Something Strange Happens When You Flatten the Earth…

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0:02 In February, 1544, a man sits inside a cell

0:06 inside a stone fortress in what is today Belgium, imprisoned by the Inquisition.

0:12 Several of his fellow prisoners have already been burned at the stake,

0:16 beheaded, buried alive.

0:20 The charge against him is not for treason or rebellion.

0:23 He's been charged with heresy for studying things

0:26 that the church had decided were not meant to be studied.

0:28 For asking forbidden questions about the world.

0:31 His name was Gerardus Mercator.

0:34 And 25 years later, in spite of all of that, he published this.

0:40 This is the most famous map in history.

0:42 You've seen it in classrooms.

0:44 It's a map most of us still use every day.

0:47 And YouTube is full of videos explaining all the ways that this map is wrong,

0:52 all the ways that it distorts reality.

0:54 I mean, Greenland is big as Africa, like Europe,

0:58 dominating the view, whatever's going on with Antarctica down here.

1:04 What this was never meant to be the map that we use for everything.

1:08 It wasn't made to be hung in a classroom.

1:10 It was built to solve one of the hardest problems of the age of exploration.

1:14 And for that one thing, this map is a masterpiece.

1:19 This is a video about something else because there's

1:22 an even more fascinating mystery hiding in Mercator's map building.

1:26 This map should have been mathematically impossible

1:29 because it requires mathematics that didn't exist

1:33 yet and that wouldn't be invented until

1:35 half a century after Mercator published his map.

1:38 So how did he do it and why?

1:42 This is the story of the best, worst map ever made.

1:52 Hey, smart people.

1:53 Joe, here.

1:54 I wanna start with the problem that that map was built to solve because

1:57 it was a real problem and it was killing people in the 14 and 1500s.

2:02 Explorers were sailing into uncharted ocean places

2:05 that no European had ever seen, let alone mapped.

2:09 And when you are in the middle of the Atlantic, no GPS, no radio, no satellites.

2:14 Figuring out where you are is sort of a matter of life and death.

2:18 This is the most accurate map of the earth.

2:21 I finally get to use that in a video.

2:23 Problem is this is pretty useless for navigation.

2:27 Let's say I wanna sail from Lisbon, Portugal to New York.

2:31 If I wanna find the shortest route, I can do this.

2:35 Notice that even though these two cities are basically the same latitude,

2:40 the shortest path actually curves kind of north.

2:43 That's because straight lines on a sphere are curves.

2:46 The shortest path between two points on the globe is curved, not straight.

2:51 And we call these great circle roots because they

2:54 always fall on circular paths positioned around Earth's center.

2:58 These roots are easy to calculate and follow

3:01 today with GPS and computers and everything.

3:03 But to follow this curve route on a ship,

3:05 you'd need to constantly recalculate your compass heading every few hours.

3:10 And on a moving wooden ship in the middle

3:12 of the Atlantic without a calculator and using 16th century math,

3:16 that was basically impossible.

3:18 Plus, do you know how much rum these guys were drinking?

3:21 If only there was a way that sailors could just set

3:23 their compass to one angle and follow that bearing without touching it,

3:29 it would be much easier and safer,

3:31 even if it wasn't the shortest route on a globe.

3:33 That path looks like this.

3:35 Notice that every time it passes a vertical line of longitude,

3:39 it crosses at the same angle.

3:41 Set.

3:42 Your cup is bearing once here and you get where you want to go.

3:46 Easy peasy lemon squeezy.

3:48 This kind of path is called a rum line, R-H-U-M-B.

3:54 Not like rum like they were drinking on the ships all the time.

3:56 Rum lines actually follow spiral paths on a globe, not straight lines.

4:02 It's also not the shortest path, which would be a great circle,

4:06 but sailors were willing to make their journey a little

4:08 bit longer if it meant not getting lost at sea.

4:11 What sailors needed was a flat paper map where these spiral rum lines were

4:17 straight and that they still met at each longitude line at the same angle.

4:22 The man who solved this problem was named Gerardus Mercator.

4:26 He was born Gerard Kremer in 1512,

4:29 the seventh child of Cobbler in what's now Belgium.

4:32 His hometown sat just a few miles down river from Antwerp,

4:35 then one of the wealthiest and most connected cities in the world.

4:38 And most importantly,

4:39 Antwerp in the early 15 hundreds was one of the busiest ports on earth.

4:43 Hundreds of ships arrived and departed every single

4:46 day loaded with goods and stories from distant lands.

4:50 And we truly can't understand why Mercator created his map

4:54 without thinking about the era that he grew up in.

4:56 He was born just 20 years after Columbus arrived in America.

5:00 Magellan circled the globe and Gerard was just around 10.

5:04 As he was growing up,

5:04 the world known to Europeans doubled in size and no one could keep track of it.

5:10 For most of the thousand years before Mercator,

5:13 maps were not tools for getting from place to place.

5:17 They were religious documents.

5:18 They were made to tell a story about how the universe was ordered.

5:21 This is one of the most famous maps of the medieval world.

5:25 It was made around the year 1300.

5:27 Look where Jerusalem sits exactly in the middle, geographically centered.

5:32 Why?

5:32 Because Ezekiel chapter five, verse five says, that's where it is:

5:36 "Thus sayeth the Lord God; This is Jerusalem:

5:39 I have said it in the midst of the nations

5:41 and countries that are roundabout her." Who is gonna argue with that?

5:45 And on these maps, east was at the top

5:48 because east was the direction of paradise.

5:50 The Garden of Eden, this is literally

5:53 where the word orientation comes from to orient

5:57 oneself or your map meant to face east toward the orient toward paradise.

6:04 And here's a map from the 12 hundreds.

6:06 It showed the world as the literal body of Christ.

6:09 Head up top in the east, hands on the sides,

6:13 feet at the bottom, maps like these hung in churches across Europe.

6:18 They were beautiful works of art, spiritually significant.

6:22 But the point was not to draw the world as it actually is.

6:26 Their point was to make scripture feel physically real.

6:30 But good luck getting anywhere with a map like that.

6:33 So while the church is making maps like those sailors in the Mediterranean,

6:37 were using something different.

6:38 These are portal land charts.

6:40 The first maps to draw coastlines accurate enough to navigate in real weather.

6:45 And they weren't oriented toward the holy land.

6:48 North is up top like we're used to.

6:51 You see those lines running in every direction.

6:53 Those are navigation lines that told sailors what compass bearing

6:56 to follow to get from one port to another around the Mediterranean.

6:59 And for the short distances of trade around the Mediterranean,

7:02 these maps worked fine.

7:04 And these two types of maps coexisted for centuries without much

7:07 problem because everyone understood that they

7:09 each had their own special purpose.

7:11 But then in 1406, everything changed.

7:13 A Greek text by Claudius Ptolemy,

7:16 written around 150 ad was translated into Latin for the first time.

7:20 And this was shocking.

7:22 Even though this was an ancient manuscript,

7:24 it showed a world bigger than just about anyone knew existed.

7:29 And it was charted with grids and coordinates.

7:32 It was based on mathematics, not faith.

7:35 Gerard Kremer was born into a world struggling to reconcile all of this.

7:40 The church's sacred geography, Ptolemy ordered mathematical view of the world

7:45 and boats full of explorers that were

7:47 coming home from places that before then no one in Europe even knew existed.

7:51 The life of the man that we call

7:54 Mercator was dedicated to figuring out one question,

7:57 what did the world actually look like?

8:00 Come with me, I need to show you something.

8:02 And when Gerard enrolled in school as was standard practice,

8:05 he translated his name to Latin Kremer,

8:08 which meant "trader" or "merchant" became Mercator.

8:13 He studied under some of the greatest mathematicians in the Low Countries,

8:16 learning how to make detailed astronomical instruments,

8:19 learning spherical geometry and how to make globes.

8:24 We're gonna get to the flat map, I promise,

8:26 but it's worth taking a second to just nerd out on globes today.

8:29 You can just order one of these on Amazon or whatever, right?

8:32 But in the 15 hundreds globes were the ultimate status symbol.

8:35 These were things that emperors and popes

8:37 would have out to flex on common people to show off your knowledge and also

8:42 let everybody know that you could afford one.

8:44 A globe was one of the most expensive things that you could own

8:47 because making these was one of the most technically demanding crafts in Europe.

8:52 It meant carving or turning or assembling a perfect sphere,

8:57 hand engraving every detail in reverse on metal plates,

9:02 and then printing these football shaped strips called gores

9:05 and meticulously gluing them to the sphere with every seam perfect.

9:10 And we think that Mercator ended up

9:13 making hundreds of these globes during his life,

9:15 but only a handful are still around today.

9:18 And this is one of them.

9:20 Just think every time that he sliced the globe and tried to squash it flat,

9:26 he had to wrestle with an almost impossible problem.

9:29 How do you take a curved world and make it play nice on a flat page?

9:34 And that question nagged at him for nearly 30 years

9:38 until he came up with the map that changed history.

9:41 But Mercator nearly didn't live to make his famous map.

9:45 See, he was friends with some followers of one Martin Luther

9:49 and the Catholic Spanish who ruled Mercator's homeland at the time.

9:53 They didn't much like that.

9:55 And well, nobody expected what happened next.

10:01 Nobody expects the Spanish Inquisition.

10:04 In 1544, he was arrested for heresy.

10:07 He spent seven months inside a fortress in the town that he was born.

10:11 Four of his fellow prisoners were executed during that time.

10:15 He survived apparently because some powerful patrons vouched for him,

10:19 but he left his Catholic homeland and never went back.

10:22 And in 1569, while he was in exile in Germany,

10:26 he published this the Mercator map.

10:30 You remember those spiral rum lines?

10:32 He had figured out how to turn rum lines into straight lines.

10:37 It made long distance navigation with a compass, a cinch,

10:40 and it spread across the world in the hands of sailors that used it.

10:45 But there is a deeper mystery hidden

10:48 in this map that took over 400 years to solve.

10:52 And it has to do with how Mercator made it because it turns

10:55 out that this map requires mathematics that didn't exist when he made it.

10:59 Let's start with the fundamental problem.

11:03 This is more or less a sphere like the earth, but let's unpeel it.

11:10 Well, part of our plan fell off.

11:12 The table is flat, but no matter how hard we try,

11:18 just can't get this orange peel to lie

11:20 perfectly flat without tearing or wrinkling it.

11:23 The fundamental problem is a geometric one.

11:28 Now I'm gonna draw a line on this globe, not on my nice globe.

11:31 I'm gonna do it on this cheap one.

11:33 Let's start at the North Pole following

11:34 this line of longitude down to the equator.

11:37 Now let's take a 90 degree turn and draw

11:40 a line along the equator until we reach this point.

11:44 Now let's take another 90 degree turn and head back up to the north pole.

11:49 I've created a triangle.

11:51 Look, there's three sides, three corners, but these angles are 90, 90 and 90.

11:56 They add up to 270 degrees a triangle where all three

12:01 corners are right angles on a flat surface that is impossible.

12:06 The inner angles on two dimensional triangles

12:08 always add up to exactly 180 degrees.

12:12 A sphere and a flat plane are fundamentally different geometric objects.

12:17 A mathematician named Carl Friedrich Gauss proved this in 1827.

12:21 You just cannot flatten a sphere without distorting it.

12:25 So what do you do?

12:27 You get stretchy.

12:29 This came outta my closet.

12:30 No comment.

12:31 Let's see what happens when we take a two dimensional

12:33 grid and try to stretch it over a three dimensional globe.

12:36 It gets all distorted and weird doing the opposite,

12:39 taking a spear and stretching it back to a flat surface.

12:43 No matter what flat surface you pick, that distorts two,

12:46 which means that every map is a lie.

12:49 The only question is what kind of lie are you willing to tell?

12:52 I think we're done with this.

12:57 What the heck?

12:58 Cut.

12:59 You can make a map that preserves area or one

13:02 that preserves distances or one that preserves shapes and angles,

13:05 but you can't do all three.

13:07 It's like against the law mathematically.

13:10 So Mercator made a choice.

13:12 His map was designed to preserve angles and shapes

13:16 that we see on the sphere in a rectangular map,

13:20 even if that means that areas get totally screwed up,

13:24 like Greenland looking as big as Africa.

13:27 So here's how a globe becomes a Mercator map.

13:30 These are those segments we were talking about earlier,

13:32 Gores, that you make a globe out of.

13:35 If I lay out these segments,

13:36 you can see that we're left with these massive gaps near the poles, right?

13:40 We need to stretch these segments out to fill these gaps.

13:45 So how do you do that?

13:47 We have to force these curved edges into straight

13:51 lines so that our lines of longitude run parallel.

13:54 But up here farther from the equator,

13:56 we have to stretch more than down here near the equator.

14:00 At 60 degrees latitude, for example, we have to stretch by a factor of two

14:04 and at the poles we have to stretch a finite point to infinity,

14:08 which is why the poles aren't visible on a Mercator map.

14:12 We can actually calculate how much we need

14:14 to stretch at any point by looking at a globe.

14:17 The equator is a circle with this circumference and any line of latitude.

14:21 Up here is just a circle with a smaller circumference.

14:25 So any segment of latitude is just a segment of a smaller circle.

14:29 Since the circumference of a circle simply scales with the radius,

14:33 we can calculate how much smaller this circle of latitude is versus the equator,

14:38 and that will give us the shrinking factor

14:41 for any latitude above or below the equator.

14:43 We want to calculate the smaller radius our prime.

14:47 We know the larger radius are, and we know the angle here,

14:52 theta, which is equal to our latitude.

14:54 The radius of our circle creates a right triangle, and in a right triangle,

14:59 the ratio of our smaller radius to the hypotonus

15:02 radius is equal to the co-sign of theta.

15:06 This tells us how much any latitude is squeezed relative to the equator.

15:10 But Mercator wanted to stretch those lines of latitude

15:14 so that vertical lines of longitude were parallel.

15:18 So the inverse of our squeezing factor tells us our stretching factor,

15:22 the secant of theta.

15:24 Now all lines of longitude are parallel on our map,

15:27 but we've introduced a new kind of distortion by only stretching sideways.

15:31 Our map looks squished.

15:33 The shapes of coastlines are all wonky.

15:35 The solution Mercator applied the same stretching factor,

15:39 the secret of theta vertically too.

15:42 Whatever he stretched east west, he stretched by the same amount north south.

15:46 This is why Greenland and Antarctica are so huge.

15:50 They've been made wider and taller, but it's not a glitch.

15:55 You see how those circles are still circles?

15:58 That's the very thing that made this map so useful for navigation.

16:03 It preserves angles so far.

16:06 That's math that most high schoolers would get today,

16:09 and it was certainly known to people like Mercator.

16:11 But here's where the map math mystery gets wild.

16:15 To figure out exactly where to put each latitude line on the final map,

16:19 you've gotta add up all of those stretch factors and not just the big ones.

16:24 All of those little secant values for every

16:26 infinitesimal step from the equator to the poles,

16:29 every tiny step, you're applying that local stretch.

16:33 You have to add them all up.

16:35 The process of summing infinitely many

16:38 infinitesimally small quantities that is called integration.

16:43 It's one of the foundations of calculus.

16:46 And when you do it for the Mercator projection,

16:48 the answer involves a special bit of math called a natural logarithm.

16:52 But here's the thing,

16:54 a mathematician named John Napier first developed logarithms in 1614.

16:59 Isaac Newton and Gottfried Leibniz didn't develop

17:02 calculus like integrals until the late 16 hundreds,

17:06 but Mercator published his map in 1569.

17:10 So Mercator made a map that requires math

17:13 that wouldn't be invented until 45 years later.

17:18 How the heck did he do it?

17:19 Well, Mercator certainly never wrote it down.

17:23 This was an unsolved mystery in the history

17:24 of mathematics for hundreds of years.

17:26 In 1599, like 30 years after Mercator first publishes his map,

17:31 an English mathematician published a book where he hand calculated how much

17:35 to stretch latitude and longitude as you go away from the equator.

17:39 It's basically a cheat sheet for making a Mercator map.

17:42 By the 1640s, people were publishing all kinds of cheat

17:45 sheets like this for multiplying and dividing large numbers.

17:48 Let's say you want to multiply these two numbers.

17:51 You can just open the base 10 logarithm table and look up this number.

17:56 These are the exponents that we raise 10 to.

17:59 In order to get those numbers, we just add these together.

18:03 Take that result, look it up in our logarithm

18:06 table and see that it corresponds to this number logarithms.

18:10 Were an easy way to multiply using simple addition.

18:14 In the days before people had invented

18:16 calculators and mathematicians realized that Mercator had basically

18:20 hand calculated one of these logarithm tables decades

18:23 before logarithms were invented to get those numbers.

18:27 Mercator had basically made tiny infinitesimal slices up from the equator,

18:31 smaller and smaller.

18:32 It turns out early mathematicians like Archimedes

18:35 did the same trick around 1800 years earlier.

18:38 To calculate a value for pie,

18:40 Archimedes sliced a circle into thin triangle wedges.

18:43 Then he compared the perimeter of this polygon

18:46 inside the circle to one that touches the outside.

18:49 Now keep dividing those into smaller wedges again and again,

18:53 the more sides that you add, your shape gets infinitesimally close to a circle

18:59 and you get a pretty darn close estimate for pie.

19:02 It took until modern times for mathematicians to figure

19:05 out that Mercator had basically done this same thing.

19:08 Infinitesimally stretch out a sphere onto his flat map.

19:12 What he figured out by hand using ancient Greek geometry,

19:15 was actually a whole new kind of math

19:18 that hadn't even been invented yet during his life.

19:21 Like I can barely solve a calculus problem with calculus,

19:24 and this dude's living on hardcore mode.

19:27 This is something that happens all the time in science and math.

19:31 We don't talk about it enough.

19:32 Mercator built a map that just worked by the time the map caught up to the map,

19:37 sailors had already been using it for decades.

19:40 But here is my favorite part of this story.

19:43 That map from 1569, you probably used it today.

19:48 Open up your phone, open Google Maps or whatever your favorite map app is.

19:53 Look at your neighborhood, your city block,

19:57 a 90 degree turn looks like a 90 degree turn.

20:01 The intersection down the street meets at the same angles it does in real life.

20:07 Now, zoom out the Mercator projection.

20:11 We use it every day because it does what we

20:14 need a map to do every day at any scale, angles are preserved.

20:18 That's what we expect when we navigate the world, right?

20:22 For the world on here to look like the world out there,

20:26 despite all the hate that the Mercator projection gets

20:29 for that, it's the fricking best map out there.

20:32 Mercator spent his whole life chasing this world

20:35 that was growing faster than he could map it,

20:37 figuring out how to create a picture of the world

20:40 on paper that was true to the world out there.

20:43 He somehow did it before the math he needed was even invented.

20:47 And what he built turned out to be wrong in exactly the right ways.

20:52 So how did Mercator's map end up being

20:54 the one that we use for basically everything,

20:57 even though Mercator never meant it for that?

20:59 Well, for starters, you can blame this guy.

21:01 This is an actual Bible printed by Johannes Gutenberg,

21:05 a German metalworker, an inventor of the movable type printing press.

21:09 Thanks to the printing press,

21:10 one map could be copied and distributed really cheaply.

21:14 These weren't valuable artifacts for rich people to flex anymore.

21:18 It could be everyone's map.

21:20 And that's exactly what happened.

21:22 Like I remember one of these hanging in most classrooms when I was growing up,

21:26 I'm pretty sure they weren't using it to teach us nautical navigation.

21:28 But the fact remains, this map isn't an accurate representation of Earth.

21:33 It makes places near the poles look way larger in area than they actually are.

21:38 Here's a website where you can go see that for yourself,

21:41 like Greenland's big, but it's not that big.

21:45 We can make this distortion visible.

21:47 A French mathematician invented this method in 1859 on a globe.

21:51 These circles are the same size and area on Mercator's map.

21:55 They stay circles because angles are preserved,

21:59 but they get huge near the poles area is not conserved.

22:04 These circles let us see how different map

22:06 projections distort the globe in their own way.

22:09 Here's another one.

22:10 Areas are closer to reality,

22:12 but things look weird because it's only stretched left to right.

22:16 And here's another one developed by a German named Arno Peters in 1967,

22:20 though it turned out it had already been invented by James Gall in 1855.

22:24 So today we call it the Gall-Peters projection.

22:26 This map was intended to be more politically correct

22:29 by showing every landmass at its true relative size to others.

22:34 Europe is way smaller than Africa because, well it is, but cartographers,

22:39 they honestly kind of hated it because

22:42 it cured one distortion by introducing new ones.

22:45 This and many other map projections were developed to give

22:49 us different perspectives on the planet each for different reasons.

22:53 But can some maps actually be harmful?

22:56 So there's this idea that growing up staring

22:58 at the Mercator projection gives us a biased sense

23:01 of the world that the map that you grow up

23:03 with shapes your view of who's big and who's small, which places matter more.

23:09 But this actually doesn't seem to be the case.

23:12 Researchers have tested this with tens of thousands of people asking

23:16 them to compare two countries and guess their actual relative sizes,

23:21 and people actually got it right most of the time.

23:24 Most of us don't really seem to over or underestimate sizes

23:28 in our mental maps of the world the way that you might expect.

23:32 Though there are some hints that if

23:34 you grew up looking at the Robinson projection,

23:36 you do slightly better than others.

23:38 The most honest thing that you can do with any map is

23:41 understand the specific ways that that particular map is lying to you.

23:45 But the fact of the matter is there's only

23:48 one totally accurate view of the earth, and that's this.

23:51 This map was built for ships,

23:54 but sometimes when a tool gets used for something other than what it was

23:57 designed for, it can help you solve problems you didn't even know you had.

24:01 It's true.

24:02 This map is bad at some things, but not all of the things that you've heard.

24:07 And it's amazing at many other things that we don't give it credit for.

24:11 It's another reminder to...

24:13 Stay curious.

24:15 Hey, here's a fun fact for sticking around to the end of the video.

24:18 Mercator is actually where we get the word atlas.

24:22 That's the name that he gave to his collection of printed navigation maps.

24:26 And we still use it today to describe a collection of maps.

24:29 And speaking of mapping earth, happy Earth Month!

24:34 PBS is celebrating Earth Month by releasing

24:36 a ton of great new episodes across our channels,

24:39 diving deep into our amazing planet, like the latest episode of Overview,

24:43 which is presented by a suspiciously handsome host.

24:47 The episode explores the Berkeley Pit,

24:49 one of America's most toxic Superfund sites that ironically

24:53 may represent the future of environmental problem solving.

24:56 There's links to that episode and the full

24:59 Earth Month playlist down in description.

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25:56 Thanks for watching this video.

25:59 I thought it was really interesting that Mercator

26:00 was saved from execution by the Spanish Inquisition.

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