Source image
Source image is a disk shaped anamorphic one shot 180° image which was presumably computer generated as a projection of a virtual 3d environment on a perfect semi sphere. After making sure it is centered and the file is perfectly square..
Filter
Photoshop -> Filters -> Distorsion -> Polar Coordinates - polar to rectangular filter is applied.
Compression
The image was compressed to fit only the northern hemisphere
Gores
Flaming Pears Flexify 2 Filter applied with Mercator as a source and 12 gores as a target. The ratio ofthis image should be stretched to be 1:2 because it is horizontally the perimeter and vertically half the perimeter. I've been confused by discrepancies about the definition of a Mercator projection (compare the illustrations on the english against the german Wikipedia articles about it).
Flower
The 12 resulting gores in a Flower Petal pattern (each segment is turned 30°)
Conclusion
The Flexify 2 filter introduces a hole in the middle and wraps the image around that. This isn't the desired function. Flexify 2 offers to add an overlap for the glue but only solid colors are available to choose, better (less visible) would be if the pattern would continue. Flexify offers other input projections like mirror ball or one shot which one would think could allow direct image procession without the polar to rectangular conversion, but those settings give inexpected results. Stitching the gores to the Flower Petal is time consuming and always slightly inaccurate, thus not the optimal workflow.