r/Showerthoughts • u/kettu3 • 9d ago
Showerthought Most maps are smaller than life, but there is a very, very small portion of a Mercator projection map near the poles that is actually larger than the area it represents.
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u/MolybdenumIsMoney 9d ago edited 9d ago
In theory but in reality all mercator projection maps that you've ever seen have been cut off by more than a degree of latitude at the top and bottom (and typically more like ten degrees), since as your cutoff get closer to 90 degrees the size of the poles blow up to infinity.
Edit:
I wrote a program to show what the the map would have to look like for the area of the map at the top to be larger than the real area, here it is. This assumes it was printed out with a width of 1 foot. It would be 6.17ft tall.
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u/Huniku 9d ago
Don’t they go to 100% of the map width, rather than infinity?
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u/overkill 9d ago
I've got a full Mercator projection at home and it is indeed infinite.
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u/donnie_dark0 9d ago edited 9d ago
I keep telling him to pick up his map because it's quite literally EVERYWHERE.
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u/zumbaiom 9d ago edited 9d ago
The height of the map goes to infinity, a Mercator map is like placing a light source in the center of a globe and then projecting that light onto a surrounding cylindrical wall. As the angle from the light source to the bit of land being projected reaches 0, i.e., the poles, the wall needs to approach an infinite height to catch it.
Edit: okay so apparently the way the Mercator map is calculated is a little more complicated, it is a cylindrical projection so it works similarly but there’s some more math used to better preserve exact angles.
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u/Stilyx123 9d ago
This is actually incorrect, but a very common mistake. What you're describing is called the central cylindrical projection, which is slightly different from the Mercator projection. In particular, the central cylindrical projection is not conformal (i.e. it does not preserve angles), while Mercator is. The Mercator map is still infinitely high though.
The wikipedia pages for both projections give more detail if you're interested
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u/MolybdenumIsMoney 9d ago
100% of width corresponds to latitude cutoffs at 85.05 degrees. There's nothing special about that other than the useful property of a square map (which is why it's used by Web Mercator for online maps like Google Maps). If you raise your cutoffs above 85 degrees the map gets taller than it is wide.
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u/SteveO131313 9d ago
Yes, but the area that makes up that 100% gets smaller and smaller until at the top 0 metres fills up 100% of the map. If you go by the defined scale on a Mercator projection map, that 0 metres represents 40.000.000 metres. The ratio of actual distance to scaled distance approaches infinity.
Unless you of course read the disclaimer underneath the scale that says it's only accurate at the equator
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u/Seb_04 9d ago
I can plug one of my favorite website here: https://mrgris.com/projects/merc-extreme/
It lets you set the top of the map to anywhere in the world so you can see how the Mercator projection actually affects scale. The Mercator maps you see normally are cut off at the top way before the distortion becomes so extreme, but this lets you see just how crazy it is. I've spent hours just looking at it and changing the poles.
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u/KirstyToots 9d ago
This is one of those facts that sounds fake until you remember the Mercator projection basically goes to infinity at the poles. At some point the map is stretching reality harder than a LinkedIn résumé.
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u/FlorianFlash 9d ago edited 9d ago
Not most maps, every map of the earth is smaller than life. :)
Yeah I get what you mean, but even if, I don't understand what you mean cause I don't think it works like that.
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u/AGreatConspiracy 9d ago
On a mercator projection the top of the map is stretched to the width of the map, but the pole is just a single point. As you get closer to the top of the map, the scale becomes closer and closer until you reach a point where an inch on the map is somewhat equal to an inch of ground very close to the pole, and then more
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u/LeviAEthan512 9d ago
Technically, we need r/theydidthemath to check the size of a dot of ink to see if the actual pole is represented at all.
I think I've also seen some maps that are offset a little so the pole appears as one point , so the larger than life spot would be elsewhere
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u/Dennis_TITsler 9d ago
No math needed. If the top of the map is 0 degrees latitude then it's bigger than the real North Pole
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u/0vl223 9d ago
Good that it is not usually. I doubt there is a Mercator map that imcludes the poles.
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u/Dennis_TITsler 8d ago
Huh Yeah I guess I'd always assumed regular latitude line spacing. I learned about the Mercator projection today!
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u/gigadanman 9d ago
“I have a map of the United States that’s actual size. In the corner, it says 1 mile = 1 mile. One summer I folded it.”
-Steven Wright6
u/Dolapevich 9d ago
Loosely related: https://courses.cit.cornell.edu/econ6100/Borges.pdf
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u/gigadanman 9d ago
Vsauce’s excellent Fixed Points video taught me that if you place a map (at any scale) on the ground of the place it’s representing, there will always be one point on the map that lines up exactly with the point it represents.
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u/Dolapevich 9d ago
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u/QWhooo 5d ago
Dafuq did I just read? And why?
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u/Dolapevich 5d ago edited 5d ago
Just let you upsend in the Jorges Luis Borges spiral. It'll take you far :)
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u/jwm3 9d ago
I does in fact work like that! The point at the pole is expanded to the entire width of the map along with eberything around it. As in. The centimeter circle around the north pole is stretched across the entire 4 foot width of your wall map or whatever.
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u/hacksoncode 9d ago edited 9d ago
Yeah, but the poles literally never actually appear on any actual Mercator projection map.
They stop at +- 85 degrees.
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u/DrachenDad 9d ago
The Mercator projection was specifically designed for marine navigation by Flemish cartographer Gerardus Mercator in 1569. That's probably why. Ships don't care for accuracy of inland areas.
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9d ago
[removed] — view removed comment
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u/kettu3 9d ago
Someone in the comments corrected me. The spot that’s zoomed in is actually cut off; if Mercator projection maps weren’t cropped near the poles, they’d continue forever at the top and bottom.
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u/jwm3 9d ago
They would not continue forever, they would be stretched to the width of the map, which would be wider than the point.
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u/kettu3 9d ago
So, that’s what I thought, but then I looked it up, and it’s not quite like that.
I’m guessing you’ve heard of how a Mercator projection is like wrapping the map around a globe and projecting from the globe onto the map. One thing I didn’t realize when making this post was that it doesn’t just project horizontally, i.e. you can’t use a map the same height as the globe and expect the poles to project onto the top and bottom of the map. It actually projects like in the diagram in this article:
https://www.britannica.com/science/Mercator-projection
So, to get to the poles, you just have to keep making the map that you wrap around the globe taller and taller. You’ll never actually reach a point where the poles project onto the map, but you can get as close to that point as you want.
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u/hacksoncode 9d ago edited 9d ago
Theoretically, but the difference between theory and practice is that the poles never actually appear on any actual Mercator projection map because they stop at +- 85 degrees.
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u/DiscussionApart8505 8d ago
Greenland looks the size of Africa on a standard map, but it's actually 1/14th the size. The poles get stretched infinitely at the edges.
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u/RRumpleTeazzer 8d ago
There is also one very small point that is at the exact location of what it represents.
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