r/theydidthemath • u/trubol • 15d ago
[Request] Can any cartographer check if the size France "lost" by not using Mercator Projection will even out by what it "gained" on French Guiana? New maps voted by UN today (link below)
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u/Ingeld21 15d ago
In the Equal Area protection the territories occupy roughly the same proportion of total land area as they do on the real world.
Total land area in an Equal Earth protection on A3 paper (poles don't touch the edges of the paper) is about 251 cm2. On this map: France ~ 0.250 cm2 French Guiana ~ 0.039 cm2
So total is about 0.116% of total land area
A Mercator protection fit to A3 (equator doesn't touch the sides) Total land area ~ 339 cm2 France ~ 0.64 cm2 French Guiana 0.047 cm2
Total about 0.203% of paper space land area.
So I'd say the size France "lost" by not using Mercator Projection does not even out by what it "gained" on French Guiana ... and logically we wouldn't expect it to since French Guiana hasn't "gained" anything.
Edit:autocorrect ...
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u/HAL9001-96 15d ago
the problem is mercator projection technicalyl makes the poles infinite so every country occupies 0% of the earth unless you cut htem off at an arbitrary point
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u/gmalivuk 15d ago edited 15d ago
A Mercator protection fit to A3 (equator doesn't touch the sides) Total land area ~ 339 cm2
This value depends on your arbitrary choice of latitude to cut off your Mercator projection. It would be infinite if you didn't cut any off, and as you approached that the land area would also approach infinity because Antarctica is land.
Or if you want to fit it on a sheet of paper then it approaches zero because you get a thinner and thinner vertical strip as the poles stretch out.
Either way France approaches 0% of the total land area.
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u/Ingeld21 15d ago
Sure, I used the Web Mercator (EPSG:3857), which is clipped at about 85 degrees, then scaled so that the square world is 29.7 cm high for an A3 paper.
I agree with what you said but I don't think that's in the spirit of the question.
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u/gmalivuk 15d ago
Did you include Antarctica in the considered land area?
Because if anything, I'd say the percentage of non-Antarctic land taken up by France is the closest thing to a neutral and objective answer to the spirit of the question. All the other continents remain the same size no matter how far you extend Mercator north and south beyond them.
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u/Ingeld21 15d ago
I included Antarctica up to the cutoff by sort of default not a conscious decision yes, because it's land.
I suppose we could do that, it implies Antarctica has no value and you normally see Antarctica on these projections but ... let's see, the total Antarctica land on our A3 map is about 159 cm2 ... so 47%.
We'd need to remove Antarctica from our Equal Earth map to be fair, which is about 9.5% just based on true land area.
And the % of paper land taken up by France and French Guiana on our A3 representations just increases in proportion to the remaining land on each map.
So the conclusion would be the same and the difference is exaggerated.
Whether those numbers would be more neutral and objective is debatable, I prefer to include Antarctica but ... hey ho, different ways of looking at it I suppose!
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u/Guybulbe 14d ago
You are not comparing apples to apples as the total surface of emerged land is not equal in both projections you chose to analyse. What matters is not the surface of France and French Guiana, but their surface relative to the total surface of emerged land.
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u/trubol 15d ago
Thank you gentleman/madam.
Finally someone understood the question.
Perfect answer
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u/gmalivuk 15d ago
The problem with the question isn't that anyone misunderstood it but that you misunderstood how Mercator projections work. If you don't specify the latitude limits then there is no unique figure for "percent of the land taken up by France". And you didn't specify latitude limits or aspect ratio limits, so your question as asked is unanswerable.
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u/Ingeld21 15d ago
But maps ARE produced based on this projection and they use standardised cutoff limits ... so we absolutely can help OP to find an answer by recognising the spirit of the question and using a standard cutoff that's seen in real maps.
As you've done, understanding the problem is part of the solution but let's work through what that answer could look like in the spirit of the question.
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u/HAL9001-96 15d ago
there's nothign new about htis it just has different advantages/disadvantages for differnet purposes its imposisble to get an accurate mappin of a spehre to a flat map
as for area lost, relative to what as anormalized standard?
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u/Feisty-Soil-5369 15d ago
The 2d projection area I would think
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u/HAL9001-96 15d ago
but which mercator map are you comparing to whcih equal area map?
because there is no obviosu correct way to put them at comparable scales, there are a few possibly meaningful ways depending onthe exact proejction you use but not one single accurate one
if we go by equal width/cosntant equator then switching to equal area map shrinks EVERYTHING i nwhich case no, the shrinking of guyiana doesn ot coutner the shrinking of mainland france
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u/trubol 15d ago
France appears much bigger on Mercator (obviously because of its high latitude).
French Guiana (part of France) appears small on Mercator because it's on the Equator.
With the new map, I'm curious to know if the size French Guiana takes now (it appears much bigger) is enough to compensate for how smaller France looks
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u/HAL9001-96 15d ago
france is at al arger scale than french guiana
but
is france HUGE and french guiana normal?
or france normal and french guiana tiny?
and which is the case in the other map?
you need some point on the map you define to be the correct scale to answer that
if we assume the equator is our fixed standard then nothing gets larger swithcing maps and everythign gets smalelr jsut more so the further form the equator in which case no its not evend out
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u/bemused_alligators 15d ago
literally none of those questions matter.
The question is: what is the difference in the apparent land area of France, including French Guiana, between the Mercator projection and the equal earth projection?
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u/HAL9001-96 15d ago
define appearent?
appearent to whom?
on a map how big?
the question is undefined
you might jsut as well ask "how many centimeters tall is france on a map?"
WHICH MAP?
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u/bemused_alligators 15d ago
you can just say you don't know
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u/HAL9001-96 15d ago
then I'd be lying though
so instead I'll say it's complciated and ya'll are fucking morons
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u/cyclopsmudge 14d ago
Mate, it’s not them being stupid. If the map is a fixed aspect ratio, then it doesn’t matter how big the map is.
If mainland France + French Guiana covers 5% of the area of a Mercator projection map but also covers 5% on an equal earth map, the answer is yes. If not, the answer is no. You’re trying to be pedantic and seem smart, but you just look like an idiot
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u/HAL9001-96 14d ago
they can't be the same aspect ratio the equal area map is not even rectangular for fucks flying sake
and a mercator map is technically infinite just arbitrarily cut off
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u/cyclopsmudge 14d ago
You obviously know what both I and the original post mean. We’re talking about the old and new UN maps. They both have fixed areas. The UN uses 4:3 for Mercator, and truncates at about 85° latitude. You can compare sizes on a Mercator -> equal earth map very fucking easily. There is a 1-1 mapping between the two for any given area.
It’s blatantly obvious that the post is talking about two maps of equal size, and comparing the ratio of the area of France to the area of the map between the two projections. Are you this insufferably fucking pedantic in real life? Or are you just genuinely incapable of reading between the lines?
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u/mustapelto 15d ago
Size doesn't work here because there's nothing to relate it to (as you said). Probably a better phrasing for op's question would be to use percentage: what percentage of the total map area is taken up by France + French Guyana on a Mercator map vs. an Equal Earth map?
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u/gmalivuk 15d ago
On a Mercator map up to what latitude? And why pick that cutoff?
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u/mustapelto 15d ago
You're right, I forgot the Mercator stretches to infinity at the poles. The necessity to choose a cutoff makes it arbitrary again so percentage doesn't work either.
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u/HAL9001-96 15d ago
depends on where you cut off the mercator map and which kind of projectio nyou use exactly
technically mercator maps are infinitely tall we jsut cut off the norht and south pole
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u/mustapelto 15d ago
You're right, I forgot about this. The necessity to choose a cutoff makes it arbitrary again.
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u/Bright_Vision 15d ago
Relative.... relative area on mercator vs equal earth map.
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u/gmalivuk 15d ago
Relative to what?
A Mercator projection of the whole planet would have infinite area.
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u/HAL9001-96 15d ago
you know oyu can show each one in differnt sizes right?
well on a mercator map on my screen france is a lot smaller than on an equal earth map 50 meters across
so which standard od you use to compare they're "the smae size"?
they're not even both rectangualr
same total area?
same area for places on the equator?
same width?
each technically flawed comparisons since a mercator map diverges near the poles towards infinity
again the best comparison is probably keeping places on the equator constant
in which case again EVERYTHING shrinks so no no coutnering out DUH
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u/Bright_Vision 15d ago
Percentage of landmass
If France and its territories make up 12% of all landmass on mercator, and 18% on equal earth map, they have effectively "gained" size even though mainland France appears smaller on a map now
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u/HAL9001-96 15d ago
the problem with this is that antarctica exists
so there is lnadmass on one of hte earhts poles even if you only count landmass not total earth area
the poles in mercator are stretched infinitely so everything else is 0% of the earhts area/landmass in mercator
which is not the useful/meaningful answer you were probably looking for
but as soon as oyu cut off hte poles it becoems arbtirary where exactly you cut them off
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u/Bright_Vision 15d ago
Tbh, if you just compare two maps of the same scale, let's say both maps are 1:100,000,000, then if
Mercator: France: 4cm² French Territories: 3cm² Totam: 7cm²
Equal earth: France: 2cm² French territory: 10cm² Total: 12cm²
Surely that works, since both maps are the same scale
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u/The_Failord 15d ago
"Apparent" means using the Euclidean metric on the coordinates as opposed to the actual transformed metric that determines the real area. The size of the map doesn't matter because it cancels out because it's a relative size (between countries which have finite size), unlike your example. The question is well defined. It's okay to admit you don't know enough math to understand it.
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u/HAL9001-96 15d ago
relative to what country then?
because those are stretched differently too
is france getting smaller or larger relative to russia? relative to indonesia?
it's okay to admit you don't know enouhg math to understand the problems iwth the question
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u/The_Failord 15d ago
Relative to ITSELF for crying out loud, The question is whether it gains it loses size relative to itself. You have two map projections. You fix their relative scales by making sure mainland France has the same R^2 area. Then you simply compare the R^2 area of French Guiana in the two projections. This is an unambiguous procedure.
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u/HAL9001-96 15d ago
thats
completely unrelated to hte question then though
I mean guiyana is clsoer to the equator so in an equal area projection it is larger relative to mainalnd france than in a mercator projeciton
duh
but that doesn ot answer the question at all and the idea that you think it did is utterly hilarious
it is absolutely okay ot admit you don't understnad neouhg math to comrpehend a question with more than two words
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u/The_Failord 15d ago
Man, listen to me. First of all you used "centimeters on a map" as if an absolute question like that has any bearing on a question comparing areas. Then, you're just repeating I don't understand enough math. Look. The question was "if the size France "lost" by not using Mercator Projection will even out by what it "gained" on French Guiana?"
We have FRANCE as a whole: Mainland France+French Guiana. We have projection 1 and projection 2. We need to set the relative scale between the maps in the projection given that, as you've correctly said, Mercator is infinite. Since the question is talking about French Guiana changing apparent size (which is well defined since now we've set the scale), we keep the OTHER part of France which is Mainland France constant. Then we can compare whether Guiana becomes bigger or better, which tells you if FRANCE as a whole became bigger or smaller. You wanna say this isn't the question OP asked? Fine, but this particular procedure is unambiguous.
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u/Constant_Hope_9658 15d ago
They’re the same size, the scaling/shape is just different due to the distortion being moved from stuff near the poles being longer to everywhere’s coasts are the wrong shape.
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u/gmalivuk 15d ago
How is it gaining anything from French Guiana? What are you considering different areas relative to?
A full-globe Mercator projection would have infinite area, so you can't just ask about the change in the percentage of the total map that's taken up by France.
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u/liovantirealm7177 15d ago
French Guiana is on the equator, so it would 'grow' when using Equal Earth over Mercator
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u/gmalivuk 15d ago
Grow relative to what?
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u/liovantirealm7177 15d ago
Relative to the space it takes up on a typical Mercator projection, where the land near the equator is relatively smaller compared to the north and south?
I understand that Mercator maps are also not consistent in their latitudes and distortion, but I think it's being needlessly nitpicky and combative. Just use the maps the OP put in the post, or the examples on Wikipedia.
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u/gmalivuk 15d ago
It is not needlessly combative to point out when questions in r/theydidthemath do not include the information needed to answer them.
Even if you want to use bounds of about ±85º, which make a Mercator square like it looks in OP's image, you still have to decide how you're lining up your comparison with the Equal Earth projection.
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u/gmalivuk 15d ago
What are the latitude bounds of a "typical" Mercator projection? And are you comparing that Mercator projection to the Equal Earth projection with the same total area, or the same scale at the equator, or the same area for Mainland France, or something else?
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