Images with provenance

Every documentary image has been checked for an explicit public-domain or Creative Commons release and is stored locally. Original mathematical images are generated by MandelKit or by the reproducible MandelKit knowledge renderer.

Seven construction stages of the Cantor set — Seven iterations of the Cantor construction: at each step, the open middle third is removed from every remaining interval.Image: Wikimedia contributors · Wikimedia Commons · Public Domain (simple geometry)

Seventh iteration of the Sierpiński triangle — Seventh iteration of the Sierpiński triangle: three reduced copies form the next stage while leaving the central gap empty.Image: PiAndWhippedCream · Wikimedia Commons · Public Domain

Overlaid first three orders of the Hilbert curve — The first three orders of a Hilbert curve show how a one-dimensional line densely traverses a square in the limiting process.Image: Geoff Richards (Qef) · Wikimedia Commons · Public Domain

Fifth iteration of the Koch snowflake — Fifth iteration of the Koch snowflake: every segment is repeatedly replaced by four segments one third as long.Image: Wrtlprnft · Wikimedia Commons · Public Domain (simple geometry)

Construction diagram of an Apollonian circle packing — Construction of an Apollonian gasket: further tangent circles are inserted into every gap between three mutually tangent circles.Image: Stevo-88 · Wikimedia Commons · Public Domain

Fifth iteration of a Sierpiński carpet as the two-dimensional analogue of the Menger sponge — Two-dimensional analogue of the Menger sponge: the central ninth is recursively removed from every square.Image: Solkoll · Wikimedia Commons · Public Domain

Color-marked iteration stages of a dragon curve — Dragon curve with color-separated orders: repeated folding or an L-system produces its characteristic space-filling line.Image: Joehms22 · Wikimedia Commons · Public Domain

Barnsley fern generated by an iterated function system — Barnsley fern: four affine maps and weighted random selection generate a mathematical image resembling a fern.Image: Farry · Wikimedia Commons · CC0 1.0

Bifurcation diagram of the logistic map — Bifurcation diagram of the logistic map: as the parameter grows, a stable state repeatedly splits before chaotic bands emerge.Image: PAR · Wikimedia Commons · Public Domain

Phased-array antenna board with repeated branching conductor structures — Experimental antenna array with repeated branching conductor geometries. Such geometries can fit multiple effective length scales into a compact area.Image: Isenberg · Wikimedia Commons · CC BY 4.0

Blue Mandelbrot deep zoom with repeatedly branching spirals — Deep zoom into a Mandelbrot region: related, but non-identical boundary structures return across many scales.Image: Björn Kindler / MandelKit · MandelKit · Eigene Darstellung · Own work

Color-rendered Julia set with winding boundary bands — Julia set in MandelKit: one iteration rule produces a boundary with structure down to the display limit.Image: Björn Kindler / MandelKit · MandelKit · Eigene Darstellung · Own work

Burning Ship fractal with red islands on a turquoise field — Burning Ship in MandelKit: the absolute value in the iteration breaks the symmetry of the classic Mandelbrot map.Image: Björn Kindler / MandelKit · MandelKit · Eigene Darstellung · Own work

Vividly colored fractal relief with layered color bands — Color design in MandelKit: the image interprets computed values; it is not the mathematical set itself.Image: Björn Kindler / MandelKit · MandelKit · Eigene Darstellung · Own work

Pink three-dimensional fractal relief on black — 3D relief in MandelKit: height, normals and lighting translate scalar fractal data into a spatial reading.Image: Björn Kindler / MandelKit · MandelKit · Eigene Darstellung · Own work

Romanesco with similarly shaped florets arranged in spirals — Romanesco repeats growth motifs over a few size levels—an instructive natural example, but not an infinitely exact fractal.Image: Jon Sullivan · Wikimedia Commons · Public Domain

Close-up of a fern with similarly branching leaflets — Fern fronds and leaflets follow related branching patterns. Biological growth limits the self-similarity to finite scales.Image: Abbas Karimi · Wikimedia Commons · CC BY-SA 4.0

Satellite image of the branching Colorado River Delta — ASTER view of the Colorado Delta: flow, sediment and terrain form branching networks whose statistics depend on observation scale.Image: NASA/METI/AIST/Japan Space Systems/U.S.–Japan ASTER Science Team · NASA / Wikimedia Commons · Public Domain (NASA)

Scanning electron microscope images of branching snow crystals — Pseudo-colored USDA scanning electron microscopy: temperature and humidity shape the finite dendritic growth of each crystal.Image: USDA Agricultural Research Service · USDA ARS / Wikimedia Commons · Public Domain (U.S. Government)

Highly branching lightning against a dark night sky — A lightning channel branches during electrical breakdown. Its resemblance to other networks does not mean the same processes are at work.Image: Brad Caldwell (Caldwbr) · Wikimedia Commons · CC0 1.0

Medical illustration of bronchi, alveoli and pulmonary vessels — Bronchi and vessels distribute air and blood through hierarchical branching. Fractal models describe parts of this geometry, not the entire physiology.Image: Patrick J. Lynch, medical illustrator; C. Carl Jaffe, MD · Yale / Wikimedia Commons · CC BY 2.5

Portrait of mathematician Benoît Mandelbrot at EPFL in 2007 — Benoît Mandelbrot at EPFL, 14 March 2007. His work connected scale analysis, computer imagery and the geometry of rough forms.Image: Rama · Wikimedia Commons · CC BY-SA 2.0 FR

Box-counting grid over a self-similar rough curve — Box counting measures how many grid cells a structure touches as the cell size changes.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Recursively branching mathematical tree with eleven levels — Each branch invokes the same construction rule again with a shorter length and rotated angle.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Two initially adjacent logistic-map orbits diverge — Two starting values only one ten-millionth apart quickly diverge in the chaotic regime.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Three approximations of an increasingly rough Weierstrass function — Each added frequency band makes the curve finer and rougher without destroying its continuity.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Complete Mandelbrot set with clearly visible main body and satellites — Black points remain bounded under z² + c; the surroundings are colored by smoothed escape time.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Side-by-side cubic Multibrot set and Multicorn — The left panel iterates z³ + c, the right its conjugate variant; the small rule change produces different symmetry.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Newton fractal with three color-separated basins of attraction — Color shows which root Newton's method reaches; the fractal boundaries respond sensitively to the starting value.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Interwoven basins of a Newton method with five roots — Five attractors divide the plane into regions whose shared boundaries remain interwoven at ever smaller scales.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Lyapunov fractal of a periodically alternating logistic map — Light regions mark stable parameter sequences; dark and blue regions indicate positive Lyapunov exponents and chaos.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Numerically integrated Lorenz attractor with two butterfly-shaped lobes — A single trajectory never settles into a cycle yet remains permanently confined to the characteristic Lorenz structure.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Simulation of a diffusion-limited, dendritically branching aggregate — Random-walking particles stick on first contact with the cluster, producing screened, tree-like branches.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Procedural cloud field assembled from layered noise scales — Several smoothed noise fields at different sizes combine into a statistically multi-scale cloud structure.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Shaded procedural mountain terrain from recursive height-field subdivision — Recursive subdivision and decreasing random offsets produce terrain with broad forms and progressively finer roughness.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Multifractal cascade with unevenly distributed mass across many scales — A multiplicative cascade redistributes weight unevenly at each step, producing several local scaling exponents.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Multi-scale synthetic time signal with a slow baseline — A spectrally generated 1/f-like signal illustrates long-range fluctuations studied in physiological time series.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Fractal image with block grid and three transformed subregions — The schematic shows the principle of describing image regions through affine copies of other regions.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Isometric rendering of a third-order Menger sponge — At every stage central subcubes are removed, leaving a perforated solid with an ever-growing surface area.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Distance estimate around the Mandelbrot set with fine distance contours — The iteration derivative provides an estimate of distance to the set; contours make this scalar value visible.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work

Mandelbrot rendering with a visible tile grid for a large export — Oversized images are divided into manageable tiles and then assembled without visible seams.Image: Björn Kindler / MandelKit · MandelKit Wissensgrafik · Eigene Darstellung · Own work