On Wrapping Spheres and Cubes with Rectangular Paper - Discrete and Computational Geometry and Graphs - page 34

Demaine et al. [ 6 ] revisit strip folding, showing that any polyhedron can be
wrapped by strip folding.
Sphere wrappings are less extensively studied than cube wrappings.

1 and 1 / √ 2 × √ 2 rectangles wrap a 1 / ( ˀ √ 2) -sphere.

Lower Bound 4 [ 5 ]. 1 ×

Demaine et al. [ 5 ] also apply strip folding to spheres but do not provide an
explicit construction.

3 Upper Bounds

The following two techniques create new upper bounds on sphere wrapping and
provide a substantial improvement over the previous upper bounds, as illustrated
in Fig. 1 . Our general approach is to reduce the problem of bounding rectangular
wrappings to simpler shapes like circles and stadiums.

3.1 Inscribed Stadiums on Spheres

As x approaches 1, Upper Bound 1 (surface area) becomes less effective, as it
fails to account for necessary “crumpling” of the paper. This technique improves
upon this by observing that a particular shape inscribed within paper must waste
a certain amount of its surface area when mapped onto a sphere.
An x

y stadium is the Minkowski sum of a length- x line segment (called
the major path ) with a diameter- y disk. Refer to Fig. 2 .

×

Proposition 1. Given an x y stadium S with major path P mapped onto a

sphere by some contractive function f ,let X be the points on the sphere within
surface distance y/ 2 from f ( P ) .Then f ( S )

×

ↂ X .

Proof. On the flat paper, all of the points in S are within y/ 2 of the major path
P . Because f is contractive, all of these distances can only decrease when S is
mapped onto the sphere.

y

dx

x

Fig. 2. x × y

stadium, dashed major dx

Fig. 3. Extension of a stadium by .

path.