So being asked to think of volume that takes advantage of the least amount of materials and that counters for the most amount of space seems easy... Then you looking into what they are really are asking you to do and came to no other real conclusion, so we rummaged in the scrap bin hoping to find our answer there.
Laying down our wepons we began storming how we could relate these two dimentional object into a from that creates space. Taking into consideration the geater project, we looked at childrens pop up books and decided to mould a crumpled piece of paper in between two more ridgid pieces that would idealistically fold out to present the self expanding structure.