Below is a slight modification of this Flapping Small Rhombicosidodecahedron (version 1), modified to cause a collision when the flaps are completely unfolded. In this case, some of the triangles nip some of the squares.
This situation (overlapping/collision) is mentioned for the Snub Dodecahedron in Croft, Falconer, and Guy's Unsolved Problems in Geometry: "We can get an overlapping net if we cut perversely." A colliding Snub Dodecahedron is HERE.
But I have yet to see this situation mentioned anywhere for other Archimedean solids, such as the one here. Admittedly, the collision here is tiny.
This "cutting" is highly symmetric, far from "perverse". It is likely that this has been noticed before but simply discarded as sub-optimal (it precudes a paper cut-out, for instance). It would be fun to find the net with the greatest fractional area of overlap. (Greatest total overlap? Greatest single polygonal overlap? Greatest fraction of overlap of a single polygon?) Are there any such nets that triply overlap?
Grab it, turn it, spin it with the mouse. Double click the figure to stop the animation. You can hold the RIGHT mouse button down and drag it HORIZONTALLY across the frame to sequence through the frames.(Please be patient while the applet processes mucho compressed data.)
.