Johnson Polyhedron #26 (gyrobifastigium)
(2 compound tri angular prisms)
8 vertices
14 edges
8 faces 4 equilateral triangles (4 tr.) 4 squares (4 sq.) 
 vertex
2 paper strips
2 x 1/3 tr. 5 x 2/3 tr. 8 x 1/2 sq. 
pattern download
2 paper strips, 2653 x 1137 pix, 128 kb: str2.gif 
plaiting scheme:

a1  a2  a3  a4  a5  a6  a7  I 
↑↑ i5 I  ↓↑ i6 I  ↑↑ i7 I  ↓↑ i3 II 
↑↑ i1 II  ↑↑ i4 II  ↓↑ i2 II  II 
↑↑ i5 II  ↓↑ i6 II  ↑↑ i7 II 
↓↑ i3 I  ↑↑ i1 I  ↑↑ i4 I 
↓↑ i2 I 

modified JP26
12 vertices
22 edges
12 faces 8 trapezoids (6 tp.) 4 rectangular isoceles triangles (4 tr.) 
 plain
2 paper strips each divided into 2 parts
4 tr. 4 tp. 4 x 1/4 tp. 4 x 3/4 tp. 
pattern download
2 paper strips, 2784 x 1116 pix, 96 kb: wst2.gif red lines: mountain folding

plaiting scheme: see above.
The quadrangles of the strips for the solid JP26 above were replaced by squares ("wrapping").
Remark 1: if a1, a3, i5, and i7 are folded as mountain along the arrows, and the grey lines as valley, the result is
a polyhedron with 12 vertices, 30 edges and 20 faces (8 squares, 12 tr.), 

Remark 2: If a2 is folded as valley along the arrow and i6 as valley across the arrow, then the result is a solid, which seems to be konvex. 
