Koch snowflake

The Koch snowflake is a fractal curve formed by repeatedly replacing each line segment with four shorter segments that create an outward triangular bump. Its perimeter is infinite, while the enclosed area is finite.

Koch curve: A fractal curve generated by repeatedly replacing each segment with four segments that form a triangular bump. Each side of the snowflake develops as a Koch curve.
Helge von Koch: A Swedish mathematician who introduced the curve now bearing his name in 1904. His construction supplies the snowflake’s repeating boundary rule.
Coastline paradox: The observation that measured coastline length increases as the measuring scale becomes finer. The snowflake gives an exact model of scale-dependent boundary length.
Peano curve: A continuous curve that passes through every point of a square. Unlike the snowflake, it fills a region rather than enclosing one with an irregular boundary.
Self-similarity: A property in which parts of a structure resemble the whole across different scales. Every stage repeats the snowflake’s basic shape at smaller scales.
Émile Borel: A French mathematician whose work helped establish modern measure theory. Koch presented his curve amid mathematical interest in pathological examples and measure.
Fractal coastline: A coastline whose irregularity persists across a range of measurement scales. Its changing measured length echoes the snowflake’s ever-growing perimeter.
Dragon curve: A family of plane-filling fractal curves generated by repeated folding or substitution rules. Its recursive construction produces a different pattern from triangular Koch bumps.
Hausdorff dimension: A dimension measure that can assign noninteger values to geometrically irregular sets. The snowflake’s boundary has dimension log 4 divided by log 3.
Continuous nowhere-differentiable function: A continuous function that has no derivative at any point of its domain. Koch’s work belongs to the era’s exploration of continuous but irregular objects.

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