From 3-D to 4-D It's difficult for us to ... to remain equidistant to every point of the circle's circumference—unless he moves into the third dimension. Alas, we don't have the four ...

Electrons moved in two dimensions when influenced by a magnetic field, hinting at how a similar effect would occur in the fourth dimension.

Paul Steinhardt. The key insight that enabled their explanation was that quasicrystals were, in fact, periodic—but in a higher dimension than the one in which they exist physically. Using this ...