I finally need to figure out how the System’s ‘sectors’ are laid out on Earth. I’ve been dreading this because math. A sector is 1/4,320th of the surface. That’s 45,586 square miles. They are constrained to 3 degrees of longitude, which means they get taller as they approach the poles. How does that actually lay out on a globe?
I didn’t get the AI to generate a globe (do you think I could?), but I did ask:
Starting at the north pole, a wedge three degrees longitude wide and containing 45,586 square miles would end at what degree of latitude?
It answered. It seems reasonable.
To find the latitude where a 3-degree wide wedge from the North Pole contains 45,586 square miles, we calculate the surface area of a spherical cap defined by that latitude.
Math follows. The equations are dubiously rendered, but not illegible.
The wedge would end at approximately 70.8 degrees North latitude.
Iterating on that:
Starting at 71 degrees North latitude, a trapezoid three degrees longitude wide and containing 45,586 square miles would end at what degree of latitude?
It went the wrong way and ended up just north of 80 degrees, not 90.
Starting at 71 degrees North latitude and extending south, an area three degrees longitude wide and containing 45,586 square miles would end at what degree of latitude?
It provides two answers: 61.2 degrees assuming parallel longitudinal lines, but points out that so far north they are not parallel (duh, that’s why I’m asking) and says 62. Less work shown, but it seems reasonable.
Two more, I think. Maybe three.
Just changing 71 to to 62 freaked it out. Pages of response. 53.65 seems to be its consensus after polling itself. And now 54… 48.6. One more should put us into South Dakota.
I can smell the data center overheating from here. 44.4.
We have the northern boundary of our intrepid werewolf’s territory. The southern one? 40.1. Yikes! That’s southern Nebraska.
Three degrees longitude made sense at the equator, where I started this idea. Our intrepid wolves ran from -100 to -105 longitude. Let’s try all this again using four degrees.
From pole: 73.42. Next south (from 73): 66.2. Next south (from 66): 60.7. Next south (from 61): 56.4. Next south (from 56): 51.9. Next south (from 52): 48.3.
This is very convenient for an upcoming plot point. It’s just north of the Missouri River in North Dakota.
44.6 North (45) is the northern boundary of our intrepid werewolf’s territory. 46 is the North Dakota/South Dakota border. Perfect.
Where’s the southern boundary? 41.7. It’s a bit far south: Alliance, NE (where Carhenge is). But it’s better than the NE/KS border. Authorial handwaving will make it work.
Which four degrees of latitude? I’d rather have a bit of Montana and Wyoming than include the Missouri River. -100.4 is as far east as they travelled. That’s where the Missouri crosses the ND/SD border. SD’s western border is -104. -105 is as far west as they went and they headed southeast from there. -100 to -104 it is.
Before I try to figure out a numbering scheme, I wonder if this will work:
Show me a globe with the Earth divided into 4,320 equal area sections, each about 4 degrees longitude wide.
And the AI breaks.
It is mathematically impossible to divide the Earth into 4,320 equal-area sections that are each exactly 4 degrees of longitude wide.
The premise contains a fundamental contradiction between “equal area” and “constant longitude width” on a sphere:
- Convergence of Meridians: Lines of longitude converge at the poles. A strip of land that is 4 degrees wide at the Equator is approximately 277 miles wide. That same 4-degree strip at 60° latitude is only 138 miles wide (half the width), and it shrinks to zero at the poles.
- Area Variation: Because the width of a 4-degree slice decreases as you move away from the Equator, the area of a fixed-height latitude band (e.g., 1 degree tall) also decreases drastically toward the poles.
- A 4°×1° cell at the Equator has roughly double the area of a 4°×1° cell at 60° latitude.
- The Requirement for Equal Area: To create cells of equal area (as calculated in our previous steps), the height (latitude span) of each cell must increase as you move toward the poles to compensate for the narrowing width.
- Near the Equator, an equal-area cell might be 1.0° tall.
- Near the poles, that same area cell might need to be 5.0° or 10.0° tall.
Your explanation of why it’s impossible included the solution, you dumb-ass piece of silicon! I didn’t say anything about “equal latitude”! Attempting to clarify got me less indignant answers, but no image.
On the other hand, I’m certainly not going to do it. I think the AI agrees with me: It’s too much bother.
The point of that: The numbering. I pulled “sector 634”, which is the number of our intrepid hero’s sector, out of my ass. Since longitude lines are arbitrary, I figure the System will just use our zero-point (Hello Greenwich!); why not? 100 is conveniently divisible by 4: It’s the 25th strip. (yes, I’m dropping the negative.)
It took seven sectors to get from 90 north to 45 north. This is obviously symmetric with the southern hemisphere. I really want this to be over, but let’s check one at the equator. Yikes! Only 2.4 degrees latitude. I think I hate spheres. I do not want to iterate that north.
A different approach: There are 4,320 sectors. Each is 4 degrees wide. 360/4 = 90 longitudinal strips. 4,320/90 = 48 sectors per strip. 24 on each side of the equator.
It seems logical to number these things from one pole to the other. The numbering can either continue on the opposite strip or the adjacent strip. The latter is easier to work out, so we’re going with that.
Sectors 1-48 are in strip 1 (from 0-3 longitude), numbered from north to south. Strip 2 (4 to 7 longitude) reverses the numbering order and contains sectors 49-96, numbered from south to north. This means that odd numbered strips have increasing numbers to the south while even numbered strips have increasing numbers to the north.
The 25th strip starts with 25*48 = sector 1,200 (+1, there is no sector 0). It’s odd numbered, so North Dakota has a smaller number and Nebraska has a larger one. We’ve already counted this one out: The sector of concern is seventh down from the pole.
s/634/1,208/
No one is EVER going to notice or figure this out. Geography of the day.
Time to make dinner.
Shit! Off by one and mod 180 errors. If strip 1 is 0-3 degrees, strip 2 is 4-7, etc… 100 is not a boundary. 99 is. 99 thru 103 does not work. Kier is at -103.5. BUT these are negative degrees. If zero is a legitimate longitude value, that means that 359 is the maximum. Since we have negative values, for whatever insane reason, and negative zero is generally frowned upon, that means -1 is 359. -180, which must be positive 179, is opposite 0. That means -104 is positive (76 from 180) 245, 243 would be the boundary. ARG!
I’m going to just ignore the whole mess and throw ‘system fuckery’ at anyone who asks.
Marked Up 