ANISOTROPIC METRIC DEFORMATIONS OF ITERATED FUNCTION SYSTEMS: A ONE-PARAMETER FAMILY OF SELF-AFFINE FRACTALS WITH RIEMANNIAN–LORENTZIAN TRANSITION

Abstract

Classical iterated function system (IFS) fractals such as the Cesàro and Koch curves are built upon a fixed Euclidean metric, resulting in isotropic geometry where the underlying space plays a passive role. This work introduces a three-dimensional hypercomplex algebra, the Trinition algebra, equipped with a real deformation parameter [Formula: see text]. The algebra generates a one-parameter family of inner products on a distinguished plane, whose signature changes from Riemannian ([Formula: see text]) to Lorentzian ([Formula: see text] or [Formula: see text]), passing through degenerate (parabolic) cases at the boundaries. The associated anisotropic metric naturally generalizes the Cesàro IFS by replacing the classical complex rotation with an Archiometric exponential that preserves the deformed metric. Restricting the IFS to two dimensions yields a continuous family of planar snowflake fractals. As [Formula: see text] moves away from the isotropic value [Formula: see text], the fractal elongates, collapsing to a nearly linear structure near the degenerate limits [Formula: see text] or [Formula: see text]. In three dimensions, applying the IFS generator to every edge of a regular icosahedron produces a network of self-affine fractal curves; an additional out-of-plane twist angle [Formula: see text] lifts the pattern into a true three-dimensional tangle. Numerical simulations reveal a sharp morphological transition: near [Formula: see text] the attractor is a compact, cloud-like object, whereas for [Formula: see text] approaching the Lorentzian regime the structure develops pronounced unbounded branching. The box-counting dimension is conjectured to follow a product formula combining the planar Cesàro dimension with the unit interval, attaining the maximal value [Formula: see text] in the space-filling isotropic limit. This study demonstrates that a single metric parameter can continuously deform fractal morphology, opening a novel route for anisotropic fractal synthesis in geometry, materials science, and procedural modeling.