Fractal

A geometric or mathematical structure with detail across multiple scales, often exhibiting self-similarity and a non-integer dimension.

Self-similarity: A property in which a structure resembles itself across different scales. Many fractals repeat recognizable patterns as magnification changes.
Mandelbrot set: The set of complex parameters for which iterating a quadratic map does not escape to infinity. Its boundary displays intricate structure at every magnification.
Georg Cantor: A German mathematician whose work founded set theory and introduced the Cantor set. The Cantor set became an early example of a strange, scale-structured set.
Coastline paradox: The observation that a coastline's measured length increases as the measuring scale becomes finer. Coastlines illustrate how scale-dependent detail complicates ordinary length.
Euclidean geometry: The geometry of flat space based on Euclid's axioms and familiar integer-dimensional figures. Fractals challenge its usual reliance on smooth shapes and integer dimensions.
Fractal dimension: A measure describing how a set's detail changes with scale, often taking a non-integer value. It quantifies how fully a fractal occupies space.
Julia set: The boundary between points with different long-term behavior under iteration of a complex function. Each fixed complex parameter produces a distinct fractal boundary.
Cantor set: A subset of the interval formed by repeatedly deleting open middle thirds. Its construction illustrates self-similarity and fractional dimension.
Brownian motion: The random motion of particles suspended in a fluid, modeled mathematically by stochastic processes. A Brownian path has fractal dimension two despite being a curve.
Smooth manifold: A space that locally resembles Euclidean space and has smoothly compatible coordinate systems. Fractal boundaries can be too irregular to form smooth manifolds.

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