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2.1 Upper Bounds
Two techniques generated the previously known upper bounds on wrappings of
spheres and cubes. The first is the surface area bound: the surface area of the
image of a contractive mapping cannot exceed the surface area of the paper.
Upper Bound 1 (Folklore).
A u
ni
t-area rectangle
o
f paper may wrap an
S
-
sphere or an
R
-cube only if
S
1
/
√
6
and
R
1
/
(2
√
ˀ
)
.
≤
≤
Catalano-Johnson and Loeb [
4
] observe that every point on the
S
-cube has an
antipodal point at least 2
S
away. Because wrappings are contractive, every point
(particularly the center) on the original paper must also have another point that
is 2
S
away, implying the paper's diagonal is at least 4
S
. Demaine et al. [
5
] apply
this argument to spheres.
Upper Bound 2 [
4
,
5
].
An
x
×
1
/x
rectangle of
paper ma
y wrap an
S
-sphere
√
x
2
+
x
−
2
/
4
and
R
√
x
2
+
x
−
2
/
(2
ˀ
)
.
or an
R
-cube only if
S
≤
≤
The surface area bound becomes tight as
x
approaches 0 and the antipodal points
bound is tight when
x
= 1. Between the endpoints, these bounds are likely far
from optimal.
2.2 Lower Bounds
Numerous lower bounds for particular rectangles, some with unclear origins,
exist in the form of physical foldings.
Lower Boun
d
1 ([
4
,
9
], Folklore).
1
/
√
7
×
√
7
paper wraps a
1
/
√
7
-cube, and
1
and
1
/
√
2
×
√
2
papers each wrap a
1
/
(2
√
2)
-cube.
1
×
Akiyama, Ooya, and Segawa [
3
] produce a series of six ecient “symmetric-skew”
wrappings which spiral the paper around the cube.
Lower Bound 2 [
3
].
x
1
/x
paper wraps an
S
-cube for each
(
x, S
)
pair:
1
24
,
37
×
,
9
,
36
,
15
,
17
,
264
120
75
,
17
,
23
,
1
92
,
45
,
36
.
120
Akiyama et al. [
3
] also invent a technique called
strip folding
, using extremely
long, narrow rectangles to come arbitrarily close to the surface area bound.
Lowe
r Bound 3 [
3
].
A strip of paper with
x
=1
/
√
24
n
2
+12
n
−
2
can wrap a
2
n/
√
24
n
2
+12
n
−
2
-cube for integers
n
≥
1
.
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