The plot of Fez begins when a two-dimensional character living in a two-dimensional world suddenly realizes that just beyond his perception lies a third dimension, filled with unexplored wonders and mystery. While it may have lacked the elegant mechanics, devious puzzles and brilliant audiovisual style of the video game, a 19th-century text by Edwin Abbott presented a story and world that is in many ways similar — a kind of spiritual prequel to the transdimensional tale dreamed up by Polytron.
This text was Flatland, a romantic satire written by Abbott in 1884 as a critique of the rigid, class-based social structure of Victorian England. Using geometric figures of varying complexity to represent different members of the Victorian hierarchy, Abbott's descriptions of a flat world given new perspective by a three-dimensional outsider, and specifically his postulation of a fourth dimension, eventually made their way into Einstein's theory of relativity. Then, in 1980, Carl Sagan famously demonstrated a thought experiment inspired by Abbott's Flatland on his legendary television program, Cosmos.
It's a challenging and fascinating concept: just as a cube can be drawn on a flat sheet of paper, consider that, to the inhabitants of a two-dimensional world, a three-dimensional being could only be perceived partially, as a two-dimensional "shadow" of its true self. Thus, a hypothetical four-dimensional being or object could only exist in our world as a three-dimensional projection.
This shadow from the fourth dimension is known as a hypercube, or tesseract, and it represents the closest mathematical approximation of the fourth dimension that we can perceive within our three-dimensional universe. Polytron's previous game, Super Hypercube, also references this concept. Going even further, just as the square and cube before it, a hypercube can then be extended at right angles to itself into the 5th and 6th dimensions, ad infinitum — but these new shapes will still be bound in three dimensions.
Or we could explore the opposite: that our three-dimensional universe is actually a two-dimensional, holographic universe, meaning that we are all actually living in Flatland, but are experiencing it with an additional, projected dimension. Wrap your head around that and solving puzzles in Fez will start to seem like playing with tinkertoys.