2911: Greenland Size
| Greenland Size |
 Title text: The Mercator projection drastically distorts the size of almost every area of land except a small ring around the North and South Poles. |
Explanation
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This explanation may be incomplete or incorrect: Created by a MAP THAT'S WAY BIGGER THAN IT'S SUPPOSED TO BE - Please change this comment when editing this page. Do NOT delete this tag too soon. If you can address this issue, please edit the page! Thanks. |
When talking about maps of the world, it's common to discuss the ways that it distorts the land areas that are depicted (all flat maps suffer from some kind of distortion, because the surface of a sphere cannot be flattened without stretching parts and possibly cutting it into pieces). This normally refers to the way that the shapes change or the relative sizes of different land areas.
As many people know the Mercator projection makes land areas near the poles look larger than similar-sized areas near the Equator. For example, a common complaint is that Greenland seems as big as Africa on the map, when Africa is actually 14 times larger than Greenland. This is because it lets the land masses keep their overall shape, and allows for easy course planning for ships on the ocean (thats a big deal in the 1800s).
The joke in this comic is that Cueball is comparing the size of Greenland on the map (usually on the order of centimeters, unless you have a really big or really small map) with its real world size (which is about 836,330 square miles), rather than with the map's other landmasses, which Cueball deems misleading. Of course, this is absurd for an argument against the Mercator projection, as any projection of a map of the same size would be erroneous by Cueball's argument. Any map that doesn't suffer from this distortion would have to be the size of the Earth's surface, which would make it useless.[citation needed]
The title text is about the fact that a horizontal line on a worldwide Mercator projection corresponds to a line of latitude. Most lines of latitude are thousands of miles (kilometers long, but they become smaller and smaller approaching the poles, and in fact there is a line of latitude in a small-diameter circle around each pole whose length would equal the width of the map that Cueball is looking at. If Cueball's map were 1 m wide, then this line of latitude would be at 89.999998568° N - that is, the line of latitude there would be a circle around the north pole with a circumference of 1 m.
Mercator projections have been mentioned previously in 977: Map Projections and 2613: Bad Map Projection: Madagascator.
Transcript
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This transcript is incomplete. Please help editing it! Thanks. |
- [Cueball and White Hat are looking at a world map on the wall showing a Mercator projection, with Cueball gesturing with his hand towards the map.]
- Cueball: This map is really misleading about the size of Greenland.
- Cueball: It's actually much bigger than that - it's hundreds of miles across.
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Anyone else really wanting to know the radius for which the title text is true? I got 356'd Rxy (talk) 20:28, 25 March 2024 (UTC)
New life goal: Go to the poles, find the ring that is mapped to-scale, and color it. Require all satellite maps to be modified to add this stripe of color. PotatoGod (talk) 22:37, 25 March 2024 (UTC)
This is clearly based on Lewis Carroll's Sylvie and Bruno Concluded (1893) which discusses a map made at a scale of 1:1. Take The A Train To Watertown (talk) 22:49, 25 March 2024 (UTC)
Which latitude of Greenland is 1660miles across? I'm noodling around and can find a spot in the northern part which is - more or less - 1660*km* wide, but nothing close to that number in miles. 172.68.144.140 (talk) 23:01, 25 March 2024 (please sign your comments with ~~~~)
Hmm. First time making a comment here, thought that the title text was referencing that the Mercator projection goes to infinity at the poles, and there would be a ring where the map’s unseen parts is 1:1 to the real world. 172.71.214.100 (talk) 01:40, 26 March 2024 (please sign your comments with ~~~~)
Yeah, I think the explanation is wrong; there is a ring around the poles which is the same size on the map as it is in real life, because the mercader projection stretches it out. 172.71.150.50 05:49, 26 March 2024 (UTC)
I'm amazed at how nitpickingly annoying Cueball can get with respect to mapmaking. 162.158.134.38 08:42, 26 March 2024 (UTC)
"there is a ring around the poles which is the same size on the map" - in standard Mercator projection 1m wide map would need to be kilomenters if not thousands km high to show 1m ring on poles. Usually cutout is at 80-85 latitude 162.158.102.110 (talk) 12:12, 26 March 2024 (please sign your comments with ~~~~)
- A 1 meter long circle would only be ~16cm from the pole, not 100's of meters or km. SDSpivey (talk) 03:16, 28 March 2024 (UTC)
- Above is saying how big the map would be to show the bit where the 1m-on-the-ground exists. (It needs to distort latitude (height), as you close in on the pole, to match the distortion of narrowing longitude (against constant width) and thus maintain shape plus rhumb-line consistency.)
- This has nothing to do with the on-ground radius around the pole. Or the distance from such a map's 'pole edge'. (Intuitively: it maintains shape, though the singularity of the pole will confound this, so the ~16cm radius will mean about 16cm from the edge to maintain an arbitrary small-scale graticule-to-rectangle geometric translation. But others have actually calculated this, it looks like.) 172.70.163.24 06:43, 28 March 2024 (UTC)
The latitude band would actually be one Earth's radius (6,378 km) high on the map. 172.69.223.158 (talk) 12:36, 26 March 2024 (please sign your comments with ~~~~)
- Not actually true; the Mercator projection is logarithmic so it would actually be a reasonable-if-large amount high.
For a small angle ε from the pole, we have sin(ε)≈ε and cos(ε)≈1, so y=R*ln((1+cos(ε)/sin(ε))≈R*ln(2/ε); the 1m ring would have a colatitude (angle from pole) of ε≈1m/40000km=1/(4e7), which means y≈R*ln(2/ε)≈R*ln(8e7)=ln(8e7)/(2*pi)≈2.896 (meters), i.e. the 1-to-1 band on a 1m wide map would be slightly under 3 meters above the equator (and a corresponding amount below).
It turns out that a map 3 meters high and 50 cm wide centered at the equator would end almost exactly at the 1:1 scale band (a few millimeters short of it). --172.68.110.80 12:37, 27 March 2024 (UTC)
I'm slightly tempted to add a list of possible uses for a 1:1 scale map of the world. All that I'm coming up with are essentially about its being a ginormous sheet of paper, with its being a map being irrelevant. BunsenH (talk) 17:43, 26 March 2024 (UTC)
- A replacement planet? Either flat (work out which mapping compromise would suit its population best) or get around the "no flat map can..." stuff by making it into an actual globe. You might need to break out artificial gravity equipment (and pursuade people not to wear sharp footwear?), or just take advantage of it being paper-thin, as well as no pesky uncrossable ocean (if you're allowed to 'step on blue') or awkward mountains (you can't actually trip on gradient lines/etc!), so the experience would be ....interesting. 172.70.163.24 19:02, 26 March 2024 (UTC)
- That assumes you make it out of paper - you could always make your map out of rocks and such.172.70.91.211 09:29, 27 March 2024 (UTC)
- Someone's building the world at 1:1 in Minecraft, does that count? Additionally, 1:1 maps of smaller things certainly do exist, though these are more usually called mockups or engineering diagrams. A 1:1 map of a mall was used in Better Call Saul to plot a heist, and sometimes historical sites have 1:1 maps of buildings and streets to show where they were once located. Take The A Train To Watertown (talk) 19:42, 26 March 2024 (UTC)
Because any map that shows a 'point pole' as a measurable length must have a latitude (or other 'small circle', in oblique or transverse versions) where map.width>ground.distance changes to map.width<ground.distance (on any map for which 'equatorial' width is not greater than the respective equitorial circumference, of course!), there are many more map-types that are guaranteed to have a 1:1 relationship. A bit more complex for some, like those with composite 'orange peel' ovals, but it takes an undifferentiatable scaling algorithm to skip over an exactly matching line, or a choice to truncate the map before showing it. (Even a technically 'half the world, only at infinity' map, like a gnomonic will have a matching circle somewhere, given sufficient extent; 'gnomonic equator' is infinitely circumferential, so somewhere between there and the gnomonic-'pole' (inclusively), is a matching distance to reality, whatever the enlargement/reduction factor.) 172.70.91.11 15:30, 28 March 2024 (UTC)
