Mandelbrotv

Mandelbrotv is a term used in speculative contexts to refer to a family of fractal sets that generalize the classical Mandelbrot set. The central idea is to extend the iteration that defines the Mandelbrot set either to higher-dimensional spaces or to modified polynomial maps, producing a locus of parameters for which the orbit remains bounded.

Formal variations include higher-dimensional versions, such as z in C^n with z_{k+1} = z_k^2 + c and c in C^n, or p-generalized versions with z_{n+1} = z_n^p + c for p > 1. Other approaches couple components of z, yielding multivariate dynamics. In all cases, Mandelbrotv refers to the set of parameter values c for which the orbit of the initial point remains bounded.

Visualization is typically done by taking slices, projections, or by fixing some coordinates to render two- or three-dimensional views. The standard escape-time algorithm and color mappings are used, with computation becoming more intensive as dimensionality grows.

Properties vary with the chosen map and dimension. Mandelbrotv-type sets often retain fractal boundaries and self-similarity in projections, but global topology and connectedness can differ from the classic Mandelbrot set.

Origin and usage: Mandelbrotv is not an established standard in mainstream mathematics; rather, it appears in speculative literature and educational discussions as a way to explore generalizations of complex dynamics and fractal geometry.

See also: Mandelbrot set, Julia set, fractal geometry, escape-time algorithm.