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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?

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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.

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  • $\begingroup$ Thanks! Aren't some prisms much better than others, though? Cubes seem to be the weakest, structurally, even if they do offer a lot of space? Overall, elliptical ceilings seem to be the strongest, but I wondered if anything offered good strength while also fitting together more efficiently. $\endgroup$
    – user38849
    Aug 7, 2022 at 8:52
  • $\begingroup$ @user38849 If the prism is vertical, then the vertical strength doesn't really change with floor plan since you're just pushing down on the vertical walls of the prism. The lateral strength will change though in the sense a triangular floor plan will be much stronger than a square floor plan if you apply sideways forces. $\endgroup$
    – DKNguyen
    Aug 7, 2022 at 13:05
  • $\begingroup$ Remember, tessellating depends the OUTSIDE of the prism. Arched CEILINGS are necessarily on the INSIDE of the prism. They can extend to the outside but if you want the floor above to be flat then that means you have a very thick dead space between the arched ceiling and the flat floor above. The purpose of the arched floor is to transmit vertical forces in the middle of the floor to the walls and there are more space efficient methods to use. The most obvious just making your floor plan smaller so the vertical beams are close so there is less sag. Or horizontal trusses like warehouse ceiling. $\endgroup$
    – DKNguyen
    Aug 7, 2022 at 13:17

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