Non-Euclidean Dungeons
"He had said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours."
-H. P. Lovecraft
A week or two ago I had a dream in which I was trying to draw a dungeon on the surface of a sphere. This got me thinking about the possibilities of laying out dungeons on spaces other than the standard Euclidean plane.
Non-Euclidean Space
'Non-Euclidean' geometry explores geometric systems that violate Euclid's fifth postulate:
That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less that two right angles.
or more succintly in Playfair's version:
Through a given point P not on a line L, there is one and only one line in the plane of P and L which does not meet L.
When this postulate is violated, seemingly parallel lines do weird things in the distance, either converging or diverging. Crucially, however, at small scales (say, that of a dungeon room or corridor) the deviations are hard to distinguish from Euclidian space.
In an old-school dungeon crawl, players are typically expected to draw their own maps based on the GM's description. Confounding their efforts with warped spaces in which, say, they arrive back at their starting location after only three 90-degree turns (or fail to do so after four) is cruel, but could be the right level of cruel for a one-off gimmick in an OSR game.
Spherical Geometry
In spherical geometry, 'lines' are 'great circles' circumscribing the spherical space, and so two straight lines will always converge at two locations. Moreover, it is possible to have a triangle with three right angles. Spherical space is finite, a dungeon drawn on a spherical surface has limited real-estate, and travel in one direction far enough leads back to the starting point.
Light in spherical space wraps back around to its source, so if a passage in the dungeon forms a great circle, the party would be able to see themselves in the distance. Permeating spherical spaces with fog might be advisable.
Dodecahedrons & Icosahedrons
Probably the easiest way to make a pseudo-spherical dungeon with traditional methods would be to draw it on the unfolded net of a dodecahedron or icosahedron, taking care to keep track of which faces meet each other.
This approach could be facilitated by making a set of pentagonal or triangular geomorphs, respectively.
Blender
The original dream that inspired this post involved insetting a dungeon onto the surface of a sphere in Blender by selecting individual faces to extrude. Attempting this in waking hours, I ran into a pretty significant issue in that a sphere in Blender is either a subdivided icosahedron ('icosphere') or made of quadrilateral faces generated by a set of parallel latitudinal circles and converging longitudinal circles ('UV sphere'). The latter makes it nearly impossible to create normal-looking rooms and passages near the 'poles', though it might work if they were ignored (though then you might was well inscribe the dungeon on a cylinder). The former has uniformly sized and shaped faces, but they're all triangles so the dungeon is limited to 60 and 120 degree angles. Given how niche this concept is that's probably fine.
I made a rough mockup of the icosphere version:
To run this, you'd need to use the UV unwrapping tools to project it onto a flat surface, or keep it open in blender to rotate as needed.
Toruses
One could also inscribe a dungeon map onto the surface of a torus. Toruses 'unwrap' somewhat more easily than a sphere; any dungeon written on a rectangular sheet of paper can be treated as a torus by having travel off the top edge wrap around to the bottom and left connecting to right in like fashion.
Hyperbolic geometry
In hyperbolic geometry, space is 'saddle-shaped' at each point, and triangles can be composed of all acute angles. I confess I don't have nearly as great an intuition for this one.
Papercraft with heptagonal geomorphs would work, though the result looks quite fragile. It is also possible to crochet hyperbolic surfaces, but according to my fiber-artist wife the colorwork necessary to encode a dungeon that way would be very difficult.