Mdthbs

It's very much a matter of what information they want to display in their primary 3d to 2d projection.

The Mercator projection that everyone is so familiar with is simply designed so that lines of latitude are straight horizontal. This makes ocean navigation easier as the important factor was knowing how many degrees north or south you needed to be.

The Mercator projection (/mərˈkeɪtər/) is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment.

Any minor details like distances or areas of countries can be neglected, that's not what this particular map projection is for. The reason north is at the top and the prime meridian goes through Britain is because the British cartographers made the best maps and they chose such a layout. There's also McArthur's corrective Mercator projection that puts Australia top centre and it looks really strange to the rest of us.

If you go far enough back in history, you'll find maps with Jerusalem at the centre, but they're not particularly useful.

What you have to do is define what it is that your map is supposed to show accurately and what can be sacrificed in turn.

Say that the primary empire of the early industrial period was landlocked, now ocean navigation isn't a defining factor but rather land distances from the capital. Now you have a map that has the imperial capital in the centre and uses something like Azimuthal equidistant projection which preserves the distances from the centre.

The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians (lines of longitude) as straight, with distances from the pole represented correctly.

This map isn't great for travelling to or from anywhere other than the centre point, but as they say, all roads lead away from Ankh-Morpork*. Perhaps regional cities would have their own local map with the city in the centre allowing local navigation.

*Sometimes people go along them the wrong way.

The aliens know some mathematics that we don't.

In the question, there is mention of a particular rectangular projection, best known as the Mercator projection. However, there are many other ways we Earth people have figured out for projecting a sphere onto something flat, including conic, conformal, gnomonic. There is a decent article at Wikipedia.

Each of these projection the techniques requires an understanding of some type of geometry, trigonometry, or related areas of mathematics, and there could easily be other possible projections we have yet to discover, but the aliens have known for a long time. Their maps won't be radically different as a result, but perhaps just different enough to handwaved it into your story.

We project our world on a bi-dimensional surface because we learned to make flat surfaces well before making spheres.

And don't forget that you can easily cut a flat surface in smaller pieces, while cutting down a sphere is more difficult. In other words, if you don't need the whole world but just the 1000 km around the Republic of San Marino, it is more easy to do it on a flat surface than on a sphere.

Moreover, flat surfaces can be more easily carried around and stored. A map can be rolled up, a ball will stay a ball.

Change those factors above and you will have a different outcome with respect to map making.

If you're dealing with cartological evolution, the answer is "pretty much nothing."

I suspect all species would develop maps in the following way:

1. They sit down with a stick and some rocks, draw the local river, put some rocks in to mark the villages, then in their own language intone the words, "grog, you take bugga and clobber dis place. Me take Wherk and pound over der."




2. A little while later, something akin to paper/vellum/papyrus is invented, and the conversation goes something like, "Hermicules, thou taketh thy holy troops and stand atop the Hilltop of Crahakia the Elder and then shalt thou count to three, no more, no less. Three shall be the number thou shalt count, and the number of the counting shall be three. Four shalt thou not count, nor either count thou two, excepting that thou then proceed to three. Five is right out! Once the number three, being the third number, be reached, then thou shalt assault the city in the name of Daiphus and bring salvation to the wicked!"¹




3. A little later, someone discovers a rock, or a pin, or something on a leaf, that happens to point in a useful direction in a predictable way. Such a person might suggest that direction to be "Gorp!" Thereafter, the "top" of the page upon which the battle plan holy consecration is thus inscribed is known as "Gorp!"²




4. And sometime after that subways are invented and some genius figures out that actually knowing the geography involved with cartography isn't actually relevant or practical when it comes to dropping down to the corner to buy a box of smokes.

My point is, cartography inevitably starts with a 2-D rendering of the local geography and becomes more complex as more data needs to be represented, but there are only so many ways simple geography (the local river, hill, and village) can be represented unless your aliens don't have eyes.

An advanced species may have a reason for not using what Humans use today, but how they came to the threshold of that decision is, IMHO, highly unlikely to be any different than Humanity experienced.

Remember, knowledge is a pyramid, ever growing in size, and your aliens are at any given moment standing atop their pyramid. However, in the beginning, that pyramid is very small, and that means very little variation compared to any other sapient species.

IMHO.

_{¹ My undying thanks to Monty Python for one of the most enduring and funniest moments of dialog in history.}

_{² It's worth noting that step #3 might come before step #2. It's a bit wishy-washy in the middle.}

As others pointed out, there are some well-explored ways to project a sphere onto a map sheet, including ones to preserve angles (e.g. Mercator) or to perserve areas (e.g. Gall-Peters). Each has advantages and drawbacks for various purposes, and there is mathematical proof that there is projection to perserve angles, distances, and areas at the same time.

• Have your aliens use an azimuthal equidistant projection centered on a holy city or capital. Then have some adventure set on the opposite end of the world, near the point which becomes the outer rim of the map. (The usual maps have two such points, but they are at the poles where few people live.)


• Have them use a Peirce projection.

But in both cases you would have a flat map and one would have to know the continents to recognize how they are distorted.

The geography of the alien planet can be very different from ours. Imagine a water world with just a single continent centered on one of the poles, like our Antarctica, with no other continents, not even islands. In this case, they wouldn't just need to map the other water-only hemisphere, and their maps will be of round shape with the pole in the center.

Or the planet can have the shape not near-spheric like the Earth's, but of irregular shape, for example it had a heavy collision with another object that deformed it. In this case, the projection of the planet's shape on a flat surface can take rather weird shapes, the ones that cannot fit effectively into a rectangle, or the planet can have even not two, but three or more "hemispheres", so where we have the map of two round hemispheres, their map will have four triangular ones.

We can go on imagining different geographies and different planet shapes with their respective maps. I'd love to see a map of a planet which has tunnels going through the core as wide as to allow the aliens to live not on the outer surface of the planet, but on the inner surfaces of the tunnels, too.