It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry"[3] by the Swedish mathematician Helge von Koch.

The Koch snowflake can be built up iteratively, in a sequence of stages.

The areas enclosed by the successive stages in the construction of the snowflake converge to

times the area of the original triangle, while the perimeters of the successive stages increase without bound.

Consequently, the snowflake encloses a finite area, but has an infinite perimeter.

The Koch snowflake has been constructed as an example of a continuous curve where drawing a tangent line to any point is impossible.

Unlike the earlier Weierstrass function where the proof was purely analytical, the Koch snowflake was created to be possible to geometrically represent at the time, so that this property could also be seen through "naive intuition".

However, the picture of the snowflake does not appear in either the original article published in 1904[3] nor in the extended 1906 memoir.

An investigation of this question suggests that the snowflake curve is due to the American mathematician Edward Kasner.

[5][6] The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: The first iteration of this process produces the outline of a hexagram.

The Koch snowflake is the limit approached as the above steps are followed indefinitely.

A Koch curve–based representation of a nominally flat surface can similarly be created by repeatedly segmenting each line in a sawtooth pattern of segments with a given angle.

If the original equilateral triangle has sides of length

an inverse power of three multiple of the original length.

times the original triangle perimeter and is unbounded, as

As the number of iterations tends to infinity, the limit of the perimeter is:

The volume of the solid of revolution of the Koch snowflake about an axis of symmetry of the initiating equilateral triangle of unit side is

It is impossible to draw a tangent line to any point of the curve.

Every point on a continuous de Rham curve corresponds to a real number in the unit interval.

For the Koch curve, the tips of the snowflake correspond to the dyadic rationals: each tip can be uniquely labeled with a distinct dyadic rational.

It is possible to tessellate the plane by copies of Koch snowflakes in two different sizes.

A turtle graphic is the curve that is generated if an automaton is programmed with a sequence.

If the Thue–Morse sequence members are used in order to select program states: the resulting curve converges to the Koch snowflake.

To create the Koch snowflake, one would use F--F--F (an equilateral triangle) as the axiom.

Following von Koch's concept, several variants of the Koch curve were designed, considering right angles (quadratic), other angles (Cesàro), circles and polyhedra and their extensions to higher dimensions (Sphereflake and Kochcube, respectively) Squares can be used to generate similar fractal curves.

Starting with a unit square and adding to each side at each iteration a square with dimension one third of the squares in the previous iteration, it can be shown that both the length of the perimeter and the total area are determined by geometric progressions.

while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve.

In addition to the curve, the paper by Helge von Koch that has established the Koch curve shows a variation of the curve as an example of a continuous everywhere yet nowhere differentiable function that was possible to represent geometrically at the time.

From the base straight line, represented as AB, the graph can be drawn by recursively applying the following on each line segment: Each point of AB can be shown to converge to a single height.

is defined as the distance of that point to the initial base, then