I’m fairly unsatisfied with that exposition because I had to impose a rule that said that there was only one way, for example, to divide the united states into contiguous regions of 2 people each. I just don’t know how to figure out which contiguous regions are possible and really explain the rich immensity of contiguous region permutations without some sort of mega-intensive multi-year project with like a team of people and incredibly powerful graphing software. I can’t imagine that there’s a theoretical solution at all, without actually plotting where people live, but I suspect that the number of possible maps, still ignoring borders, is likely closer to the trillions or quadrillions than to 144
Note: I crashed my computer twice writing this post, because I wrote a very silly and useless script that was a) not helpful and b) the least robust monstrosity ever to terrorize my RAM. I hope the result was worth the pain.
The problem of dividing the United States into contiguous regions of equal population is actually quite interesting, in a way that this map does not reflect at all.
I’m going to take (imho) the minimum assumptions (1) that I need to engage with the problem in an interesting way. These assumptions basically boil down to two things: i) a contiguous region with N people can only include the N most closely grouped people, and ii) border shapes don’t determine what a unique region is, all that differentiates regions is which people are grouped together. Given these assumptions, we can ask, how many maps like that map could you create?
Allow that there are 313900000 people living in the United States. We can then divide the map into any number of regions that 313900000 is divisible by; eg, we can have 1 region of equal population, 2 regions of equal population, but we cannot have 3 regions of equal population, for 313900000/3 = 104633333.33… and I do not condone the subdivision of a person into 0.33… people. There are 144 divisors of 313900000, and as such we can create 144 distinct maps which depict the subdivision of the United States into regions of equal population. This is under our incredibly restrictive rule, but, remember, there’s no reason that you should only have to fit people into regions with the very closest people. I only took that restriction because I can’t tell how many contiguous regions you could create without that restriction; I would have to know where people are specifically, because, for example, there are ways to arrange 10 people so that you can have tons of possible permutations of 5 contiguous regions of 2 people each, and there ways to arrange them so that you can have very few possible permutations of contiguous regions of 2 people each. This is also assuming that borders are irrelevant; if every possible way of drawing the borders matters and we ignore that there is an absolute minimum distance scale in this universe then there are infinite possible permutations of maps :( So let’s not allow that. Regardless, we already see that the mapsontheweb map was distinctly un-special. Even with my unnecessary rules, it is only one of 144 possibilities.
Looking at the mapsontheweb map, I guess (but refuse to verify) that there are 50 regions. I asked myself, how many possible regions are there, across every map? In other words, how many possible regions can exist in the set of all hypothetical subdivisions of the United States into regions of equal population? For the population of the United States N = 313900000, that’s just the sum of the divisors of 313900000, or 801229968: nearly a billion possible regions. Each of the regions on that mapsontheweb map is exactly as interesting as one billion other regions would be, each fitting into that same space in a different way, each with tremendous variety in what shapes you could make them and how you could position them.
To conclude, here’s a photo of my computer trying hard to solve this problem:
(1)The motivation of the problem is this: We seek to plot where each individual in the United States lives, and then sort them into every possible contiguous region of equal population size. We adopt a method of determining where an individual lives which takes into account their proximity on the minimum possible scale, i.e. a quantum scale, to the precision of plank’s length. It is assumed that, as a result of this method, for any individual I, the ordered set S of distances to any N individuals contains only unique values; that is, no two individuals are equally distant from any one individual. From this assumption, then, the ordered set S of the distance D between all individuals in the United States and any individual I will only contain unique values. To make the problem solvable without intensive geometrical methods, we adopt as a rule that each contiguous region of population N will contain only the N nearest individuals (choosing which individuals are the nearest N depends on the metric chosen, but for the number of combinations of maps which can be produced, which is the motivating question, it does not matter which specific indivuals are chosen, only that it is possible to choose them. Perhaps a methodologist would seek to minimize the mean of the mean distances between all N individuals in each region, but, again, the specific method does not matter in simply addressing the solvability of this problem).
cat surgeon: pat anywhere but my paws. My paws are my livelihood
- star citizen release
- Sims 4 released
- beyblades now unignorable
when I open a restaurant, you can bet it will have Chicken Wing Morning
as an appetizer, and as an entree you can offer Chicken Wing Mourning
- fingerless gloves
- chicken wing night
- only the fingers of a glove
- chicken wing morning
it’s also particularly strange that of the handful of words that Romans borrowed directly from Greek, so many of them were for yards or green spaces. It’s like they were all scratching their heads, trying to think of what to call a park, and someone suggested, maybe, Aristotle has a word for this ?