Before we continue with the 3D model and choosing a placement, I think it’s time to take a closer look at the Hollow World Set world maps. This treatment is going to get long and complicated, so I’m going to split it up and present it over the next few days.
The Hollow World Set came with world maps for both worlds, inner and outer, in a pseudo-Robinson projection. I’ve already mentioned the problems with the Outer World map, but it’s worth going over them again now.
Hollow World Set Outer World map
The Hollow World Set Outer World map
The world maps in this set were full of detail, in terms of both place names and overall terrain, making them invaluable sources for expanding the hex maps.
The main problem with this map is that it stretched the Master Set map’s coastlines out in order to fill up the pseudo-Robinson projection grid. In so doing, it broke the connection with the hex maps, which of course remained unstretched (since they were based on the Master Set map). This was all done by hand, of course, and it turns out they did a really excellent job of it, as we’ll see in a moment.
But before we can compare these maps, there’s something we need to do: we need to return the map to a standard latitude/longitude grid, aka an Equirectangular or Plate Carrée projection. (This seems like the most likely projection to assume for the Master Set map and the hex maps, because it’s just a regular grid, with all the cardinal directions where they should be.) I actually did this some years ago.
I used a GIS program called Manifold to do this. First, I added extra latitude and longitude lines to the map, so that it had a line for every 10º. Then, I painstakingly added a control point to every single intersection:
Hollow World Set Outer World map with control points
703 points in total — it’s not a terribly fun job… But when it’s done, you can then georegister the image and change the projection, pulling the pixels into new patterns based on the control points.
Here’s the result of changing it to Equirectangular projection:
Hollow World Set Outer World map in Equirectangular projection
This is the projection needed to drape the map on a 3D sphere. It must be in a 2:1 ratio, or it will not cover the whole sphere. That’s essentially the problem the designers were facing when they decided to stretch the coastlines. (And in Mystara’s case, the polar openings mean that a 2:0.9 or 2:0.8 ratio is needed to cover the regular spheroid/ellipsoid areas of the planet’s surfaces.)
Now compare this with the full extent of hex maps we’ve assembled so far, and the stretching should be quite obvious:
Lining Up Mystara hex maps, 20th February 2016 Progress Snapshot
Here’s what happens when we overlay this:
Outer World hex crawl overlaid on Hollow World Set map
Note the stretching of the hexes east-west, or the north-south squashing if you prefer. The hex maps would need to be changed quite drastically to fit.
Also note, however, that the fit is actually pretty good. Let’s squash the map and overlay it with Master Set coastlines to see just how good a fit it is:
Master Set coastlines (in red) with Hollow World Set map squashed to fit
I’d say that’s pretty amazing for a reprojection done completely by hand. Once again, awesome respect for the original artists who did the cartography on these maps!
Now let’s do some analysis. I’ll try and keep it short and sweet.
The Outer World latitudes were squashed to fit the projection template, but the land itself is mostly compatible with the Master Set version.
The regions from 60-90º especially were shown smaller than the rest of the map. This was likely done to represent the approach to the polar openings, but the precise configuration remains vague. Notably, the land shapes are not different from the Master Set — they haven’t been squashed or stretched.
The regions that were cut off past 90º were placed in the polar openings on the polar opening maps in the Hollow World Set.
However, the latitudes on this map will not fit with our model, as shown in my exploration of Placements 1-3. They are also quite radical, placing Alpha at about 63ºN, which I think most will agree is too far north.
The longitudes can all be safely ignored too, due to squashing.
We can cut this map up in this squashed state and use it as a source to expand the hex maps.
First, thank you very much for the wave of feedback since my posts last weekend. I think I have finally caught up with all the messages — apologies if yours slipped through the net. I hope this article answers all of your comments and questions.
I’m going to examine three more placements, all based on your feedback.
Placement 4
Both Paul Dupuis and Michele Carpita advocated a tunnel-shaped polar opening, to solve the problems with Mystara’s size. Here’s what I’ve come up with. First, the latitudes are the same as placement 2:
Placement 4 full world view; the red line shows 76º latitude, where the polar lip begins
This of course means that the size of the world is the same 31,752 miles as placement 2, too, with 12.25 hexes to every 10º:
Placement 4 close-up showing 10º of latitude
But the polar lip now starts a full 10º higher, at 76º. The Hollow World remains largely the same, with the polar lip starting at 66º, although the curve of course is different.
Placement 4 cross-section
I used the official figure of 774 miles for the width of the polar openings at their narrowest point. Of course they could be a lot smaller if we wanted them to be, but I think the terrain fits pretty well like this anyway; the land beyond the red lines on the map above will easily fit onto the curve before the tunnel begins.
This seems like the best we can do using the Princess Ark/Hollow World Set equator. But what about that other one — the equator shown on Poor Wizard’s Almanac II‘s map?
Placements 5 & 6
Sheldon Morris proposed that we use the PWA2 equator. Back in 2012, he used it and 72 mile per hex maps to calculate an Outer World circumference of 25,920 miles. That is small enough to solve our problem with polar openings, even using the gradual curve model, I believe. This circumference comes with the bonus of 1º being equal to 72 miles.
However, Sheldon’s calculations were based largely on 72 mile per hex maps, and as we have already seen in this project, those unfortunately are not to be trusted. Our model now is primarily based on 8 and 24 mile per hex maps, with the 72 mile per hex maps filling in just a few gaps with Davania, Skothar and Bellissaria. The main difference is that the 8 and 24 mile per hex maps include more space between the various areas.
Personally, I don’t think the neat mathematics of 1º = 72 miles is as important as getting a good fit with the latitudes, so I’m going to present my own take on this, too.
Placement 5
My take on the PWA2 equator is that it could perhaps work best using the official figure for Farend at 60ºN. Take a look:
Placement 5 full world view; the red line shows 66º latitude, where the polar lip begins
Applied to the gradual curve model, this will mean that Frosthaven sits at the top of the world, near or on the point of absolute north. Farend is at 60ºN, Sundsvall 45ºN, Landfall 44ºN, Thyatis 31ºN, and the eastern Thanegioth Archipelago around 18ºN. These numbers are all rather close to the official figures.
The main difference with this model, unsurprisingly, is that a large chunk of Davania will be moved further into the polar opening. It’s actually less of a chunk than it may appear to be, because (due to the shape of the world) southern Davania is nowhere near as large as it looks. But it’s still quite a lot. And of course Davania itself will be smaller, since it straddles the equator less.
It’s still huge, nonetheless.
I reckon that Davania’s southern tip will find its way far into the polar opening.
It’s worth mentioning that a tunnel model could still be used with this, but if a gradual curve model is an option at all, I’d definitely prefer to go with it.
Let’s do some counting:
Placement 5 close-up showing 10º of latitude
This is 10.5 hexes precisely. 10.5 x 72 = 756. 756 / 10 x 360 = 27,216 miles circumference.
Placement 5 cross-section
We need to see some mock-ups of this on the 3D model, but I think this is the best option so far. Mystara is quite a bit bigger than Earth, but not nearly as much as it was with the other equator. The official latitudes are largely in place. The only potential problem is with how much of Davania and indeed northern Brun and Skothar may fall into the polar openings.
Placement 6
Last for your consideration today is Sheldon’s placement, albeit adapted to the Atlas of Mystara‘s hex map composite.
Placement 6 full world view; the red line shows 66º latitude, where the polar lip begins
As you can see, it’s not all that different from placement 5 — just a bit more compact. The main difference is that reducing the length of 1º to 72 miles makes the world a bit smaller. Count the hexes, ten to each 10º:
Placement 6 close-up showing 10º of latitude
Everything is a bit further north: Farend is at 63ºN, Sundsvall 47.5ºN, Landfall 46.5ºN, Thyatis 32ºN, and the eastern Thanegioth Archipelago around 20ºN. Frosthaven looks as if it may be past absolute north, on the inward slope of the polar opening. The gap between the sides of the opening will be around 250 miles.
I prefer placement 5 to this one. The latitudes worked out better, with everything a little further south, and slightly less land in the polar openings. I don’t think it’s worth shrinking things just to make the maths rounder — especially since I believe that 72 miles per hex is an inappropriate scale. Hex maps work better at larger scales such as 8, or 24 at most. Smaller scales should be topographical maps, not hex maps.
In any case, you can all make up your minds for yourselves. I await your feedback. 🙂
Thanks for the feedback from yesterday’s posts. Today I’m going to concentrate on applying the different placements of Davania and the resulting latitudes to the 3D model.
Size Problems
Unfortunately, it turns out that this is not as easy as I would have liked. Why? Because of the polar openings.
The size of the Hollow World is reasonably fixed. Official sources give a circumference of 11,908 miles, and the full extent of official maps fits quite neatly into this space. The only question for me is how much land to fold into the polar openings; recently I have been thinking that the majority of land shown on the Hollow World poster map should not fold into the openings. Official maps seem to support this.
The problem is, if there is a large difference in size between the inner and outer worlds, the polar openings get smaller and smaller, and eventually disappear. This can be fixed by lowering the latitude of the beginning of the polar lip, but that of course means folding even more land into the openings.
The problem lies in the amount of space the hex mapped areas take up on the Outer World.
Let’s take a look at the placements I proposed yesterday, and see how they do.
Placement 1
Count the hexes between the two horizontal graticule lines here:
Placement 1 close-up showing 10º of latitude
I count roughly 12.75 hexes from line to line; this represents 10º of latitude. At 72 miles per hex, we can do a simple calculation to find out how much land 1º of latitude on the Outer World takes up: 12.75 x 72 / 10. 12.75 x 72 = 918. 918 / 10 = 91.8, so that’s 91.8 miles per 1 degree of latitude.
In comparison, earth has roughly 69 miles per degree. So this Mystara is going to be significantly larger than earth. How large? Simply multiply by 360 to get the circumference of the spheroid: 33,048 miles, compared to earth’s 24,860 miles.
Some of you may at this point think, “Wait a minute, isn’t earth an ellipsoid, not a sphere?” Yes, it is, and the circumference I just quoted is the polar circumference (since we’re measuring latitudes). The equatorial circumference is about 24,902 miles. So yes, there’s a difference of around 42 miles — a pretty minor difference, and one that’s not hugely relevant to this discussion. I will likely define Mystara as an ellipsoid later in this project, when I create a coordinate system, complete with its own reference ellipsoid. But note that this minor equatorial increase in circumference is not really enough to help us solve the problems at hand, so let’s ignore it for now.
Getting back to Placement 1, what would this look like on our model, with the polar lip starting at 66º in both the Hollow and Outer worlds?
Placement 1 cross-section
Oh dear. So there would be no polar openings at all.
In order to fix this, we need to do one or all of the following:
Decrease the size of the Outer World.
Increase the size of the Hollow World.
Move back the lips of the polar openings in one or both worlds.
Placement 2
Happily, we have already done this with Placement 2: by rescaling the latitudes, I shrunk the dimensions of the Outer World. But is it enough? Let’s count the hexes and see.
Placement 2 close-up showing 10º of latitude
I count roughly 12.25 hexes. 12.25 x 72 = 882. 882 / 10 x 360 = 31, 752 miles circumference. Radius 5,053 miles, diameter 10,107 miles.
It still seems rather big, but let’s mock it up anyway, just to be sure:
Placement 2 cross-section
Still no good.
Talk about hitting a brick wall!
Options
I don’t like option 3 (bringing back the polar lips to lower latitudes), which leaves us with just the first two options: further shrink the Outer World, and enlarge the Hollow World. Maybe we can try and make things meet halfway.
We can probably squeeze the Outer World a bit more. These circumference numbers in the thirty thousands were unexpected, because previous fan estimates have been in the twenty thousands, with a Mystara just a little bit bigger than Earth. The true size of the world doesn’t really matter, as long as everything fits, so we can play with this further.
It actually also deals with an issue I had, which is that at 30ºN, the Sea of Dread coast with Karameikos and Thyatis seems awfully tropical. However, shifting latitudes will also result in northern Alphatia and northern Norwold being at a much higher latitude. Perhaps this fits, too, though.
But let’s start with the easier part of this next attempt.
The Hollow World
Back in 2012, there was a great discussion at The Piazza about World Dimensions. We didn’t come to any firm conclusions, due to all of the issues I’ve been working through on this project. But there was a lot of discussion of the Hollow World, which we can draw from here.
Here’s the composite I came up with to measure the extents of the Hollow World’s landmasses:
Hollow World Set Iciria hex map overlaid on the same set’s world map, with hexes marked off in fives from the equator for counting
A couple of interesting things to note: the latitudes near the equator are larger north-south than the extremes near the poles; and the map shows 90ºN and S, despite the lack of an actual pole.
Regardless of the marked latitudes, it seems clear from the hex map that all of this terrain is supposed to be on a constant north-south scale. In other words, the closer together lines from 60º outwards can be safely ignored. I’m not at all worried about having to reassign latitudes, either, as it may be a necessary evil in order to get the world working right.
A little hex counting will reveal 62.5 hexes (at 40 miles per hex) from the equator to “90ºN” and 62 from the equator to “60ºS”. The north shows more water, so let’s ignore that and call it 124 hexes. 124 x 40 = 4,960 miles. Or we could think of it as 125 hexes for a nice even 5,000 miles.
At this point in the 2012 map I went on to extrapolate the circumference, assuming these figures stretched from imaginary pole to pole; the “extra” lands could then be folded into the actual polar openings. (Given the above figures, this would be 9,920 or 10,000 miles.)
But that’s not what I’m thinking this time. First, the compressed latitudes are a big clue that these lands are not supposed to extend to the imaginary poles, and therefore are not supposed to fall into the polar openings — at least not very much. Further, my reproduction of the Hollow World map, which “fixed” these squashed latitudes, turning them into regular latitudes, was probably a mistake. I now believe that these regions should be left unstretched.
The biggest point is that I think these lands should largely fit into the unwarped space between the polar lips, i.e. between 66ºN and 66ºS — or at least a smaller area.
We can calculate an ideal area by extrapolating the circumference in the same way we did with the Outer World: by counting hexes and finding out the number of miles per degree. If we do this for the equatorial area, I think we will reach a larger figure than we did before.
I count roughly 8 hexes, so: 8 x 40 = 320. 320 / 10 * 360 = 11,520 miles. (This is also what Hugin came up with in the 2012 thread.)
Unfortunately, this is not helping: the official measurement was 11,908 miles, and our estimate has actually ended up smaller. If that’s the case, we’d be better to go with the official number, because making the Hollow World smaller at this point is simply not an option.
Darn.
Placement 3
So we’re left with shrinking the Outer World. I’m beginning to feel increasingly desperate… Are we going to be able to make this work at all? Keeping Davania in the same Champions of Mystara-based location, I’ve squashed the latitudes as much as I dare to.
Placement 3 full world view; the red line shows 66º latitude, where the polar lip begins
Lands past the red line start to fold into the polar openings, but until around 70º it’s not really obvious that this is starting to happen. Nevertheless, Frosthaven will be squarely within the opening.
Ambur, Ar and Frisland now fall between 60 and 63ºN or so, and Alpha is at about 60ºN too. Farend looks to be almost at 65ºN. Take a close look at the map to see all the other latitudes.
So how does this measure up? Time for some more hex counting:
Placement 3 close-up showing 10º of latitude
I count about 10.75 hexes. 10.75 x 72 = 774. 774 / 10 x 360 = 27,864 miles circumference. Better, but is it going to work?
Placement 3 cross-section
We do at least have polar openings this time, but I don’t know if I’d call this working.
And yet I can’t see shrinking the Outer World any more than this — we’re already going into dangerous territory by shrinking it this much.
What other options could there be?
Should we move the equator north to the PWA2 location? I somehow doubt that would allow us to shrink the world much more, if at all, because southern Davania can’t all be shunted into the polar openings.
But what else is there to do?
Should we enlarge the Hollow World? We may be able to squeeze another thousand miles or two in there, in order to open up the polar openings a bit. That may be the best bet — and I can stretch the northern and southern areas to compensate a little, or rather leave them as they are on my reprojected Hollow World map.
