Given 3 polygons which all are defined as flat (when seen from the side they have no points out of line with the rest, so essentially they are 2D shapes in 3D space), made of exactly 4 points, convex and with the additional information that each poly attaches to the other two at exactly two points each...
Starting from having these polygons "laying flat", that is: their values all lie on the x,y plane and all have a Z value of 0...
...how can I work out how to position these polygons in 3D space such that they are connected at their connecting points?
Label the polygons A, B, C. Select A such that B and C already each share an edge with it before folding, and consider it as remaining in the X-Y plane throughout the transformation.
Before folding, either there is exactly one point to which all three polygons attach, or there is not (in which case the resulting shape after folding contains a hole through it).
In the former case, consider that point (d). One of the edges attached to it will become shared by B and C after folding. Let e be the point on B that shares an edge with d but does not touch A. Similarly f for C. After folding, e and f are the same point. Consider the circle described by rotating e around the edge shared by A and B, and similarly for f about the edge shared by A and C. The circles intersect at exactly two points (one above and one below the X-Y plane). Write out the circle equations, solve, and arbitrarily select one of the two solutions. You now now the angles through which B and C have been rotated about the edges they share with A and the rest of the mesh is fully constrained.
In the latter case, find an edge of A such that one end attaches to B and the other to C. As before, consider the points on B and C that share edges with those points but not with A, rotate about the edges of A and solve for intersection.
Draw a diagram; it helps.