Circle diagrams of common achiral map operations

Circle diagrams for common achiral map operations: identity, dual, subdivide, parallel, radial, medial, etc.

Map operations that are achiral (as opposed to chiral operations such as snub) can operate on non-orientable as well as orientable maps. Since the edges of maps are also undirected, an achiral map operation must have two perpendicular lines of mirror symmetry allowing the truchet tile to be played in either rotation, flipped over or not. For an achiral map operation, we only need to see one quadrant of the quadrilateral tile—the rest follows from symmetry.

A quadrant of a quadrilateral can be deformed into a circle having three 120° arcs: an arc representing the primal half-edge on the left, an arc representing the dual half-edge on the right (both these arcs are also mirror lines) and an arc representing a hypotenuse edge (quadrilateral edge) at the bottom. The hypotenuse arc at the bottom is not a mirror line.

Map operations are drawn on these diagrams as graphs with vertices along the circumference of the circle (and possibly also in the interior.) Black vertices are the real vertices, white vertices are simply where lines continue across the boundaries of the representation.

A map operation, O(), is associated with three other map operations: O*() = Du(O()) and O'()=O(Du()), and also O'*=Du(O(Du()). Two map operations O() and Q() are dual if Q = O'* = Du(O(Du()). For example, Ki and Tr are dual operations.