Yes, game designers regularly take advantage of physically impossible models. Look at this map of the level "5-D Space" from Bungie's Marathon.[1] Like Doom, the map is made of 2D sectors, but without the restriction that sectors can't overlap. Bungie took advantage of this to create an impossible space. The crossing solid lines are part of different rooms that can't be seen from the other. Only sectors connected with the light colored two-sided lines are visible. So you could turn left four times but be in a different room than where you started.
Overlapping was used in Tomb Raider III[2] to create a UFO that is larger on the inside than it is on the outside. (This had the unfortunate outcome where I was once killed by an enemy that was inside the UFO while I was outside, but the game thought we were in line-of-sight.) The original Tomb Raider games also used a trick to make the environment dynamic. When they needed to show a wall moving, or ceiling collapsing, or a room being filled with water, you'd be instantaneously teleported to a different map. Sneaky camera positioning made sure you never saw the change taking place. Teleporting is a common technique to make a map appear larger or more complex than it really is. One I learned recently is in The Witness[3].
The best example of a game built around physically impossible spaces is Antichamber[4]. You might enter a room, walk around a pillar, and exiti the same door but end up in a different place than where you started. Antichamber uses portal rendering to show areas that are larger on the inside than the outside, similar to the overlapping in "5-D Space". But also I think it has some teleporting.
Tea for God is a VR game that uses whatever physical space you have and generates Euclidean Orbifold levels based on that. It feels like you're progressing further and further into the game, but the levels are just folding in amongst each other, you 'get on elevators' that take you somewhere else, etc, and in actuality you're just walking around a bunch pretty much in circles.
There's a really interesting video documenting and explaining the development of higher-dimensional Rubik's cubes and the partially successful efforts of constructing such puzzles in the real world:
Is this a categorical question? I.e. "is it possible to create a 3D twisty puzzle that works on computers but not in the real world?"
I can easily imagine a 3D twisty puzzle which allows self-intersection. (To be less cheesy, you could make the self-intersection an objective of the game, perhaps "same color cubes will combine into one cube when they intersect" and then trying to accomplish a final target 3D model.)
I cannot imagine a computer simulation being more robust than reality without relaxing the rules of the physical world (like exclusivity of space/extent). Perhaps something like the Banach-Tarski Paradox[0], where a mathematical sphere can be split into two identical-to-the-first spheres?
"Moncage is a stunning vignette puzzle adventure developed by Optillusion.
The game takes place inside a mysterious cube, with each side of the cube housing a unique world: be it an old factory, a light tower, an amusement park, or a church, etc. At first sight, they may seem random and unrelated, but upon closer look, you will become mesmerized by the subtle and intricate ways of how these worlds connect."
Quadratis [0] is a set of 2D twisty puzzle inspired by algebraic topology. It is mostly an experimental game so far. Here are some slides that give some insight on the principles of the game [1].
It was once believed that 6x6 and higher Rubik's Cubes would be impossible to physically make. In the real world, the outermost edges have to be a little thicker or the faces rounded in order for the puzzle to stay together while being turned. Nowadays they're widely available on Amazon.
The Megaminx puzzle is a dodecahedron, having 12 faces. Would it be possible to make a similar puzzle for an icosahedron? What about the stellated polyhedra? Or semi-regular polyhedra? There's no reason we have to stick to the platonic solids, right?
I'm no expert in 3d geometry or twisty puzzle manufacturing but maybe this helps?
possible in a physical form? certainly, but you would never know how much the sculptures are painted: The scoring would be completely impractical for a physical version
Overlapping was used in Tomb Raider III[2] to create a UFO that is larger on the inside than it is on the outside. (This had the unfortunate outcome where I was once killed by an enemy that was inside the UFO while I was outside, but the game thought we were in line-of-sight.) The original Tomb Raider games also used a trick to make the environment dynamic. When they needed to show a wall moving, or ceiling collapsing, or a room being filled with water, you'd be instantaneously teleported to a different map. Sneaky camera positioning made sure you never saw the change taking place. Teleporting is a common technique to make a map appear larger or more complex than it really is. One I learned recently is in The Witness[3].
The best example of a game built around physically impossible spaces is Antichamber[4]. You might enter a room, walk around a pillar, and exiti the same door but end up in a different place than where you started. Antichamber uses portal rendering to show areas that are larger on the inside than the outside, similar to the overlapping in "5-D Space". But also I think it has some teleporting.
[1] https://www.lhowon.org/level/marathon/30
[2] https://www.tombraiderforums.com/showthread.php?t=166545
[3] https://old.reddit.com/r/TheWitness/comments/47pr3h/spoilers...
[4] http://www.antichamber-game.com/