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Here the folding lines are not diagonals of the outer/inner quadrangles of the paper strips;
the cube is "(2|1)-wrapped", see:
http://www2.eng.cam.ac.uk/~sdg/preprint/weaving.pdf
Cube

platonic solid

8 vertices

12 edges

6 faces
6 squares
(6 sq.)

"(2|1)
wrapping"

6 strips

10 x 1/5 sq.
pattern download,

6 paper strips
2641 x 1413 pix,
312 kb,

w21.jpg

plaiting scheme:
a1a2a3a4a5
I↑↑i4V↑↓i3II↑↓i3IV↑↓i1V↑↑i5II
II↑↑i4VI↑↓i3III↑↓i3V↑↓i1VI↑↑i5III
III↑↑i4IV↑↓i3I↑↓i3VI↑↓i1IV↑↑i5I
IV↑↑i5V↑↓i2II↑↓i2V↑↑i4II↑↓i1I
V↑↑i5VI↑↓i2III↑↓i2VI↑↑i4III↑↓i1II
VI↑↑i5IV↑↓i2I↑↓i2IV↑↑i4I↑↓i1III
Cube

platonic solid

8 vertices

12 edges

6 faces
6 squares
(6 sq.)

"(2|2)
wrapping"

8 strips

12 x 1/8 Q.
pattern download,

8 paper strips
2787 x 1723 pix,
171 kb,

w22.gif

plaiting scheme:
↑↑a1a2a3a4a5a6
Ii3IVi2IIIi1IIi2VIIIi1VIIi4VI
IIi3VIIIi2VIIi1Ii2IVi1IIIi4V
IIIi5Vi4VIIIi3IIi6Ii1IVi2VI
IVi3VIi6Vi3IIIi2IIi5Ii6VII
Vi5IIIi6IVi1VIi4VIIi5VIIIi4II
VIi5VIIi6VIIIi1Vi4IIIi5IVi4I
VIIi5VIi4IVi3Ii6IIi1VIIIi2V
VIIIi3Vi6VIi3VIIi2Ii5IIi6III
plaiting page