>>7785500A modern variant is Circular Chess, with different initial setup and movement rules that allow you to traverse the center.
Byzantine chess, BTW, is like playing on a rectangular board that is 4 files wide by 16 ranks long ... but if you get to the end of available ranks, you "teleport" for continuous movement to the opposite edge and continue using the ranks as if they are infinite. I mention this in case anyone wants to try it but can't find a round board. You have the option of a 4 x 16 rectangular board to get the identical gameplay, although visual perception of play will be a different experience.
Anyway, pic related is Circular chess, with movement rules to follow.