of each one being the length of the segments in the previous stage. Hence, the length of the curve after iterations will be times the original triangle perimeter and is unbounded, as tends to infinity. Limit of perimeter As the number of iterations tends to infinity, the limit of the perimeter is: since . An -dimensional measure exists, but has not been calculated so far. Only upper and lower bounds have been invented. Area of the Koch snowflake In each iteration a new triangle is added on each side of the previous iteration, so the number of new triangles added in iteration is: The area of each new triangle added in an iteration is of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration is: where is the area of the original triangle. The total new area added in iteration is therefore: The total area of the snowflake after iterations is: Collapsing the geometric sum gives: Limits of area The limit of the area is: since . Thus, the area of the Koch snowflake is of the area of the original triangle. Expressed in terms of the side length of the original triangle, this is: Solid of revolution 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 Other properties The Koch snowflake is self-replicating with six smaller copies surrounding one larger copy at the center. Hence, it is an irrep-7 irrep-tile (see Rep-tile for discussion). The fractal dimension of the Koch curve is . This is greater than that of a line () but less than that of Peano's space-filling curve (). The Koch curve is continuous everywhere, but differentiable nowhere. Tessellation of the plane It is possible to tessellate the plane by copies of Koch snowflakes in two different sizes. However, such a tessellation is not possible using only snowflakes of one size. Since each Koch snowflake in the tessellation can be subdivided into seven smaller snowflakes of two different sizes, it is also possible to find tessellations that use more than two sizes at once. Koch snowflakes and Koch antisnowflakes of the same size may be used to tile the plane. Thue–Morse sequence and turtle graphics A turtle graphic is the curve that is generated if an automaton is programmed with a