# The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area

*Communications in Nonlinear Science and Numerical Simulation*31 (1-3):21–29 (2016)

**Abstract**

The Koch snowflake is one of the first fractals that were mathematically
described. It is interesting because it has an infinite perimeter in the limit
but its limit area is finite. In this paper, a recently proposed computational
methodology allowing one to execute numerical computations with infinities
and infinitesimals is applied to study the Koch snowflake at infinity. Numerical
computations with actual infinite and infinitesimal numbers can be
executed on the Infinity Computer being a new supercomputer patented in
USA and EU. It is revealed in the paper that at infinity the snowflake is not
unique, i.e., different snowflakes can be distinguished for different infinite
numbers of steps executed during the process of their generation. It is then
shown that for any given infinite number n of steps it becomes possible to
calculate the exact infinite number, Nn, of sides of the snowflake, the exact
infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn,
of the Koch snowflake as the result of multiplication of the infinite Nn by
the infinitesimal Ln. It is established that for different infinite n and k the
infinite perimeters Pn and Pk are also different and the difference can be infinite.
It is shown that the finite areas An and Ak of the snowflakes can be
also calculated exactly (up to infinitesimals) for different infinite n and k and
the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed
starting from different initial conditions are also studied and their
quantitative characteristics at infinity are computed.

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Archival date: 2018-12-17

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