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

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.

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