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5.6 — Maps, Projections and Finding Yourself
Every flat map of the world is wrong, and it is wrong for a reason that was proved mathematically two centuries ago. The only choice a mapmaker has is which kind of wrong to accept.
Why maps lie
Why can no flat map be accurate?
Because a sphere cannot be flattened without stretching, and this is a theorem rather than an engineering limitation.
Gauss proved it in 1827, in a result he called the Theorema Egregium — the remarkable theorem. It says that the curvature of a surface is an intrinsic property, measurable entirely from within the surface without reference to anything outside it. A sphere has positive curvature; a flat sheet has zero. Since curvature cannot change without stretching, no piece of a sphere can be laid flat without distortion.
You can feel this with an orange peel: it will not lie flat without tearing, and pressing it flat splits it. The same is why a pizza slice holds itself rigid when you fold it lengthwise — folding forces a curvature in one direction, and the theorem then forbids it from curving in the other, so it cannot droop.
Every projection therefore chooses what to sacrifice: shape, area, distance or direction. You can preserve any one of them and never all of them.
What does the Mercator projection get wrong, and why is it still used?
It gets area badly wrong and it gets one specific thing exactly right, and that one thing is why it exists.
Gerardus Mercator published it in 1569 for navigation. Its property is that a straight line drawn on the map is a line of constant compass bearing. A navigator could rule a line from port to port, read off the angle, and hold that heading. Nothing else available did that, and it made ocean navigation tractable.
The price is that it stretches everything towards the poles, and the stretch grows without limit. Greenland at 2.16 million square kilometres appears comparable to Africa at 30 million. Alaska looks the size of Brazil and is about a fifth of it. Antarctica becomes an infinite band along the bottom, and the poles cannot be shown at all.
The political criticism is real and partly overstated. Mercator did not choose it to inflate Europe; he chose it for compass bearings and the distortion is a mathematical consequence. But the effect of centuries of classroom Mercator maps on how people picture relative sizes is genuine and measurable.
And it is still everywhere, because online mapping uses a variant of it. The reason is that Mercator is conformal — it preserves local angles and shapes — so a street corner has the right angle at any zoom level, and the tiling and rotation-free rendering that web maps need come out simply. At street scale the area distortion is invisible.
What are the alternatives, and what do they sacrifice?
Gall-Peters preserves area exactly, so Africa looks its true size relative to Europe. It does so by distorting shape severely — countries near the equator are stretched tall and thin, those near the poles squashed. It was promoted from the 1970s on explicitly political grounds by Arno Peters, and cartographers objected as much to his claim of originality as to the map itself, since James Gall had published the same thing in 1855.
Robinson and Winkel Tripel are compromises: neither area nor shape is exactly right, and both are wrong by only a little across most of the map. National Geographic uses Winkel Tripel, and it is the sensible default for a general world map.
Dymaxion, designed by Buckminster Fuller, unfolds the globe onto an icosahedron so that area and shape distortion are both small — at the cost of a map with no up, no continuous ocean, and interruptions wherever the faces separate.
AuthaGraph, a Japanese design from 1999, divides the sphere into 96 triangles and produces a near-accurate rectangular map that can be tiled in any direction. It has won design prizes and remains unfamiliar because it looks wrong to eyes trained on Mercator.
Why is north at the top?
By convention, and not a very old one.
Medieval European mappae mundi usually put east at the top, towards Jerusalem and the rising sun — which is where the word orient comes from, and why orienting yourself originally meant finding east. Islamic maps of the same period frequently put south at the top. Chinese maps often put south at the top too, because the compass was understood as pointing south.
North won for practical reasons in the age of European navigation: the magnetic compass points north, Polaris marks north in the northern hemisphere, and the European mapmakers who set the standards were themselves in the north.
There is no physical reason for it. "Upside-down" world maps are sold as novelties in Australia, and they are exactly as correct.
Coordinates
What are latitude and longitude, and why was longitude so hard?
Latitude is easy and longitude was a crisis.
Latitude is your angle north or south of the equator, and it can be measured from the sky alone. The height of the pole star above the horizon, or the height of the sun at noon adjusted for the date, gives it directly. Ancient navigators could do this.
Longitude is your angle east or west of an arbitrary line, and the sky offers no reference for it, because the Earth's rotation carries all the reference points round once a day. What longitude actually measures is a time difference: the Earth turns 15 degrees per hour, so if you know the local time where you are and the time at your reference meridian at the same instant, the difference gives your longitude.
So the problem was a clock. A pendulum clock does not work at sea; the ship's motion, temperature and humidity destroy its accuracy. Ships routinely did not know where they were, and in 1707 four British warships ran onto the Isles of Scilly and around 1,400 men died, because the fleet's position was wrong.
The British Parliament offered a prize of £20,000 in 1714 — an enormous sum. The scientific establishment expected the answer to come from astronomy. It came from a Yorkshire carpenter. John Harrison spent decades building a series of marine chronometers, solving temperature compensation with bimetallic strips and eliminating lubrication problems with self-lubricating wood and new escapements. His H4, a large watch, was tested on a voyage to Jamaica in 1761 and lost about five seconds in 81 days.
The Board of Longitude, dominated by astronomers, resisted paying him for years. He received most of the money eventually after intervention by George III, at the age of nearly eighty.
Why does the prime meridian go through Greenwich?
Because of shipping tonnage, as described in 2.7 — around two thirds of the world's ships already used charts based on it.
There is a modern postscript. The line marked on the ground at Greenwich, where tourists stand with a foot in each hemisphere, is about 102 metres west of the actual prime meridian used today. The original was defined by a particular telescope; the modern reference frame used by satellites is defined by the Earth's centre of mass, and the two differ slightly because of the local deflection of gravity. Zero longitude on your phone is in the park, not on the brass strip.
Finding yourself
How does GPS work, and why does it need relativity?
Around thirty satellites orbit at about 20,200 kilometres, each carrying an atomic clock, each continuously broadcasting its own position and the exact time.
Your receiver picks up several of these signals. Because it knows the speed of light, the delay in each signal tells it how far away that satellite is. One distance places you on a sphere. Two distances place you on a circle where the spheres intersect. Three narrow it to two points, one of which is absurd. A fourth satellite is needed for a different reason: your phone does not have an atomic clock, so its own clock error is a fourth unknown, and the fourth measurement lets it solve for time as well as position. That is also why your phone's clock is extremely accurate whenever it has a fix.
The relativity part is not a curiosity, it is a working correction. Two effects act in opposite directions. The satellites are moving fast relative to you, and special relativity says a moving clock runs slow — by about 7 microseconds a day. They are also higher in Earth's gravity well, and general relativity says a clock deeper in gravity runs slow, so theirs runs fast relative to yours by about 45 microseconds a day.
The net is around 38 microseconds a day fast. That sounds negligible until you convert it: light travels about 300 metres in a microsecond, so an uncorrected system would accumulate roughly 10 kilometres of position error per day. The satellites' clocks are deliberately set to tick at a slightly offset rate before launch, and further corrections are applied continuously.
GPS is the most widely used practical application of general relativity in existence, and it is a clean answer to anybody who asks what relativity is good for.
Does GPS need the internet?
No. This is worth being clear about, because the two are constantly confused.
A GPS receiver only listens. It transmits nothing, and it needs no network — which is why a handheld receiver works in the middle of an ocean or a desert with no signal of any kind.
What the internet provides is two separate things. The map — the images and street data drawn around your position — is downloaded, which is why navigation apps appear to stop working offline, and why downloading a map area in advance fixes it. And assistance data: a cold receiver with no idea where it is or what the satellites are doing can take several minutes to find them, and a quick download of the current satellite positions cuts that to seconds. That is why your phone gets a fix faster than a car navigation unit from 2005.
There are now four global systems: the American GPS, Russian GLONASS, European Galileo and Chinese BeiDou, plus regional ones including India's NavIC, which covers the subcontinent and a surrounding region. Modern phones use several at once, which is why accuracy in cities has improved so much — more satellites means more chances of seeing enough sky between buildings.
How did people navigate before any of this?
By dead reckoning, and by reading things most people no longer notice.
Dead reckoning is arithmetic: known starting point, plus heading, plus speed, plus elapsed time, gives an estimated position. Errors accumulate, so it was corrected whenever a landmark or a star sight was available. It is still taught, because it is what you fall back on when the electronics fail.
Polynesian navigators crossed thousands of kilometres of open Pacific with no instruments at all, using a body of knowledge that included star paths, the direction and shape of ocean swells refracting around islands, cloud formations that sit over land, and the flight lines of birds at dawn and dusk. The knowledge nearly died out and was deliberately revived from the 1970s, most famously by Mau Piailug of Satawal, who navigated a traditional canoe from Hawaii to Tahiti in 1976 without instruments to demonstrate that it could be done.
Arab and Indian Ocean navigators used the kamal, a rectangle of wood on a knotted string, held at arm's length to measure the altitude of a star — the knots setting the distance from the eye, so each knot corresponded to a latitude. It is about as simple as an instrument can be and it works.
What comes next
That is the end of the map. The next Part is about what people do on it for fun — the rules, the history and the numbers of the games the world watches, starting with the one that takes five days and can still end in a draw.