Their article is mainly about the maths, which looks very elegant.
It seems to be a bit confused about the bee aspects though, which is strange considering they do mention the use of bubbles to check what happens when surface tension, alone, determines shapes.
Not surprisingly, the bubble mixture 'builds' hexagons, and even 'builds' the bottom of the cells (where they interlock with the reversed layer of cells) in the most efficient shape, predicted by the maths.
This isn't anything being clever, or 'learning' to build things efficiently - it's just physics. The molecules pull towards each other and any configuration other than the 'least tension' one will tend to move molecules in the direction of the 'least tension' one until everything reaches a balanced position. The only trick is to see the bee's comb as a liquid, rather than a solid. If it were a liquid, like the soap bubbles, the shapes would 'make' themselves.
Bubbles don't sit around thinking/considering and neither (in this case) do the bees. In Tautz's book, he points out that if a bee makes each cell by constructing a tube, using itself as the template, the tubes will naturally stack in a hexagonal array - try it with loo-roll middles. If those wax cylinders are then heated up a bit, to a temperature where they can flow - just slowly, and not for very long - they will form beautiful flat sided hexagons. The same happens at the bottom of the cells.
The bee has (apparently) 850,000 neurons in its body, most of which are involved with its complex eyes - at least 3 colour pixels per eye segment. It's not thinking about maths, it's following a simple rule-set to do this building. This might work:
1) need more space;
2) go to the edge of the comb;
3) climb into the cell;
4) if I fit in, start building a new cell next to this one;
5) add to the walls into I fit in;
6) if a cell I inspect has round walls and it isn't close to the edge, heat it up for a bit;
I think they do some strengthening of the top surface of the cell to give better rigidity too, but the hexagonal perfection of the comb is brought about by simple rule following, heating and basic physics.
This is the really clever bit, in my opinion: how instinctive behaviour coded into the bee's DNA sequence can produce something so efficient and perfect.
FG