Sometimes reasoning about balloon twisting patterns can have interesting non-balloon results. In this case, I was thinking about a rhombic triacontahedron--a ball that has a surface of 30 identical rhombi. Doodling 3D things on paper can be tricky so I normally flatten them to 2D by distorting the shapes but keeping the connections (unlike a net this is like a stereographic projection putting usually one face as the perimeter of the rest: here the outer curvy lines represent one rhombus). From a doodle in Notablity (I like this iPad app's grid), I created some interesting incidental patterns that may suggest cubes etc. Since I was in Kathmandu, where there are a lot of T-shirt embroidery folk, I got them to stitch this up for me for less than $10. I really like the colouring of the regions. I may give it some more thought but I'm happy for now.
The pattern that you can look for is the alternating of vertices with degree 5 and vertices with degree 3. Precisely, every degree 3 vertex (where three black lines meet) is surrounded by degree 5 vertices and vice versa.
Perhaps, I'll post the original pattern as well.
BTW, here is an example of a rhombic triacontahedron:
