Fractal glossary
Fractal mathematics brings together geometry, dynamics, numerical methods and computer graphics. This compact glossary keeps their central terms distinct.
Geometry
Fractal dimension: a scaling exponent describing growth of detail. Self-similarity: related structure under equal-direction scaling. Self-affinity: related structure under direction-dependent scaling. Lacunarity: a measure related to gap distribution and texture. Support: the set on which a measure is carried.
These terms describe different layers of the same problem. Self-similarity is a relation across scales, dimension is a quantity derived from that relation or from measurements, and lacunarity adds information about the distribution of gaps. Two sets can have similar dimensions while looking very different. When reading a study, it is therefore worth asking whether the subject is shape, measure, probability distribution or texture.
Dynamics
Orbit: the sequence produced by iteration. Attractor: a set approached by nearby states. Basin: starting states reaching the same attractor. Julia set: the instability boundary for a complex function. Fatou set: regions of stable family behavior. Bifurcation: a qualitative change as a parameter varies.
Orbit and attractor should not be confused either. An orbit is one path through state space; an attractor is the set approached by many such paths over time. A basin collects the corresponding starting states, and its boundary may itself be fractal. Julia and Fatou sets apply this stability question to complex functions. The vocabulary forms a hierarchy of trajectory, long-term set and initial condition.
Rendering
Bailout: a condition proving escape. Iteration limit: finite computational cutoff. Smooth iteration: interpolated exterior color coordinate. Distance estimate: derivative-based boundary-distance approximation. Perturbation: nearby-orbit calculation around a high-precision reference. Supersampling: combining multiple subpixel samples.
In rendering, the terms point to different sources of error. An unsuitable bailout changes the mathematical test, a low iteration cap stops it too soon, and insufficient precision corrupts the coordinates themselves. Smooth iteration and distance estimation shape computed evidence; supersampling reduces sampling error. Perturbation addresses another problem: it enables deep coordinates by relating many nearby pixels to one high-precision reference orbit.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- FractalWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
What is a fractal?
A clear introduction to fractals, self-similarity, scaling and fractal dimension—with mathematical examples and careful comparisons to nature.
Fractal dimension
What fractal dimension measures, how box counting works and why one object can have more than one meaningful dimension.
Escape-time rendering
How Mandelbrot and Julia renderers turn bounded-orbit tests, iteration limits and smooth counts into pixels.