Doodling with a traditional birdbase, I noticed a way to open it up so there were 4 equal sized partitions that I could get to go flat:

Researching a little, it seems Jun Maekawa pioneered this, and also suggested a lovely corner locking mechanism that made the whole structure quite stable.

I can see this as actually useful – it is really efficient as a use of paper so you could feasibly make one that is the right size for nuts, lollies etc for parties.