This map is not upside down. In fact, it retains a northern bias in a way I’ll soon mention. Down is obviously towards Earth’s centre around these parts, so an upside down map would be something like one of our core, or perhaps a mirrored projection of the surface. North is not at the top except on many maps. And as I said, this map retains a northern, or perhaps I should say Atlantic, bias, because it still puts Greenwich in the middle. To make it even fairer it needs to be rolled round a bit. This would be an example of a Pacific-centred map, with north at the top:
It’s surprisingly hard to find an example of a map which puts both south at the top and Australasia in the middle. One thing which I think this Pacific-centred map does is emphasise the impression that in terms of the human population, the Americas are in a sense the Far East, not the West. Humans spread out of Afrika, across Eurasia and then across Beringia (the land bridge between Siberia and Alaska) into North America, followed by South America, meaning that Native Americans are genetically relatively close to East Asians. It might even go further than that, because for example the abacus and paper existed in Mesoamerica in similar forms to the Chinese versions and there are cultural similarities around the Asian and American Pacific, such as the use of a two-pronged fishing spear and the cultivation of sweet potatoes on Pacific islands before European contact.
It has also been suggested that the Solutrean people of Europe colonised North America during the last Ice Age by walking across the ice, but genetic and linguistic issues rule this out. Although it’s been a respectable hypothesis in itself, it tends to be adopted by White supermacists, which is ironic because although the Solutreans were Western European they were also dark-skinned.
Putting Europe at the top influences how we think of the planet. In particular, people tend to talk about “Sub-Saharan Africa”. Afrika south of the Sahara is of course not literally under the desert and “sub” carries with it notions of inferiority, so I would prefer to avoid this. A similar phenomenon exists in the use of “cis” and “trans” to refer to different parts of the world with respect to locations in Western Europe, so for example we talk about Transylvania and Cisalpine Gaul, or Transnistria, when in fact for the people born in these places they are cis. Getting back to the North-South divide, this actually works to the advantage of this country because it means we can talk about “coming up to see” someone in York from the South of England, for instance, and it means there’s a sense of superiority about Scotland and the North of England which is absent from the South of England, although Southerners already feel superior. In Kent, we used to talk about people being from “under London”, meaning they were “furreners” from Sussex or Surrey, or perhaps just from the other side of the Medway. Kent, incidentally, holds the distinction of being the home of Greenwich, the origin of the Prime Meridian, except that in administrative terms that location became part of Greater London quite some time ago:
This means, of course, that Kent is almost entirely in the Eastern Hemisphere, a distinction it shares with Essex, Cambridge, Norfolk and Suffolk. Since the division in Kent is between East and West according to which side of the Medway one is on, the creation of Greater London means that West Kent is now sadly reduced in terms of area and population. However, this division is more a reflection of our imperialist past than anything “real”, as is our tendency to put North at the top.
What kind of planet is this “upside down” world? Well, it’s bottom-heavy in terms of land. There’s a large uninhabited wasteland of a continent at the top and most of that side is ocean, or rather, there’s less land. I remember hearing that the Southern Hemisphere has 20% of the land and 80% of the population, but whereas I think the former is so, I don’t think the latter is because there’s Antarctica and the sparsely-populated Australia to be taken into consideration. Even so, most of the land is definitely at the bottom in this map. This contrasts with Pangæa, which was mainly in the South. Things haven’t always been this way. This clustering has led to an almost landlocked ocean in the north, which is therefore less saline due to rivers discharging fresh water into it, and consequently more frozen over than it would otherwise be. The Sun doesn’t care that we think of Earth as this way up, and carries on doing its thing with El Niño and the rest, which is not tucked away down there mainly on the southern side of the Equator where it can safely be ignored. I wouldn’t want to overemphasise it, but it certainly seems to contribute to our blasé attitude that we’re able to think of that, for example, as happening far away in a place we don’t really need to think about.
El Niño, of course, is also an oceanic phenomenon, and that’s another bias we have in these maps: they’re very much focussed on land rather than the water. This makes a limited amount of sense given that humans tend to live on land, although it does also mean that Pacific nations tend to be ignored. Here is a way of looking at this planet which puts Aotearoa/New Zealand at the centre:
This still isn’t quite the same as centring on the oceans, because for example the Arctic on this map is relegated to the top left hand corner, but it does show quite effectively how much of the planet is covered in water. In fact it even comes close to showing the hemispheres of land and water, which used to be believed in very firmly in mediæval times and whose literal existence is instrumental in the formation of the British Empire.
I started this post with the Gall-Peters Projection, which was popular in the last couple of decades of the twentieth century as a kind of counterpoint to the imperialist-seeming Mercator Projection. However, it actually dates from the mid-nineteenth century, when it was invented by the Scots cleric James Gall in 1855, meaning that there’s been something of an intellectual property dispute. The idea of it being a fair representation of the sizes of land in comparison to Mercator is kind of a marketing ploy, because although it may do this, it’s been overhyped as a projection which shows the true relative sizes of the land on different parts of the globe. There was a time when other map projections looked distorted to me compared to Gall-Peters, and Mercator is still like that. It looks very top-heavy and the fact that it could be infinitely tall is quite jarring and absurd.
Arno Peters himself was not a cartographer, and professional cartographers have treated the projection he promoted with considerable disdain. Now the question here is how to balance the opinions of experts and the hostility towards the other. If an outsider to a profession devises something new and influential, it might well meet with a poor reception from within that profession, but on the other hand there’s the Dunning-Kruger Effect, that the more one knows about a subject, the more one realises there is to know and that one does not know. Not being a cartographer, all I can really do is report what real cartographers felt about his projection. Peters claimed that the projection preserved compass direction and distance when in fact it clearly doesn’t. Mercator’s projection actually does preserve compass direction, which made it very practically useful on long sea voyages when the European empires were being founded, and this does carry a lot of negative baggage with it in a similar way to institutions, streets and statues commemorating slave traders might, and it would be fair to reject Mercator simply for that reason – it’s a tainted projection in a way. However, it’s very obvious to anyone who can be bothered to look that a horizontal line on the Equator of that map of the same length as a horizontal line passing through Patagonia is not going to be the same distance, and that a 45° diagonal drawn between the Gulf Of Guinea and Buenos Aires might be close to a NW-SE compass bearing but the same angle between Sydney and the South Island of Aotearoa New Zealand is definitely not SW-NE.
From my experience in pressure groups of the 1980s, when this map became popular, there was a particular style of rhetoric which was quite unhelpful to the causes they tried to promote. I’m not expert in rhetoric by any means, but two aspects of it are λογος and παθος. The former uses fact to persuade and the latter attempts to elicit an emotional response. Particularly in the ’80s there was a tendency for groups such as Greenpeace (not the real group but the one which stole the name and is well-known) and CND to use fairly dubious factoids to present their case. One which comes to mind is a popular counterpoint to nuclear power focussing on the half-life of the waste as ” deadly” for however long that half-life is, which doesn’t make sense. It’s a big number used for rhetorical purposes, and yes it’s persistent in the environment and very harmful, but the half-life is not a measure of how dangerous it is. This is not an attempt to defend, or for that matter attack, nuclear power, but it seems unwise to make such claims when they only have a thin veneer of scientific respectability. It seems to me that the Gall-Peters Projection is of that ilk. It isn’t actually all that marvellous and is easily proven not to live up to its claims just by looking with a dispassionate eye. I don’t know to what extent this has changed since, but there often seems to be an inappropriate use of science, or in this case maths, following from an appropriate emotional response. For this reason I tend to argue from an emotive rather than a logical point of view, since this, not rationality, is the true root of our beliefs.
It’s said to be impossible to map a sphere onto a plane without distortion. This is technically untrue, although distance can’t be preserved in the case I’m thinking of. Imagine the globe divided into a large number of strips like a Chinese lantern and flattened out. There is no distortion along the midline of any of the strips by definition, but the distortion tends to increase towards the edges of all of the strips. However, the strips are also linked to each other along a great circle, along with there is again no distortion, and the strips taper towards their ends, reducing the proportionate length. There would still be distortion, but it would be smaller the narrower the strips become and the more of them there are. There is a limiting case with infinitely many infinitely thin strips which would be absolutely faithful. Another way of doing this is to draw a spiral beginning at one point on the surface, widening out to a great circle and diminishing again at the antipodes of the start, and again, the narrower this spiral becomes the less distortion there is until finally, and infinitely long straight line can be formed along which all the distances are exactly in proportion. Both of these maps are of course useless, and in fact even the projections which are not at the limits are not very useful because they constantly interrupt the surface of the globe. They might be useful for plotting out routes which are mainly linear, if a custom map for each route was generated anew.
Things get considerably easier if you abandon the aim of preserving direction and long distances. This can be done by projecting the globe onto a convex polyhedron in various ways. The role-playing system GURPS has used maps based on regular icosahedra, since it’s easier to work out distances in this way for dice throws. That produces a map which looks like this:
Other polyhedra are available, and if they consist of mixtures of faces they can approach the sphere more closely, but of course my personal favourite Platonic solid is the regular dodecahedron, whose net and map look like this:
It isn’t entirely clear which of these is closest to a sphere as it depends on what’s meant by that. Inscribed in a sphere, the dodecahedron occupies more of the sphere’s volume than a regular icosahedron at 66.49% to the icosahedral 60.55%. However, in relation to a sphere inscribed within an icosahedron that polyhedron is the most spherical as it occupies 89.635% of its volume. The midsphere of an icosahedron, which is the sphere touching each edge at exactly one point, is only 1.66% larger than the icosahedron itself, and the midsphere is important for map projections because it shares out distortions for areas whose diameter is smaller than the polyhedron and those whose are larger, thereby reducing the distortion. However, I always feel strongly drawn to the dodecahedron so that’s what I’m going to cover now.
I am not afraid of maths. That said, neither am I particularly good at it. I enjoy it but it isn’t my strongest point, and my opinion isn’t based on bad teaching, as it so often seems to be. I mention it here because I’m about to present a number of equations and do a bit of my own maths to address the question of a secant-based dodecahedral projection. This is probably best approached visually. The Mercator projection is what you’d get if you had a luminous globe wrapped in an infinitely long cylinder. You get this:
Actually that’s only part of what you get. What you actually get is an infinitely tall map, meaning that another very minor drawback of the Mercator is that you can never get the whole of the globe onto it. Equations germane to this projection include:
This is for angles expressed in degrees, hence the values 180 and 45. I’m not entirely enlightened as to the meaning of these equations, but the tan in the second must indicate that that’s the vertical component of the projection, as does the y, because tangent functions go to infinity. λ is longitude and φ latitude. The Gall-Peters projection uses these:
The difference between these two makes me wonder if I’ve got the first one right. A simpler version of these is x=Rλ and y=2R sin φ, which presumably means that the sine function squishes the latitudes and ensures they never get to infinity. I’m guessing here, but I imagine this is like putting a globe inside a cylinder and using horizontal knitting needles perpendicular to its axis to work out where to draw the map features, although if that’s true it surprises me that Afrika, for example, gets so spaghettified, so maybe there’s something else going on.
I would imagine that that attempt to set out the maths behind these has failed to inspire confidence. Nonetheless I will continue. Dodecahedra have the interesting and unique feature that an infinite number of lines can be drawn from any vertex back to itself without crossing any other vertex. This means that something approaching a great circle can be drawn in a large variety of directions. If the projection is from the midsphere, there will be twelve circles along whose circumference there will be no distortion of distance or direction. Assuming Earth to be a sphere with a diameter of exactly 40 000 kilometres, each pentagonal face would cover an area of 42 441 318.16 km2. The total distance covered by the circumference of the circles where there is no distortion is something like 251 327 kilometres with spurious accuracy. They also extend to five edges on each face and are in contact with five other circles on adjacent faces, meaning that there is also no distortion at the five midpoints of the edges. Maximum distortion is at the vertices and the centres of each face, a total of thirty-two points. Nonetheless, aligning this projection such that the poles are at the centres of two opposite faces would place them in what seemed to be the correct places.
I wish I had the νους to follow this through to the end. The main point here is that this comes very close to being an undistorted flat map of the world, and in thinking about it another issue has been raised in my mind. Earth is in fact not a perfect sphere, and therefore a perfectly spherical globe is not an accurate representation of its surface either. The equator is slightly elliptical and it’s 12 756 kilometres in diameter at the equator (on average, presumably) and 12 717 kilometres in diameter at the poles. Each hemisphere is also slightly off from even that difference, meaning that the South pole is slightly closer to the centre than the North. There are also some other irregularities. Mapping this shape onto a sphere will distort it very slightly, and as far as I know globes, although they aren’t perfect spheres, are not designed to be geoid in shape (Earth-shaped). Another question is how much map projections onto flat surfaces distort the shape not because they’re flat, but because they assume Earth is perfectly spherical.
I feel that to some extent I’ve let you down because my mathematical abilities as they stand, along with the time available to me, makes it implausible to do the calculations to work out exactly how distorted this map is. Also, I’ve concentrated on a dodecahedral projection (of which there could be several which distribute the distortion in various ways) when an icosahedral one is more promising. Nonetheless I hope I’ve made it clear that sometimes maps are adopted for apparently rational motives when in fact they’re more to do with rhetoric, and my preference for the dodecahedron is also like that. So I’m just like everybody else really.
A Box Of Chocolates 