On Wrapping Spheres and Cubes with Rectangular Paper - Discrete and Computational Geometry and Graphs - page 37

Inital Wrapping Gluing

Partition

Fig. 4. The transformations used in Lower Bounds 5 (top) and 6 (middle and bottom).

Wrapping f can be transformed into a wrapping from a different aspect ratio
rectangle as follows. Partition an x × 1 /x rect angle into some set of pieces P .

Notice that one can still wrap a 1 / 6+2 √ 2-cube with P by applying f to
each part.
To create a new rectangular wrapping, we will “glue” P back together by
identifying edges of elements of P such that they form an x × 1 /x rectangle. To

ensure this gluing produces a valid wrapping, identified points must map to the
same point under f . Then f applied t o each p iece of P results in a contractive
mapping f from x × 1 /x to the 1 / 6+2 √ 2-cube.

1 /x rectangle,
Montucla's dissection from [ 10 ] suces. This is visualized in step 2 of Fig. 4 .
To glue the parts back together:

1 /x rectangle and glue it into an x × To partition an x ×

1. Identify the original left edge and the original right edge.
2. Identify the left half of the original top edge and the right half.
3. Identify the two small parts of the bottom edge with the large part.

Figure 4 illustrates this transformation. The first and second identifications
correspond directly to special properties 1 and 2. The last one is valid because
the entire bottom edge is mapped to a single point.
Varying the angle of the diagonal cut in the dissection creates rectangles of
any aspect ratio.

The eciency of Lower Bound 5 can be increased by starting with a different
wrapping. This causes the cuts to become more constrained, resulting in a dis-
crete set of wrappings. This technique is similar to the tetrahedral wrappings
in [ 2 ].