On Wrapping Spheres and Cubes with Rectangular Paper - Discrete and Computational Geometry and Graphs - page 41

This bound is the first explicit sphere strip folding. It becomes optimally
ecient as x tends to 0. In addition, because of how it handles the ends of the
strip, it provides a powerful lower bound for larger values of x . When x =1we
recover the optimal lower bound from [ 5 ].

5 Relating Cubes and Spheres

Given contractive mappings f : A f constitutes a

valid contractive mapping from A to C . With this as inspiration, we present
mappings between spheres and cubes, allowing upper and lower bounds for one
to be translated to the other. Upper Bound 3 for inscribed stadiums translates
particularly well.

ₒ B and g : B ₒ C , g ⓦ

Theorem 2. S -cubes can be contractively mapped to (2 S/ˀ ) -spheres.

Proof. Let f be our contractive mapping. Consider the Voronoi regions on a
sphere induced by the six x -, y -, and z -extremal points. f will contractively map
each face of the cube into one of these regions.

S

Fig. 7. Onefaceofacubewiththeareausedby g drawn in.

It suces to examine one face F and the corresponding sixth of a sphere F .
RefertoFig. 7 . Let the center of F be (0 , 0) and the center of F be (0 , 0 ,R ). Let
g : F ₒ F be the map that sends a point with spherical coordinates x =( R, ʸ, ˆ )

to the polar point g ( x )=( Rˆ, ʸ ) on the paper.
Now we will show g is expansive by looking at an infinitesimal neighborhood
of an arbitrary x .Let x =( R, ʸ, ˆ )and

x =( R, ʸ + dʸ, ˆ + dˆ ). Now let dl 1 =

. These are known as line elements. It is well-
known that the sphere metric yields dl 1 =( R sin ˆdʸ ) 2 +( Rdˆ ) 2 .Doingthe
same about g ( x ) with the metric on the paper gives us dl 2 =( Rdˆ ) 2 +( Rˆdʸ ) 2 .
Because sin 2 ˆ

x − x and dl 2 = g ( x ) − g ( x )

ˆ 2 , dl 1 ≤ dl 2 . These distances can be integrated into arclengths

to show that all pairwise distances on the sphere are less than their images on
the paper. Thus g is expansive, so f = g − 1 is contractive. The image of g is just
a subset of F , but we can extend the domain of f to all of F by mapping the
unused region to the boundary of F .
The map f sends the line going through the centers of four faces of the cube
to an equator of the sphere without any contraction. If S is the sidelength of the
cube, then the resulting sphere will have radius R =2 S/ˀ . This also shows f is
optimal: no contractive mapping can produce larger spheres from a cube.

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