did. Wikipedia's article on the paper does.) That is, the length of a coastline depends on the size of your ruler. Furthermore, I would suspect that the fractal dimension of some states' coastlines is larger than others. Wikipedia states that "Measurements were made using large-scale nautical charts" which seems to imply that all the measurements were done at the same length scale, but if you did the measurements at a smaller scale, the states whose coastline have higher fractal dimension would move up the list.
So last night I spent twenty seconds yelling this at the TV, and ten seconds getting out the answer Maryland. Which is wrong. Also wrong: Maine, New York. (Maine used to be part of Massachusetts; the wording is a bit ambiguous.) It appears that only the south shore of Long Island counts as "coast"; the north shore, which borders Long Island Sound, doesn't.
And of course Chesapeake Bay doesn't count either.
Showing posts with label geography. Show all posts
Showing posts with label geography. Show all posts
30 March 2011
22 April 2009
Constrained tourism
Does there exist a Hamiltonian tour of the graph whose vertices are the 48 contiguous United States and whose edges connect states which border each other? More geographically, can you drive through all of the lower 48 states passing through each exactly once? (From 360.)
Here's one possible solution, although I ignored the constraint implicit in the story that provoked the question, which required a start in Michigan. Note that you have to start or end the tour in Maine since it only borders one other state.
Here's one possible solution, although I ignored the constraint implicit in the story that provoked the question, which required a start in Michigan. Note that you have to start or end the tour in Maine since it only borders one other state.
04 April 2009
Which two states are closest together?
Fix two sets X and Y in the plane, each the interior of a curve, such that their closures don't intersect. How would one go about finding points x in X and y in Y such that the distance between X and Y is minimal?
Furthermore, say we have a bunch of sets X1, ..., Xn, each the interior of a curve, with nonintersecting closures. Let d(i,j) be the minimal distance between Xi and Xj. How can we find the minimal nonzero d(i,j), that is, the minimal distance between any two sets with nonintersecting closures? (In particular, there should be a faster algorithm than computing all the d(i,j). I suspect O(n log n) of the d(i,j) need to be computed although I have no idea why I'm saying this.)
The problem that inspired this is the following geographic one. What two states in the US that don't border each other are closest together? I think I know the answer; I'll post about that later.
Furthermore, say we have a bunch of sets X1, ..., Xn, each the interior of a curve, with nonintersecting closures. Let d(i,j) be the minimal distance between Xi and Xj. How can we find the minimal nonzero d(i,j), that is, the minimal distance between any two sets with nonintersecting closures? (In particular, there should be a faster algorithm than computing all the d(i,j). I suspect O(n log n) of the d(i,j) need to be computed although I have no idea why I'm saying this.)
The problem that inspired this is the following geographic one. What two states in the US that don't border each other are closest together? I think I know the answer; I'll post about that later.
09 February 2009
Brian Hayes on the continental divide
Brian Hayes recently made a fascinating post about determining the location of the Continental Divide. Apparently that's what those pictures on the cover of this month's Notices were. This is to go with David Austin's review of Hayes' Group Theory in the Bedroom, and Other Mathematical Diversions
. The book consists, apparently, of reworked versions of Hayes' essays for The Sciences and American Scientist, essays which in general I highly recommend; the post by Hayes that I linked to is his reflections on redoing this map with the data that's available now, which is better than the data he had in 2000 when he wrote the original column. (Not surprisingly, the major discrepancy between the divides obtained then and now is in Wyoming, where there are some pathological areas that don't drain to either ocean.)
. The book consists, apparently, of reworked versions of Hayes' essays for The Sciences and American Scientist, essays which in general I highly recommend; the post by Hayes that I linked to is his reflections on redoing this map with the data that's available now, which is better than the data he had in 2000 when he wrote the original column. (Not surprisingly, the major discrepancy between the divides obtained then and now is in Wyoming, where there are some pathological areas that don't drain to either ocean.)
19 October 2007
The chain of cities
From radical cartography -- a map of Europe with cities that are near each other (within 150 km) and with populations of more than 100,000 connected by lines. In some places (southern England, the Netherlands, Germany, northern Italy) the lines are very dense; in others they are not. The graph has one large connected component, but there are smaller components -- in Scotland, southern Italy, and the coasts of Spain -- which are separate from the main one.
I suspect you'd get similar-looking maps at smaller scales (that is, if you reduced both the population threshold and the distance threshold). I've heard that the number of cities of population at least N scales like 1/N. To get a similar-looking map with smaller cities you'd want the number of cities "near" each one to be the same. So, for example, say you looked at cities of size 25,000 or greater; there should be four times as many of those "near" a given point as cities of size 100,000 or greater. If we then shorten the distances over which we're willing to draw lines by a factor of two, then we should get the same average degree in the resulting graph. What I'm saying is that I suspect the graph of cities of size 25,000 or greater within 75 km of each other has similar qualitative features. (Of course, at some point you run into the problem of how to define distinct "cities", by which I mean centers of population; should distinct neighborhoods of the same legal municipality be counted differently?)
I'd be interested to see a similar map for the United States (since I'm more familiar with the settlement patterns in the U.S. than in Europe), but not interested enough to make it.
In particular, I see a lot of things that look like the fragments of triangular lattices, with the triangles involved being roughly equilateral. (This seems most visible in Poland.) Now, if you imagine cities as circles -- which isn't a horrible approximation, because you don't expect to see two cities too close to each other -- and you pack them in as tightly as possible on the plane, you should see exactly a hexagonal lattice. Of course, cities aren't all equivalent, and you can't just slide them around arbitrarily...
This makes me wonder -- are there any cities which have street plans that are triangular lattices?
I suspect you'd get similar-looking maps at smaller scales (that is, if you reduced both the population threshold and the distance threshold). I've heard that the number of cities of population at least N scales like 1/N. To get a similar-looking map with smaller cities you'd want the number of cities "near" each one to be the same. So, for example, say you looked at cities of size 25,000 or greater; there should be four times as many of those "near" a given point as cities of size 100,000 or greater. If we then shorten the distances over which we're willing to draw lines by a factor of two, then we should get the same average degree in the resulting graph. What I'm saying is that I suspect the graph of cities of size 25,000 or greater within 75 km of each other has similar qualitative features. (Of course, at some point you run into the problem of how to define distinct "cities", by which I mean centers of population; should distinct neighborhoods of the same legal municipality be counted differently?)
I'd be interested to see a similar map for the United States (since I'm more familiar with the settlement patterns in the U.S. than in Europe), but not interested enough to make it.
In particular, I see a lot of things that look like the fragments of triangular lattices, with the triangles involved being roughly equilateral. (This seems most visible in Poland.) Now, if you imagine cities as circles -- which isn't a horrible approximation, because you don't expect to see two cities too close to each other -- and you pack them in as tightly as possible on the plane, you should see exactly a hexagonal lattice. Of course, cities aren't all equivalent, and you can't just slide them around arbitrarily...
This makes me wonder -- are there any cities which have street plans that are triangular lattices?
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.
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.
20 July 2007
Harry Potter and the pre-orders
As I wondered earlier: there are prediction markets in which you can bet on Harry Potter living. NewsFutures puts his probability of living at [redacted; click on the link if you want to know]. You can see a chart going back to March here. It's currently half past six in England; the answer will be publicly available in five and a half hours. Interestingly, the contract pays $100 (of virtual money) if Harry lives, $0 if Harry doesn't live, and $50 in "any other case".
Also, Amazon.com reveals that the Harry-est town in America is Falls Church, Virginia, as measured by the largest number of pre-orders per capita. (Only towns with more than 5,000 people were included.) The top twenty-two cities on this list are all within fifty-two miles of a major city. (Why 52? I was saying 50 at first, but Fredericksburg, VA was just outside the cutoff.) They are:
The Harry-est states in America is a different story; you'd expect from the first list that the states with lots of suburban population -- New Jersey or Virginia comes to mind, both states with no really large city within their borders but with one just outside -- would appear high up? Perhaps -- but the winner is actually the District of Columbia. (As a city, however, D.C. doesn't even make the top 100.) The six New England states are all high up -- Vermont is the highest at #2, Rhode Island the lowest at #16.
The moral of the story is that depending on how you sample you get very different results. Of course, the whole "Harry-est cities/states in America" thing is just a silly Amazon promotion.
Also, Amazon.com reveals that the Harry-est town in America is Falls Church, Virginia, as measured by the largest number of pre-orders per capita. (Only towns with more than 5,000 people were included.) The top twenty-two cities on this list are all within fifty-two miles of a major city. (Why 52? I was saying 50 at first, but Fredericksburg, VA was just outside the cutoff.) They are:
- Falls Church, VA (Washington, 10 miles)
- Gig Harbor, WA (Seattle 44)
- Fairfax, VA (Washington 21)
- Vienna, VA (Washington 16)
- Katy, TX (Houston 29)
- Media, PA (Philadelphia 22)
- Issaquah, WA (Seattle 17)
- Snohomish, WA (Seattle 31)
- Doylestown, PA (Philadelphia 40)
- Fairport, NY (Rochester 11)
- Woodinville, WA (Seattle 20)
- Princeton, NJ (Philadelphia 45; New York 51)
- Webster, NY (Rochester 15)
- West Chester, PA (Philadelphia 37)
- Williamsville, NY (Buffalo 11)
- Fredericksburg, VA (Washington 52)
- Port Orchard, WA (Seattle 22)
- Decatur, GA (Atlanta 6)
- Larchmont, NY (New York 27)
- Downingtown, PA (Philadelphia 39)
- Canton, GA (Atlanta 41)
- Woodstock, GA (Atlanta 31)
The Harry-est states in America is a different story; you'd expect from the first list that the states with lots of suburban population -- New Jersey or Virginia comes to mind, both states with no really large city within their borders but with one just outside -- would appear high up? Perhaps -- but the winner is actually the District of Columbia. (As a city, however, D.C. doesn't even make the top 100.) The six New England states are all high up -- Vermont is the highest at #2, Rhode Island the lowest at #16.
The moral of the story is that depending on how you sample you get very different results. Of course, the whole "Harry-est cities/states in America" thing is just a silly Amazon promotion.
12 July 2007
why Four Corners is in the middle of nowhere
Mark Chu at ScienceBlogs writes about fractals. The classic example of a fractal is the idea that it is difficult to measure the length of the border between two countries -- Chu's example is Portugal and Spain -- because the length of the border depends on the scale at which you measure it.
In technical terms, the border is nonrectifiable. For those of you who don't know that word, basically what you should know is that the only thing we can "really" find the length of is a straight line segment. To determine the length of any curve* which isn't a straight line segments, we approximate it by straight line segments and add up their length to get an approximation of the legnth of our original curve. For a circle, this means that we approximate the circle as a square, an octagon, a 16-gon, a 32-gon and so on. The square has length the ancient Greek mathematician Archimedes knew how to do this, and he famously showed that π, the circumference (i. e. length) of a circle of diameter 1 was between 3+10/71 and 3+1/7. To four decimal places, these are 3.1408 and 3.1429, respectively; the actual value is about 3.1416.
The total lengths of the square, octagon, 16-gon, and 32-gon are 2.8284, 3.0615, 3.1214, 3.1365 -- you can see how these approach 3.1416.
But for a lot of the objects that exist in "real life", that sequence doesn't approach anything. You might see that each time you halve the length of the line -- which corresponds to doubling the number of sides in the example above -- the length increases by 10%. As you measure on finer and finer scales, the length gets longer and longer.
Natural features tend to have this sort of behavior -- mountains are not cones, clouds are not spheres, and so on -- while artifical features tend to be straight lines or curves with simple mathematical definitions. Take a look at a map of the United States, for example. The eastern states for the most part have boundaries which are natural features -- rivers, the crests of mountain ranges, and so on -- because rivers and mountains were the natural barriers that separated population centers. The western states, which were divided up for the most part before they were settled and by people who didn't have much of an idea of the population patterns anyway, are often straight lines. There's a reason that "Four Corners", the point where Utah, Colorado, Arizona, and New Mexico, is remote. (Link goes to a Google map/satellite image.)
* The mathematical use of "curve" includes what normal people would call "curves", but also lots of other things. For example, a straight line is a "curve" to a mathematician. It is an exceedingly boring curve, but it is still a curve.
In technical terms, the border is nonrectifiable. For those of you who don't know that word, basically what you should know is that the only thing we can "really" find the length of is a straight line segment. To determine the length of any curve* which isn't a straight line segments, we approximate it by straight line segments and add up their length to get an approximation of the legnth of our original curve. For a circle, this means that we approximate the circle as a square, an octagon, a 16-gon, a 32-gon and so on. The square has length the ancient Greek mathematician Archimedes knew how to do this, and he famously showed that π, the circumference (i. e. length) of a circle of diameter 1 was between 3+10/71 and 3+1/7. To four decimal places, these are 3.1408 and 3.1429, respectively; the actual value is about 3.1416.
The total lengths of the square, octagon, 16-gon, and 32-gon are 2.8284, 3.0615, 3.1214, 3.1365 -- you can see how these approach 3.1416.
But for a lot of the objects that exist in "real life", that sequence doesn't approach anything. You might see that each time you halve the length of the line -- which corresponds to doubling the number of sides in the example above -- the length increases by 10%. As you measure on finer and finer scales, the length gets longer and longer.
Natural features tend to have this sort of behavior -- mountains are not cones, clouds are not spheres, and so on -- while artifical features tend to be straight lines or curves with simple mathematical definitions. Take a look at a map of the United States, for example. The eastern states for the most part have boundaries which are natural features -- rivers, the crests of mountain ranges, and so on -- because rivers and mountains were the natural barriers that separated population centers. The western states, which were divided up for the most part before they were settled and by people who didn't have much of an idea of the population patterns anyway, are often straight lines. There's a reason that "Four Corners", the point where Utah, Colorado, Arizona, and New Mexico, is remote. (Link goes to a Google map/satellite image.)
* The mathematical use of "curve" includes what normal people would call "curves", but also lots of other things. For example, a straight line is a "curve" to a mathematician. It is an exceedingly boring curve, but it is still a curve.
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