Von Koch’s snowflake curve

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$Koch snowflakeSwedish mathematician Niels von Koch published the fractal that bears his name in 1906. It begins with an equilateral triangle; three new equilateral triangles are constructed on each of its sides using the middle thirds as the bases, which are then removed to form a six-pointed star. This is continued in an infinite iterative process, so that the resulting curve has infinite length. The Koch snowflake is noteworthy in that it is continuous but nowhere differentiable; that is, at no point on the curve does there exist a tangent line.$

Von Koch’s snowflake curve

mathematics

Also known as: Koch snowflake, Koch triangle

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Last updated

July 15, 2026

Quick Summary

The Von Koch snowflake curve is a mathematical fractal discovered by Helge von Koch in 1904. It is a continuous curve that is nowhere differentiable, meaning it has no tangent lines at any point.1

The curve is constructed by starting with an equilateral triangle and repeatedly replacing the middle third of each line segment with two sides of a smaller equilateral triangle that points outward. This process is continued indefinitely, resulting in a complex, self-similar shape.21

A key characteristic of fractals, including the snowflake curve, is their fractal dimension, which is a non-integer value. For the snowflake curve, this dimension is approximately 1.26. This concept of fractal dimension is a departure from traditional Euclidean geometry and helps describe complex, irregular shapes found in nature, such as coastlines and mountain ranges.2

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number game
Pathological curves
Von Koch’s snowflake curve, for example, is the figure obtained by trisecting each side of an equilateral triangle and replacing the centre segment by two sides of a smaller equilateral triangle projecting outward, then treating the resulting figure the same way, and so on. The…
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Von Koch’s snowflake curve, for example, is the figure obtained by trisecting each side of an equilateral triangle and replacing the centre segment by two sides of a smaller equilateral triangle projecting outward, then treating the resulting figure the same way, and so on. The…
Read More

fractal
…considering a specific example: the snowflake curve defined by Helge von Koch in 1904. It is a purely mathematical figure with a six-fold symmetry, like a natural snowflake. It is self-similar in that it consists of three identical parts, each of which in turn is made of four parts that…
Read More
…considering a specific example: the snowflake curve defined by Helge von Koch in 1904. It is a purely mathematical figure with a six-fold symmetry, like a natural snowflake. It is self-similar in that it consists of three identical parts, each of which in turn is made of four parts that…
Read More

$Koch snowflakeSwedish mathematician Niels von Koch published the fractal that bears his name in 1906. It begins with an equilateral triangle; three new equilateral triangles are constructed on each of its sides using the middle thirds as the bases, which are then removed to form a six-pointed star. This is continued in an infinite iterative process, so that the resulting curve has infinite length. The Koch snowflake is noteworthy in that it is continuous but nowhere differentiable; that is, at no point on the curve does there exist a tangent line.$
Niels Fabian Helge von Koch
…for his discovery of the von Koch snowflake curve, a continuous curve important in the study of fractal geometry.
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…for his discovery of the von Koch snowflake curve, a continuous curve important in the study of fractal geometry.
Read More