On Wrapping Spheres and Cubes with Rectangular Paper - Discrete and Computational Geometry and Graphs - page 32

R

0.28

0.27

0.26

0.25

0.24

0.23

0.22

x

0.2 0.4 0.6 0.8 1.0

Upper Bound 1 Upper Bound 2

Upper Bound 3 Upper Bound 4

Lower Bound 8 Cube Lower Bounds via Theorem 2

Tetrahedron Lower Bounds via Theorem 3

S

0.41

0.40

0.39

0.38

0.37

0.36

0.35

0.34

x

0.2 0.4 0.6 0.8 1.0

Upper Bound 1 Upper Bound 2

Upper Bound 3 via Theorem 2 Lower Bound 1

Lower Bound 2 Lower Bound 5

Lower Bound 6 Lower Bound 7

Fig. 1. A summary of the upper bounds (thick lines) and lower bounds (thin lines)
on sphere (top) and cube (bottom) foldings. The horizontal axes indicate the smaller
dimension of the unit-area paper. The vertical axes denote radii and side lengths,
respectively. The shaded regions indicate the gaps between the bounds. See Sect. 6 for
discussion.