With shared walls between cells, why not just use squares? Less material, less calculation.
Hexagons resist crushing much better than squares.
Racking and shear forces are greater in squares. Further, the corners do not facilitate even evaporation of the nectar they way that the more rounded hexagonal cells do. Lastly, the queen gauges fertilzation of eggs by the size of the cell as it fits her distended abdoment. Also, corners are wasted space in the development of essentially cylindrical larvae.
Answered my own question.
From article: "...hexagonal cells require the least total length of wall, compared with triangles or squares of the same area. So it makes sense that bees would choose hexagons, since making wax costs them energy, and they will want to use up as little as possible—just as builders might want to save on the cost of bricks."
Squares use more material per volume stored.
Hexagons use the least material per volume stored in cells that have common walls. Triangles are even less efficient than squares.
Less stress by weight on top. Downward force gets distributed is my guess.
It seems to me that squares (rectangles) would require offset rows for strength (like bricks). Hexes are offset anyway and they're closer to cylinders.
I believe for the same volume of storage and material in the walls a hexagonal structure is stronger than one based on squares.
less calculation, yes. Less material, no.
With 19 walls, 10 ft. each, you can enclose an area of 1039.23 square feet in 4 hexagonal rooms.
Now, that same 1039.23 square feet in a square pattern is 32.24 feet on a side. To get 4 rooms you need 12 walls of 16.12 feet each. 12 x 16.12 = 193.42 feet of wall.
With hexagons you need only 190 feet of wall.
Squares would need more material. The compartment inside is a circle. And with the extra weight in the corners they'd need to reinforce everything a lot more.
I suspect that hexagons are naturally more stiff than squares - although I do not understand the math well enough to prove it.