Spoiler alert, I cheated and made each clamshell segment shift -250 mm on the Z-axis (i.e. out/aftward) and +132 mm on the Y-axis before rotating +/-22.5° (the doors on my Enterprise are 150 mm thick), starting with the middle pair of doors of course. As each segment rotates in front of its neighbor, each paired stack shifts out/up -250/+132 mm before rotating another +/-22.5°. This repeats one more time with three segments stacked on top of each other.
I consider it a cheat because Richard Datin’s interior maquette built for “The Galileo Seven” clearly shows the doors firmly anchored to the deck and not shifting up as they retract. Another cheat is that I had to slightly shorten the vertical height of the “control booth” to keep the upper edges of the clamshell segments from scraping the walls (or outright punching into the booth!). But after testing a couple of alternatives which looked really weird (test #1: the clamshells descend into the floor before they rotate, starting with the outermost pair; test #2: the doors stay anchored to the floor but slide out aftwards and tilt at the bottom before rotating), at least this solution
- leaves the doors looking like they look on the 11 foot miniature when closed
- looks similar to the Richard Datin model from the inside (except for the middle pair of doors flying 396 mm above the deck by the third rotation)
- is mechanically possible (i.e., the segments don’t scrape/punch through walls or collide into each other). But holy hell, the weird cam-like connection between the top of each segment and the center axle underneath the control booth is just bizarre.
I feel for you @Professor Moriarty !

There are also these alternatives

https://movies.trekcore.com/gallery/albums/xihd/trekxihd0984.jpg
https://movies.trekcore.com/gallery/albums/xihd/trekxihd0081.jpg