[tiitqghi] tesseract rooms

connect 8 cubical rooms in tesseract topology, for a 3D game.  when using a door, you see at most two rooms at a time, so you never directly see the weird non-Euclidean geometry/topology (S^3, 3-sphere) of the hypersurface of the tesseract, namely 4 cubes connected like a tetrahedron around a vertex (vertex figure).  previously.

assuming you can't fly, provide an elevator in each room to travel to vertically adjacent rooms.

you can indirectly experience the weird geometry if you travel in certain loops: your orientation becomes off, probably rotated by 90 degrees (fun with parallel transport and the curvature tensor).  also, a loop of only 3 rooms can return you back to where you started, in contrast to needing at least 4 in flat Euclidean space (and loop lengths are always even numbers in flat space).  game encourages mastering this weirdness.

gravity stays consistent with the previous room.  one can get gravity to change orientation by traveling in a loop; let this be the key game mechanic.  you can't get gravity to change orientation traveling only horizontally in a loop through doors; you must use an elevator at least once.  (I think this is true, reasoning by analogy about the square faces of a 3D cube.)

8 rooms = 48 internal faces (wall, floors, ceilings).  although 3D game is the obvious implemention, 2D overhead view is also possible.  you only see the floor (face) that you are on.  leave things at the edge of a room to transfer from a floor to a wall.  (of course, for a 2D game, e.g., Zelda 1 dungeon, one could get similar weirdness with 6 2D rooms with cube topology.)

divide each cube orthogonally into 8 sub-cubes.  the tesseract then has 64 total rooms (but with adjacency very different from the 64 squares of a chess board).  each sub-cube has one special corner surrounded by weird geometry (4 cubes meet at a point).  I think geometry is locally flat (8 cubes meet at a point) around all the other corners.  64 rooms = 384 faces, counting both sides of faces.

most generally, a collection of polygons connected as a graph.  graph edges between pairs of polygon edges (signifying doors), and each polygon also has "up" and "down" edges (signifying elevator) connecting to other polygons.  perhaps bundle polygons into an abstract polyhedron to allow transfer of objects at an edge.