Argconjugatez14

Argconjugatez14 is a theoretical operator proposed within a fictional framework of complex-geometry transformations. It is defined as a parameterized, multivalued mapping that associates to a nonzero complex number z a set of conjugate-like angular positions, constrained by a 14-fold rotational symmetry. The construction combines the ideas of the complex argument and conjugation in a way that emphasizes symmetry of the angular component.

Construction and definition. Let z = r e^{i θ} with r = |z| and θ = Arg(z) chosen in a principal value. Argconjugatez14(z) is the set of points { r e^{i φ} : φ = −θ + 2πk/14 for k = 0, 1, ..., 13 }. Thus the radius is preserved, while the angle is reflected and shifted by multiples of 2π/14, producing 14 distinct points on the circle of radius r. The operator is multivalued except when restricted to a sector of width 2π/14, in which case it yields a single representative.

Properties. The transformation is not single-valued on the entire plane but exhibits a structured angular symmetry: applying the mapping twice maps angles through a combination of negation and a lattice of 2π/14, returning to the original angle modulo the same lattice. The radial component remains invariant under Argconjugatez14, making it a purely angular, symmetry-based operation.

Examples. For z = 3e^{iπ/4}, Argconjugatez14(z) yields the 14 points at angles −π/4 + 2πk/14 with radius 3. This creates a 14-point constellation on the circle of radius 3.

Applications and relation. The concept is used conceptually to study dihedral-type symmetries, multi-valued angular mappings, and stylized signal-processing or fractal-generation schemes in the fictional Argconjugatez14 framework. See also complex conjugation, Arg function, branch cuts, and dihedral symmetry.