Even if the core and mantle of the planet were so impossibly rigid that they could hold their shape against gravity, erosion would strip the crust at the edges and wash the sediment into the oceans. This happens on Earth, too, and tectonic movements push the sediment piles up into mountains in a never-ending cycle, but the scale of vertical movement does not come close to what would be needed to turn the planet into a polyhedron.
To prove how absurd the Dodecian view is, consider what Dodec would be like if it had the same mass and volume as the Earth. Every edge would be 3,250 miles (5,200 km) long, and each face slightly larger than Asia. That is big!
For simplicity, assume that the north pole is located exactly at the northernmost vertex of Vigaelia. Three faces must meet at every vertex, so there would be an arctic region of three “polar” faces and a matching triplet of antarctic faces. The remaining six, “tropical,” faces would form a chain around the middle, pointing alternately north and south. Vigaelia is therefore polar, and Florengia tropical.
Because a plane presents a uniform angle to the sun, the faces would lack the north-south insolation gradient we take for granted on a spherical planet. Each face would show a radial change in climate, because the atmosphere would be much less dense near the edges. There would be edge ice caps.
The six tropical faces would be inclined at an average 10° to the “equator,” equivalent to the latitude of, say, Panama. The six polar faces would have a tilt of 52°, about the same as London, England. Since Dodec is known to have seasons, its axis of rotation must be tilted with respect to the ecliptic, but all we know about the amount of tilt is that it is less than 38°, for otherwise Vigaelia would have no daylight at midwinter. The changing tilt would cause seasonal effects, but the hours of daylight would remain the same year-round, so seasonal climatic change would be much less than we are used to. Seasonal variation would be negligible on the tropical faces-but note that midwinter on a tropical face would coincide with midsummer for its neighbor on either side.
We are rarely aware that the Earth is round. We do not see large bodies of water as dome-shaped, and yet if you look at a calm sea or large lake, standing with your feet at water level, your horizon is less than three miles (five kilometers) away. In other words, we perceive the globe as flat. I suspect that a flat world would seem to be concave. On Dodec the center of each face would be much closer to the center of the planet than the edges are, so water would tend to run to the middle and pile up. If you stood at an edge and looked across that enormous plane, you might well feel you were looking down into a bowl with a heap of ocean in the center. Dodec has no moon, because the mere thought of tides gives me the willies.
On polar faces, such as Vigaelia, the dominant climatic factor would be seasonal variation acting on the central ocean, which would warm up and cool down more slowly than land would. In summer, when the ocean was relatively cool, air would rise over land areas and sink in the center, giving an outward radial flow at the surface. In winter, when the ocean was relatively warmer, the flow would be reversed. Coriolis forces would make the prevailing winds blow clockwise in summer, counterclockwise in winter.
Not only would tropical faces, like Florengia, lack much seasonal variation, the small angle they would subtend to the planetary axis would make their Coriolis forces weaker, although not completely ineffectual. (Note that the “equator” would have no direct significance to those forces. On the Florengian Face, the southernmost vertices would be the farthest points from the axis and thus have the greatest rotational velocity. Fluids would move as if the entire face lay in the northern “hemisphere.”) Florengian weather, like Vigaelian, would be controlled by heat exchange between the hot, humid maritime center and the cold, dry uplands. Coastal areas would be jungles, racked by cyclonic storms, while the rims would be cold desert. The “fertile circle” in which Celebre is reported to stand must be a benign temperate zone in between. Such a zone could be relatively narrow but still enormous in actual area.
The weirdness goes on. Because the oceans would be domed, the sun would be able to set behind them, or reflect off them. It would be nice to think of the horizon flaming up bright green like an iceberg, but water on such a scale is not transparent. The domes would be opaque.
Air is not perfectly transparent either. From the window of a commercial jetliner in flight, the horizon is a blur, and yet it is only about 200 miles (300 km) distant, which is a trivial vista on Dodec. Rising hundreds or thousands of kilometers away from you, the sun would have to climb over the limit of transparency of the atmosphere, the “wall of the world.”
Where water goes, so goes air. The maximum topographical variation on Earth, from the top of Mount Everest to the bottom of the Mariana Trench, is 12.4 miles (20 km). By coincidence, the deviation of the terrestrial geoid from a perfect sphere is of similar magnitude-ice floes at the terrestrial north pole are 13 miles (21 km) closer to the center of the planet than a surfer at the equator. The mean radius of the Earth is 3,966 miles (6,378 km).
On a polyhedral Dodec the ocean bed at the midpoint of each Face would be 3,600 miles (5,800km) from the center. Sea level would depend on how much water the planet had, how much was tied up in ice caps, and how equitably it was distributed between the twelve faces, but if Dodec had about the same amount of water as the earth, the ocean would dome up roughly 50 miles (80 km) deep at the center and be 1,200 miles (1,900 km) across. Nardalborg Pass would then lie 600 miles (960 km) above sea level, and each vertex would be 350 miles (560 km) higher yet. Breathing would certainly be a problem.
Even if the planetary mass and volume were the same as the Earth’s, nowhere on the surface of Dodec would gravity be as strong. This conclusion is counterintuitive, but a sphere is the most stable shape simply because it maximizes gravitational force at its surface. Lesser surface gravity would produce a lesser density gradient in the atmosphere. Furthermore, those enormous vertex “mountains” would produce complex gravity fields of their own, dragging some air away from a purely spherical shape. Even so, holding your breath long enough to cross the Nardalborg Pass would not be advisable.
I hope such technical quibbles did not spoil your visit.