Iterated Maps & Fractals

The Wolfram Language has flexible capabilities for handling iterated maps, as well as highly optimized algorithms for common objects of investigation such as Julia sets and the Mandelbrot set.

ReferenceReference

Iterating Arbitrary Functions

Nest iterate a function

NestList  ▪  NestGraph  ▪  NestWhile  ▪  NestWhileList  ▪  FixedPoint  ▪  FixedPointList

ReplaceRepeated do repeated substitutions

Substitution Systems

SubstitutionSystem string, list, or array substitution system

ArrayFilter  ▪  ArrayFlatten

Geometric Iteration

AnglePath compute a "turtle graphics" path from turns and moves

Complex Iterated Maps

JuliaSetPlot plot Julia sets of arbitrary rational functions

JuliaSetPoints  ▪  JuliaSetIterationCount

MandelbrotSetPlot plot the Mandelbrot set at any resolution

MandelbrotSetMemberQ  ▪  MandelbrotSetDistance  ▪  MandelbrotSetIterationCount

JuliaSetBoettcher  ▪  MandelbrotSetBoettcher

Fractal Functions

CantorStaircase  ▪  MinkowskiQuestionMark

Discrete Recurrence Relations

RecurrenceTable create a table of values from recurrence relations

Substitution Sequences

ThueMorse  ▪  RudinShapiro

Iterated Boolean Functions

CellularAutomaton arbitrary cellular automaton rules in any number of dimensions

Iterated String Substitutions

StringReplace

StringReplaceList generate a multiway system with string replacements

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