Your top concern is strength in the vertical direction, correct? Are you sure you aren't tunnel visioned on exotic tessellations? Because I think the actual shape that meets your requirements is rather boring....you are just talking about an underground skyscraper which is a stack of prisms.
In a skyscraper, all the vertical force is taken up by a vertical steel beams (or rather multiple vertical steel beams riveted or welded to each other in practice) that run the length of the skyscraper from top to bottom. This is the strongest ideal configuration for that load. Then to deal with non-idealities you cross brace in an X-shape between vertical beams on the faces to prevent buckling.
If you are concerned with lateral rigidity as your second priority, then you can use a triangular floor plan for maximum lateral strength and rigidity against sideways forces.
Or if you need to balance lateral strength against usable floor space then use a hexagonal or octagonal floor plan. Or just go with square or rectangular floor plans. Honestly it doesn't matter what 2D tessellating polygon you use for your floor plan because if your walls are a vertical planar trusses it will accommodate them all. You just have a bunch of triangular, hexagon, octagonal, or square skyscrapers adjacent to each other.
EDIT: Except for the ceiling of the topmost floor...that should be arched, domed, or elliptical. Like @Fred mentioned. That way the weight of the ground in the middle of the ceiling gets efficiently transmitted to the vertical truss walls.