2911: Greenland Size

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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

Because the Earth is curved, all flat maps have some distortion. (A common comparison is flattening an orange peel, which cannot be done without tearing and wrinkling it.) Different map projections can distort different metric properties, such as distances, areas, and angles, while leaving others intact. It can be desirable to preserve different metrics in different applications.

The Mercator projection, depicted in the comic, prioritizes depicting correct angles. This allows for easy course planning at sea, and makes shapes fairly accurate. In exchange, Mercator is often criticized for distorting size: distances near the poles look larger than the same distance near the equator. A common complaint is that Greenland appears as big on the map as Africa, when Africa actually has 14 times as much area as Greenland. When these size distortions are presented out of context, they can create bias and misconceptions about different places.

Cueball's dialogue leads the reader to expect this complaint. However, instead of comparing relative sizes of two landmasses within the map, Cueball compares the absolute sizes of the depiction of Greenland and the actual Greenland. On a typical world map, Greenland might be centimeters or inches across. Judging from the human characters, the mapped Greenland in this comic might be 10 cm across. In real life, Greenland is about 650 miles or 1,050 km across from east to west (source). Cueball deems this difference misleading, presenting it as a failure of this specific map or projection.

Of course, this is absurd. The purpose of any map is to present information at a scale (usually much more compactly) at which it is easy to read and interpret. Any actual-size world map would have to be the size of the Earth's surface, in which case it would have few uses. In addition, if a map includes a scale, it enables the user to use the ratio to calculate the actual size of the places depicted (though this would not be possible on a Mercator projection, since the map-to-reality scale is not constant).

The title text is about the fact that regardless of the size of the map there is a certain point where the area on the map is equal to the area at the actual pole at that latitude. This is because a horizontal line on a worldwide Mercator projection corresponds to a line of latitude. While most lines of latitude are thousands of miles (kilometers) long, they become smaller and smaller approaching the poles. As long as the projection (and choice of how much map to print) includes the pole (a point of zero length) expanded out as a measurable edge of the map, there will be a line of latitude around each pole whose length would equal the width of the map that Cueball is looking at (though the specific line would be different depending on the size and precise geometry of the map). If Cueball's map were 1 m wide, then this line of latitude would be at 89.999998568° N or S - that is, the line of latitude there would be a circle with a circumference of 1 m around each of the poles. Of course, in order for the map to actually include (say) the northern of those latitude lines as well as the equator, it would have to be over 3 meters tall.

A map at a scale of 1:1 was discussed in Lewis Carroll's "Sylvie and Bruno Concluded":

"What a useful thing a pocket-map is!" I remarked.
"That's another thing we've learned from your Nation," said Mein Herr, "map-making. But we've carried it much further than you. What do you consider the largest map that would be really useful?"
"About six inches to the mile."
"Only six inches!" exclaimed Mein Herr. "We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!"
"Have you used it much?" I enquired.
"It has never been spread out, yet," said Mein Herr: "the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well."

The same idea was expanded in Jorge Luis Borges's "On Exactitude in Science".

Mercator projections have been mentioned previously in 977: Map Projections, 2082: Mercator Projection, and 2613: Bad Map Projection: Madagascator. The misleading size of Greenland on the Mercator projection is also the object of 2489: Bad Map Projection: The Greenland Special.

Transcript

[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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Discussion

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)

It depends on the size of the map. SDSpivey (talk) 03:16, 28 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)