It might seem straightforward to pull out a map and measure the distance between two distant points, however, the larger the distance the bigger the distortion caused by traveling on the curved surface of the Earth as opposed to flat 2D space. So while the distance you measure to your neighbouring town won't be too bad, if you're measuring between London and Rio the curvature of the Earth will make a big difference to the distance that you'll travel. To help figure out the correct distance there's the haversine formula.

The haversine formula allows you to calculate the shortest distance between two points on a sphere using their latitudes and longitudes — this will be the arc between them on the great circle that includes both points. A great circle is a circle on a sphere with the same centre as the sphere, like the Equator. The haversine formula isn't perfect in practice, as Earth isn't a perfect sphere — see the 3 tallest mountains.

It was invented around two hundred years ago, together with tables to speed any calculations, to help sailors navigate.

The haversine formula

Jared Diamond’s classic book Guns, Germs and Steel, proposes some remarkable hypotheses about how the pattern of history played out, notably trying to answer the question of why it was generally the Europeans that invaded much of the rest of the world rather than the other way around. One hypothesis, striking in its simplicity if you’ve ever looked at a world map, is the significant contribution of the general orientation of the major continents.

The Americas are largely lined up north–south, while Europe and Asia (Eurasia) are largely lined up east-west. An east-west alignment means the land generally has a more similar climate, day-length and seasonal variations. The general logic of the hypothesis is that the similar latitude of an east–west alignment means animals, crops, as well as travel and trade, more easily spread across the continents than across those with a north–south alignment — of the Americas and Africa — which naturally encounter more different climatic zones.

As a simplified example, horses and other large animals and crops spread throughout the shared temperate latitudes of Eurasia. In contrast, South America’s only large domesticated animal, the thick-coated llama (and related species like the alpaca), couldn’t easily spread to North America through the impassable tropical barrier of the isthmus of Panama. They just aren’t built for it.

There are a lot of viewpoints on the theory, and Diamond includes many more factors of geography and others — like never being able to domesticate the zebra — that go towards answering his question of why Spain invaded the Americas and not vice-versa. But ever since reading his simple hypothesis, available to anyone with a world map, I’ve found it hugely intriguing.

The continental axis hypothesis

This is the fascinating observation that it’s not straightforward to say how long a coastline is. If you were to measure the coastline of a country by using a ruler on a globe, you would come out with a vastly different number than if you were to pace around the edge. The closer you look, the more wiggles and squiggliness you come across and instead of converging on a more accurate length, the coastline just keeps getting longer. The smaller your ruler, the longer it gets.

This was originally spotted, incredibly, in the 1950s, by an Englishman, Lewis Richardson, when trying to check a theory he had that the likelihood of war between countries depended on the length of their shared borders. Remarkably, he found that the quoted lengths of borders varied significantly. While measuring on maps at different scales he saw that the smaller scale map he used, or the smaller the width of his calipers he was measuring with, the length systematically increased. When looking at coastlines, instead of borders, some countries had wigglier coasts and so the length increased at a faster rate with the scale — for instance, Norway’s coastline, with it’s crinkly fjords, increases faster than Britain’s, which in turn increases faster than South Africa’s, as he zoomed in. The rate of this increase later became known as its fractal dimension.

Long after Richardson’s research, Benoit Mandelbrot published a paper How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension that discussed how the wiggliness of something like a coastline at one scale can be repeated at smaller and smaller scales. The work led to the later term fractals. Many other things exhibit fractal-like behaviour such as river networks, borders, brains, frequencies, lightning or even the stock market.

More mindbending mapping, and the difference between Great Britain, the UK and the British Isles.

There’s a super section on this in Scale, by Geoffrey West.

The coastline paradox

All maps involve decisions. Whenever you compress real life onto something smaller and less detailed you have to choose what to keep in and what to leave out. And in the case of printing a map of the Earth, you have to figure out how to get something that is curved onto something that is flat, and this involves trade-offs and decisions as well. 3 dimensional space just doesn’t flatten to 2 dimensional paper without choices.

In 1569 Gerardus Mercator created a map that brilliantly solved a pressing problem — that of being able to follow a straight line while sailing and it correspond to a straight line on the map. His map also did a good job of preserving the shape of countries. But to do this you have to stretch the areas at the top and bottom of the Earth. Because most of the land on Earth is in the Northern hemisphere, and because that land is generally further north than the land in the southern hemisphere is south, it has the effect of enlarging Northern countries such as Europe, the US and Canada, Scandinavia and Russia as compared to countries closer to the equator and in the Southern hemisphere.

Most world maps you see are still similar to Mercator’s projection. In fact Google Maps desktop used it until just recently. It could well be that this distorts our worldview by emphasizing Northern countries in size and therefore importance. When I see a more size accurate Gall-Peters projection I have to say it’s hard to argue.

The Mercator projection

From the prefix iso meaning equal and khronos meaning time it literally means equal times. And whereas maps are usually put together with lines of equal distance you can either construct maps, or overlay maps as here, with lines of equal time. The overlay view gives you time contours rather than, say, altitude contours. Handy for checking things like commuting distance. I like how you find little islands where there is fast transport in and out like trains or planes.

You can explore time maps of London at mapumental.

Isochrones

This is Robert Plutchik’s wheel of emotions. I like it because it tries to make sense out of the range of emotions we have by relating his 8 primary emotions to related ones where two primary emotions meet. For example, remorse is like feeling both disgust and sadness, awe is surprise and fear. At opposite sides of the wheel are opposite emotions eg Joy >< Sadness. Other versions of his framework go 3D and map the intensity of emotions as well, for example, rage > anger > annoyance, or ecstasy > joy > serenity. Neat.

Mapping emotions.

This is generally the source of much confusion. Even amongst those who live over here. If you look carefully at the description of the United Kingdom, you’ll see it’s the United Kingdom of Great Britain and Northern Ireland. And the British Isles is the group of islands as a whole, including all of Ireland too.

The source I liked best for this explaining this and much more was Project Britain.

For much further confusion, think for a moment why there’s a UK passport, Great Britain in the Olympic games, and in football, we compete as England, Wales, Scotland and N. Ireland separately. 🤔

You might be interested in the Union Jack: The making of a flag

Sketchplanations tagged with maps