What is the most structurally supportive 3D Tessellation? The sort of thing you'd use to build chambers for an underground mansion

Question

Hexagons are considered the strongest grid pattern, I believe, and that's why bees use them? They're an efficient use of space, with maximal structural integrity. I was wondering what a 3-dimensional equivalent would be, for a shape that fits together into a tessellation that's an efficient use of 3D space while having the best structural support.

The sort of application I was thinking of was if you were digging chambers (rooms) for an underground mansion. For this application, you probably want a lot of smallish rooms of a practical shape that'll hold up large amounts of dirt and rock above you.

I was thinking something like a truncated octahedron might work best?

Answer 1

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.