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Proposition 2. An x
y stadium of flat paper mapped onto an R -sphere may
occupy no more surface area than
A ( x, y )=2 R ˀR
×
.
y
2 R + x sin
y
2 R
ˀR cos
Proof. To bound A ( x, y ) we will first establish A (0 ,y ) and then bound the deriv-
ative dA/dx . This will allow us to bound the areas of all mapped stadiums.
First, consider an 0
y stadium: a radius- y/ 2 disk. By definition, the disk
must fall within y/ 2 of its center on the sphere. A radius- y/ 2 spherical cap has
area 2 ˀR 2 (1
×
cos 2 R ), proving the claim for A (0 ,y ).
Now consider an x
y stadium S with major path P .Let f be a contractive
map to the sphere and A be the area of points within distance y/ 2of f ( P )on
the sphere. From Proposition 1 , it suces to bound A . Extend S by some length
dx (see Fig. 3 ). For suciently small dx , the extension of f ( P ) runs along a
geodesic. Call the added area dA . In Fig. 3 , this is the dotted region.
Now let ʸ = dx/R . This is the central angle corresponding to a geodesic of
length dx on the sphere. Extending P by dx will affect the latitudes within y/ 2
of our geodesic. Each latitude can be extended by at most an angle of ʸ .Let r a
be the radius of a circle of latitude at a spherical distance a from the equator.
It is well-known that r a = R cos R . This yields:
×
ʸr a da =
y/ 2
y/ 2
dx
R R cos
a
R da =2 R sin
y
2 R dx
dA
−y/ 2
−y/ 2
so dA/dx
2 R sin y/ (2 R ). For a stadium of length x , the area on the sphere is
bounded by A ( x, y ) as desired.
A(1 /x
x )
×
x stadium can be inscribed within any x
×
1 /x paper rectangle.
By Proposition 2 , this stadium only occupies A (1 /x
x, x ) area on the sphere.
The remaining paper only has an area of x 2
ˀx 2 / 4.
Upper Bound 3. x
×
1 /x paper can wrap an R -sphere only if
4 ˀR 2
x 2
ˀx 2 / 4+ A (1 /x
x, x ) .
3.2
n
Circumscribed Circles on Spheres
Cutting the paper or adding more material can only increase the ability of the
paper to wrap a sphere. With this as inspiration, we transform the paper into n
congruent disks, and then relate upper bounds on spherical cap coverings back
to rectangular sphere wrappings.
Proposition 3. If an R -sphere can be wrapped by an x
×
1 /x paper, then it can
also be wrapped by n congruent disks of diameter x 2 +( nx ) 2 .
Proof. Consider a arbitrary wrapping from an x
×
1 /x paper to an R -sphere.
Partition the flat paper into n small x
×
1 / ( nx ) r ectangles. C ircumscribe each
1 / ( nx ) rectangle to get n disks of diameter x 2 +( nx ) 2 . These disks can
contractively map into the original paper.
x
×
 
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