Abby Warré readily admitted to a square being less ideal than a cylinder, (although much imprived over a rectangle,) but which was impractical to build. Now we have $129.95 mini table saws. Toolry makes a difference. Still, a square is simple and rugged as compared to an octagon.
To this end I fitted my square design with corner infills. As this robs volume, it was first increased by the addition of a 9th topbar and accomodating size increase. Eco-mindful of waste, a common 4x4 was split diagonally, and then again. This design used a basic square internal dimension of 12.5"/317.5mm. I use metric well but as a rebel, did americanize it with imoerial dimensioning. Topbar spacing was continuously variable and succeeded well, but has been changed.
Lumber is "2x" so thickness is 1.5". A standard Warré box will be 14 lbs and mine, 16. But now topbars will be evenly spaced at 1.375"/ 1 3/8" (the most common US spacing,) ~35mm to Warré's 36mm. Basic internal square will be 13" versus the present 12.5, (versus 12" which is close to Warré's 11.81"/300mm.) 18 liters? 5 gallons. 2 boxes make for the american "10 gallon hat," (which, FYI, never actually got past 3 gallons.) The 4x4 may only be split once to create the cirner infills, in this design upgrade, I think. The infills are rebate height so ALL 9 topbars match exactly!
A sister hive is a nuc, as it were. Same lumber. 10" versus 13" and only 7 topbars. This approaches Seeley & Morses' 1976 "typical tree cavity," where they saw, "up to 8 combs..." which to me says, usually 7 combs.
This smaller box is easier for small swarms to initially warm. In colder areas, it can drop into the 9bar-box if corner infills (held in by a single screw, each,) are removed, or set atop of the corner infills and topbars of an existing 9-bar [internal-only] octagon.
It must be said, it will be a very tight fit. Do your best work. Ideally, 1.5" lumber would be planed down 1/16" so there was more clearance. But 9-bar boxes can be unscrewed a bit to de-encapsulate a 7-bar box which could become stuck. Or, a dimension could be altered here or there.
Seeley and Morses' drawing shows a 6 or 7" thick tree-wall. Wood is about R1 per inch. Near San Francisco, my marine influenced climate does not get "cold," and bees have, untreated, continuously inhabited my 9-bar "octagon," for about 6 seasons.
Some term this, "bee-having," as opposed to "beekeeping." Ok.
I have been too busy to harvest, (secondary to me wanting bees in my area after "progress" displaced 4 nurseries and bees...) so they stayed at 3.5 boxes for 5 years. (A few beekeepers have noted that some colonies seem to like this volume and do not tend to swarm.)
The smaller 7-bar is the result of a desire to have a 2nd hive, and experiment with a "tree cavity sized" version, which ought to grow-down, more quickly, making for harvesting more often, and a hive within a hive, if as an option, it is desired.
I did have severe DWV the 2nd season. It was hard to watch. Superceedure occured. Almost no DWV since. A study showed that powdered sugar was useful as a census-tool but not useful as a treatment.
I use 8" foundation strips as guides. The central strips are 1-2" deep, decreasing to almost nothing near the sides. I plan to consider meat-skewer "side frames" about 4-5" long in 8" tall boxes in the 9-bar box size. Box height for the 7-bar design has not beed decided... various thoughts are in play.