Single Comment

Your 'Circular King' is not exactly analogous to the 'Circular Knight' that the Rose is, because of the necessity to choose (root 2 being irrational) between an equilateral and an equiangular octagon. The Rose move is composed of up to eight moves of the Knight, an uncombined piece, and is therefore equilateral and not equiangular. Your 'Circular King' alternates between Wazir and Fers moves, and is therefore equiangular and not equilateral. The equilateral pattern of Wazir moves would be e.g. e1-e2-e3-d3-c3-c2-c1-d1 and that of Fers moves e.g. e1-f2-g3-f4-e5-d4-c3-d2. The repeated moves in the same direction are because, for example, the 'two most forward' Wazir moves are the same move. An equiangular version of the Rose would alternate Knight and Camel moves - a 'Circular Gnu' in your terminology.