On Wrapping Spheres and Cubes with Rectangular Paper - Discrete and Computational Geometry and Graphs - page 31

On Wrapping Spheres and Cubes
with Rectangular Paper

B

Alex Cole 1(

) , Erik D. Demaine 1 , and Eli Fox-Epstein 2

1 MIT, Cambridge, MA, USA
[e-mail], [e-mail]
2 Brown University, Providence, RI, USA
[e-mail]

Abstract. What is the largest cube or sphere that a given rectangular
piece of paper can wrap? This natural problem, which has plagued gift-
wrappers everywhere, remains very much unsolved. Here we introduce
new upper and lower bounds and consolidate previous results. Though
these bounds rarely match, our results significantly reduce the gap.

1 Introduction

The problem of minimizing the amount of paper necessary to wrap a given
3-dimensional surface arises naturally from the economics of any factory packag-
ing physical items. We study two closely related cases of this problem: given an
x 1 /x unit-area rectangle of paper, what is the largest possible cube or sphere

it can wrap?
Informally, we consider wrappings that do not stretch, cut, or intersect the
paper with itself. We allow multiple layers of paper in the folding, and unlike [ 3 ],
do not differentiate between the front and back of the paper. As in [ 5 ], to formally
capture what it means to wrap a surface with non-zero curvature, we define a
wrapping (a.k.a. folding )tobea noncrossing, contractive mapping from a 2-
dimensional rectangle of paper to a subset of Euclidean 3-space. A contractive
function ensures no pair of points move apart under their image. This definition
captures the “crinkling” that you observe when, for example, you physically
wrap a billiard ball with a sheet of paper. The noncrossing condition is dicult
to formalize (see [ 7 ]); intuitively, it prohibits wrappings that cause surfaces to
strictly intersect.
We present a variety of novel techniques that improve upper and lower bounds
for wrapping both spheres and cubes. Figure 1 graphically summarizes these
results. A sphere wrapping (respectively, cube wrapping ) is a wrapping whose
image is a sphere (cube). We denote cubes of sidelength S as S -cubes and spheres
of radii R as R -spheres. Throughout, assume that 0 <x

×

≤ 1sothat x is the

smaller side of our x × 1 /x paper.

Eli Fox-Epstein: Supported in part by NSF Grant CCF-0964037.