# 'Trapping' 3D regions with sheets of paper

Given a square sheet of paper, how does one create a bag (a closed surface) with it such that the 3D region contained within this closed surface has maximum volume (operations allowed include wrinkling and sewing/gluing at the edges but not stretching or tearing)?

The closed surface need not be smooth. This question might be related to the 'paper bag (or teabag) problem'. Further, by varying the shape of the paper sheet (to say a circular or elliptical sheet), one has a range of questions.

Apart from solving each specific case, are there 'global' properties? For instance one could ask if these claims hold:

1. "given any convex sheet, to produce a bag that can hold max volume, all sewing/gluing operations are done necessarily at the edges."
2. "for any convex sheet, the bag formed with it holding maximum volume cannot be smooth when filled to capacity"
• – Josiah Park Nov 28 '19 at 8:12

A section of Geometric Folding Algorithms: Linkages, Origami, Polyhedra reports on an exploration1 of all the convex polyhedra that can be folded from a unit square (by "gluing" the perimeter to itself without overlap or gaps). The polyhedra fall into six interconnected continua. We found that the largest volume was achieved by this folding to an irregular octahedron:

Volume $$\approx 0.056$$.
Joe Malkevitch posed the max-volume question (for squares and convex polyhedra). Unfortunately we gained no geometric insight into why that folding achieved the max volume (and our calculations were not exact).

The max volume polyhedron lies on ring $$A$$ near 12 o'clock.

1 By myself and two students, Rebecca Alexander and Heather Dyson.

• MattF.'s pyramid is on ring $B$ near 3 o'clock. – Joseph O'Rourke Nov 30 '19 at 13:25

There is some recent fascinating relevant work in the physics-of-materials community:

Paulsen, Joseph D., Vincent Démery, Christian D. Santangelo, Thomas P. Russell, Benny Davidovitch, and Narayanan Menon. "Optimal wrapping of liquid droplets with ultrathin sheets." Nature materials 14, no. 12 (2015): 1206. Journal link.

"ultrathin sheets automatically achieve optimally efficient shapes that maximize the enclosed volume of liquid for a fixed area of sheet"

Concerning the OP's mention that "operations allowed include wrinkling," Paulsen et al. say,

"our experiments and simulations, as well as previous work, indicate that the shape is wrinkled but has no folds."

• Interesting... Does this work assume the presence of a gravitational field? – Nandakumar R Dec 3 '19 at 8:30
• @NandakumarR: Their experiments are carried out under a gravitational field, but I don't think gravity plays a role. – Joseph O'Rourke Dec 3 '19 at 10:43

As an upper bound, a sphere with surface area $$1$$ has radius $$(4\pi)^{-1/2}$$ and volume $$(4\pi/3)(4\pi)^{-3/2}=(4\pi)^{-1/2}/3=.094$$

As a lower bound, we can fold the unit square along the solid lines, into a pyramid with vertices at $$(0,0,0),(1,\frac12,0),(\frac12,1,0),(\frac23,\frac23,\frac13)$$. This has base $$3/8$$ and height $$1/3$$, so volume $$1/24=.042$$