Koch curve and snowflake
Helge von Koch’s 1904 curve replaces the middle third of every segment with two sides of an equilateral bump. Repeating that rule creates an everywhere rough limit.
Four segments replace one
Each iteration turns one segment into four segments, each one third as long. The number of segments grows by four while their length shrinks by three, yielding dimension log 4 / log 3.
Each replacement multiplies the segment count by four and reduces every segment to one third. Total length therefore grows by 4/3 per stage and equals (4/3)ⁿ times the starting length after n stages. The smallest feature simultaneously shrinks as 3⁻ⁿ. In a raster image, the useful stage is reached when that feature approaches pixel size; further iterations mostly increase sampling error.
Infinite perimeter, bounded area
Starting from an equilateral triangle produces the Koch snowflake. Its perimeter is multiplied by 4/3 at every step and diverges. The added triangular areas form a convergent geometric series, so the enclosed area approaches a finite value.
For the snowflake, each stage adds ever smaller triangles. Their area contributions form a convergent geometric series while perimeter grows as (4/3)ⁿ. The apparent paradox disappears when different quantities are separated: many extremely thin protrusions can add unlimited boundary length while occupying only a bounded total area. The curve demonstrates that edge and content need not grow together.
Why it resembles but is not a coast
The curve is exactly self-similar and has unlimited construction depth. A real coast has tides, rock grains and measurement conventions. The Koch model illuminates scale-dependent length without claiming that every coastline follows its precise dimension or rule.
A coastline has neither fixed sixty-degree turns nor four identical pieces at every scale. Its relation to the Koch curve concerns measurement: a shorter ruler follows more bays and reports greater length. The ideal model isolates the mechanism; natural data require an estimated exponent and a finite scale interval. Keeping those cases distinct preserves the value of the comparison without replacing geology with a substitution rule.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Koch SnowflakeWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
The coastline paradox
Why smaller rulers measure longer coasts, how Richardson plots estimate scaling and where the paradox stops physically.
Fractal dimension
What fractal dimension measures, how box counting works and why one object can have more than one meaningful dimension.
When curves became ‘pathological’
Why continuous nowhere-smooth and space-filling curves disturbed classical intuition—and later became central examples.