Perhaps you know of the invention of chess with ancient India's Chaturanga, together with the variants it later engendered in medieval Arabia's Shatranj, as well as in China's Xiangqi, Japan's Shogi, and Korea's Changgi. Those traditional variants have been recently been added to with a Master's Capablanca Chess and the later Gothic Chess. And to this add the historic contribution by the late Icelandic citizen, Bobby Fischer, creating Random Chess, (perhaps to outdo the former) Capablanca Random Chess, not to mention far, exotic America's Gettochess. There are even deviations into Hexagonal Chess and the heresy of supposed Grand Chess! Well, chess-wise, you a'int seen nuthin' yet!
All the latter share the fact of being played with variously shaped pieces on a flat two-dimensional board. Not all variations of chess conform to these limits. Even businessmen have seen the value of 3-D Chess, with others supplying the basic theoretical foundations, still others spontaneously inventing the many 3-D variations, such as Hyperchess (otherwise similar to conventional chess), as well as a variant inspired by the theme of Star Trek; Even more Trekkie intoxication derives from 3-D obsessions with the Borg Queen, no doubt using the transporter pad, with the Enterprise traveling at warp factor through curved space on a mission of first contact into the stellar Federation's Neutral Zone. Yea, this is chess worthy of our recently begun third Millenium. All of which is to say that the three dimensions of daily life can be a challenging chess venue. And - it doesn't stop there!
Physicists say there are four dimensions - and you CAN play chess in them, such as in both Chesseract and also Wormhole, not to mention the equally fascinating Tessa. All are played utilizing the basic four-dimensional figure, the tesseract.
Now, here is where things start getting more interesting with the transition to a chess game played as 5D Chess, the creation of aesthetic patterns of chess beauty, as in Hyperspace, and the exciting hyperspatial moves of Skava, moving always along five axes.