On the maximum volume solid wrappable by a given sheet of paper

Abstract: We consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid wrappable by any given sheet: the maximum is always achieved (or approached) by a non-convex body. In other words, for any convex solid wrappable by a given sheet, there exists a non-convex solid of strictly greater volume that the same sheet can wrap. We discuss related work, a key subquestion involving the sphere, and several further directions.

• The paper shows that the maximal volume wrappable by a given sheet is attained by a non-convex solid rather than any convex body.
• It introduces a rigorous isometric embedding framework that permits folding and pleating to maximize enclosed volume.
• The work analyzes singularities, multilayered wraps, and open problems to extend geometric and origami optimization theories.

Maximum Volume of Solids Wrappable by a Planar Sheet: Non-Convex Optimality

Problem Statement and Formalization

This work addresses the geometric optimization problem of determining the maximal volume of a three-dimensional solid $B \subset \mathbb{R}^3$ that can be "wrapped" by an arbitrary given connected planar sheet $D \subset \mathbb{R}^2$ of positive area, using isometric embeddings (i.e., permitting only folding and wrinkling, not stretching or tearing). The notion of "wrapping" is formalized rigorously: any path from the boundary $\partial B$ to infinity must intersect the image of the deformed sheet. In particular, the mapping must be 1-Lipschitz, and the sheet's boundary must glue to itself to fully enclose the volume.

The paper distinguishes between "neat" wraps—where every point of the paper's image is in contact with precisely one point on $\partial B$ (so the surface areas coincide)—and general wraps, which may involve folding, crumpling, or increased local sheet overlap, making the resulting surface area of $B$ strictly less than that of the original sheet.

Main Conjecture

A central conjecture is proposed:

For any connected, non-degenerate planar region $D$, the supremum of the volumes of all three-dimensional solids $B$ wrappable by $D$ is never achieved by a convex body. That is, for any convex $B$ that can be wrapped, there exists a non-convex $B'$ (also wrappable by $D \subset \mathbb{R}^2$0) with strictly larger volume.

This conjecture challenges intuitions from isometric embedding theory and connects to the behavior of crumpling and the geometry of non-convex bodies. The shift from the "neat" (convex) case to the "crumpled" (potentially highly non-convex) case is not encompassed by existing results.

Connections to Existing Literature

Prior work by Bleecker and Pak has addressed isometric deformations and inflations of polyhedral surfaces, predominantly within the neat wrapping regime. In these formulations, the maximal volume is provided by convex bodies, as any non-convexity cannot be created without stretching, if the mapping is isometric everywhere. The crumpled, non-neat scenario considered in this paper, however, lies outside the scope of these results, relying crucially on the possibility of self-contact, pleating, and metric singularities.

Karasev showed that for the reverse problem—wrapping a given convex body with a disk—the minimal disk has radius equal to the intrinsic diameter of the body's surface, by virtue of Alexandrov's curvature bounds and the exponential map. This gives no bound in the forward direction of maximizing volume given a fixed sheet, especially outside the convex class.

Spherical Case and Open Subquestions

As a critical test case, the author examines wrapping a unit sphere $D \subset \mathbb{R}^2$1 with a disk of radius $D \subset \mathbb{R}^2$2. The exponential map generates a disk covering $D \subset \mathbb{R}^2$3 with area ratio $D \subset \mathbb{R}^2$4, implying significant excess paper beyond the surface area of the sphere. The central open question is whether this surplus allows wrappability of some non-convex body with volume exceeding that of the sphere ($D \subset \mathbb{R}^2$5).

Solving this instance would constitute explicit evidence for the main conjecture in the case of disks, but the problem is currently unresolved. The challenge lies in determining whether the available excess area enables feasible geometric folding and closure to increase the internal volume while maintaining the wrappability constraints.

Singularities, Multilayered Wrappings, and Attainment

The discussion elaborates on several advanced geometrical and topological concerns arising from the maximal volume question:

• Singularities: In the non-neat setting, surfaces of optimal solids may exhibit high-order singularities and local self-contact. Classification and analysis of these singularities are open problems that require techniques from geometric measure theory and the calculus of variations.
• Multilayered wrapping: The idea of $D \subset \mathbb{R}^2$6-layered wraps, where the sheet must cross every path from infinity to the boundary at least $D \subset \mathbb{R}^2$7 times, provides a natural extension. Stronger constraints may simplify the problem structure and yield more tractable subclasses.
• Attainment of the supremum: Whether the maximal volume is realized by an actual solid or only approached by a sequence (i.e., the supremum is not attained) remains open. The latter scenario would itself corroborate the non-optimality of convex solids.

Implications and Future Directions

This paper raises new questions at the intersection of geometric topology, convex analysis, and mathematical origami. The conjecture, if resolved affirmatively, would have implications for understanding the structure of optimal isometric embeddings from two to three dimensions under folding. It potentially informs engineering applications involving material utilization, packaging, and morphable surfaces. The mathematical techniques required to advance in this direction likely involve analytic and combinatorial understanding of crumpled surfaces, singularity theory, and possibly computational geometry if constructive methods are discovered.

On the theoretical front, resolution of the singular case for disks or other canonical domains could yield new principles for the volumetric efficiency of origami and crumpling processes, with applications to physical modeling and structural mechanics. Future research may focus on explicit construction techniques for non-convex bodies exceeding the sphere's volume in the disk wrapping case, or in general domains via variational and numerical approaches.

Conclusion

The study presents a significant conjecture on the maximal volume of solids wrappable by a given planar sheet, asserting non-convexity as a necessity for optimality. Through careful problem formulation and connection to known results, the paper situates the problem in a broader mathematical context. The explicit open questions regarding singularity formation, attainment, and test cases offer a pathway for future research in geometric analysis and related applied fields.