Not all origami creations spring into existence in a perfect form or a with a perfect folding sequence. Mine are no exception.
The original Barber's Pole Box had two minor flaws.
1. The under side of the base looked slightly untidy because of the double edges which showed at some of the seams.
2. The lid was difficult to assemble.
Since I thrive on spatial problem solving I have spent several days working on ways to overcome these problems. The search produced several variations along the way, some of which survived.
THE BASE. The displayed crease pattern produces a base which is neater on the underside and removes some of the bulk of the sides by tucking them under the floor rather than triple folding them up the walls. The pattern of the floor changed slightly and became a little looser. My first attempt to rectify this problem was to halve the outer edges and tuck them under. Unfortunately this distastefully exposed the tucked under wall sections. Quartering the sections produced a more aesthetic result and that is what is diagrammed here.
THE LID. The following crease pattern produces a lid which looks the same on the outside as the original lid but is much easier to assemble. The pattern on the inside is the same but the depth levels are reversed: the outer edges are now over the center section instead of under it.
Unfortunately understanding how the floor of each unit is interconnected is not obvious from the Crease Pattern and not particularly intuitive. The floor and the wall ends are tucked around each other to form a secure unit without loose and floppy edges. You may come across the method on your own but you will probably have to wait for 3D diagramming or photographic examples to achieve enlightenment.
The CP includes a variation to the apex of the stripe which is now narrowed to a finer point. Different looks can be created by tucking the two halves under the adjacent section together or tucking one half differently from the other. Pictures explain this much better than words.
Narrowing the apex by folding in the other direction results in exposure of the underlying pattern at the center and can give rise to a number of interesting textures.
Still another alternative is to construct a lid using a modification of the pattern for the underside of the base section. Here is the crease pattern for that also.