About the 3D chess they play on Star Trek They play a type of 3D chess on Star Trek ( the 60's series) not on a 8x8x8 3D board but on some smaller variation. Yet the game as they have it is supposed to be complicated. Are there real rules and a 3D board set-up for this?
Andrew Bartmess was the first Star Trek fan to reverse-engineer the game of tridimensional chess from the TV series (which never explained the rules). He sells his version of the rules for $9.95: Tri-D Chess Rules.

In the most popular version of Tri-D chess (which Andrew calls the “The Federation Standard 5.0” rules), the game is played on seven boards, three of which are the “main” boards and four of which are “repositionable attack boards”:



Players can reposition one of the four attack boards by repinning them to the static boards at different points, provided that one of their pieces occupies that board (and no other piece occupies it). This is instead of moving a piece in that turn. The boards are, in effect, treated as mobile “attack craft”.

The notation system uses a z-index from 1-7 to indicate which board the piece is moving to:



Unfortunately there’s no free version of Bartmess’ rules available online.

Jens Meder maintains an alternate but free online version of Tournament rules for Three-Dimensional Chess which are inspired by Star Trek and tied more closely to standard FIDE chess rules than Bartmess' version, albeit with the same wonderful board stand and complex notation: