Really, the business of Euclidean constructions involving apothems and polygons does not touch what's going on here. It is trivial and doesn't generalize. And t

Really, the business of Euclidean constructions involving apothems and polygons does not touch what's going on here. It is trivial and doesn't generalize. And there's no reason why a square with apothem r is more natural than a square with side r. See Tong's paper.