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Orbiting Components of the Athena's Helmet 

The plate's left shows the Cone's top circle, with the symmetry of its two original arcs limited by the green lines. The two arcs are combined. The  top-circle of the Cone shown in red is concentric with a larger cyan-colored circle. In our experiment, we move the arc along with its appendages, which floats in open space) to the cyan circle, and we let it orbit in the cyan circle's track towards the head. Subsequently, the Zigzag-arc snaps onto the hat.
For contrast, the zig-zag's color was changed to magenta in the area of the fit, which enhances the hat considerably, showing it in more detail.. My impression is, we're looking at a helmet with a flap near the back. This is another trick to be associated with the Cone's properties -  another episode of reassembly.