Operational Mathematics of Geometric Transformations: Extending the Iteration Count of Plane Similarities and Their Inverses to the Complex Domain

This paper systematically transplants the core methodology of Operational Mathematics—the extension of the repetition count of fundamental operations from natural numbers to integers, rational numbers, real numbers, and ultimately complex numbers—onto a new class of binary operations: geometric transformations of the plane, specifically translations, rotations, scalings, shears, reflections, and projections, together with their inverses. A complete set of seven axioms is established; integer-order, fractional-order, real-order, and complex-order iterations are rigorously defined; and the existence and uniqueness of iterative roots at each level are proved by means of Schröder’s equation, Abel’s equation, and a suitably adapted Kneser construction that, for linear transformations, reduces to the matrix logarithm-exponential method. Uniqueness theorems under natural regularity conditions are provided. The singularity structure of complex-order geometric iterations is analyzed in depth, revealing a phenomenon that depends on the algebraic type of the base operation: for translations, rotations, scalings, and shears, the complex iteration is an entire function of the iteration count, possessing no finite singularities; for reflections, the iteration is expressed through trigonometric functions and remains entire; for projections, a ‘soft-projection’ semigroup provides a continuous interpolation. The only branch points arise from the multivaluedness of the matrix logarithm when a transformation possesses negative real eigenvalues. Natural boundaries do not appear in finite parts of the complex plane of the iteration count.