torusfrakturs

Torusfrakturs are fractal patterns generated on the surface of a torus, a doughnut-shaped two-dimensional manifold. The term describes self-similar sets or images that lie embedded in the torus and arise when fractal generation techniques are adapted to the torus’s periodic, boundaryless geometry. The concept draws on broader fractal geometry and on the study of dynamical systems on compact manifolds.

Construction methods commonly involve iterated function systems (IFS) defined on the torus T^2 = R^2 / Z^2, where each map f_i: T^2 → T^2 is a contraction. The unique attractor of the IFS yields a torusfraktur that exhibits self-similarity under the torus’s identifications. The alternative approach uses dynamical maps F: T^2 → T^2 and investigates invariant or Julia-like sets produced by repeated iteration from a seed region, with wrap-around behavior due to modular arithmetic.

Properties of torusfrakturs reflect the topology of the torus. They are compact and may display a range of fractal dimensions between those of smooth curves and surfaces, depending on the rule set. Their patterns can be highly symmetric under torus translations and cylindrical or longitudinal wraps, a consequence of the torus’s identifications. Visualizations often emphasize color maps of orbit structure, convergence, or escape time.

In practice, torusfrakturs appear in mathematical research on fractal geometry on curved spaces and in computer graphics for texture generation on toroidal surfaces. Examples include torus-adapted Cantor-like dust or Sierpinski-type perforations forged within the fundamental domain. The term is used variably in literature and online discussions, with definitions and conventions continuing to diversify. See also fractal geometry, torus, iterated function system.