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

TL;DR

This paper applies a novel computational methodology using infinities and infinitesimals to precisely analyze the Koch snowflake's infinite perimeter and finite area, revealing distinctions at infinity and differences based on initial conditions.

Contribution

It introduces a new computational approach with the Infinity Computer to exactly compute properties of the Koch snowflake at infinity, including perimeter and area, and explores variations due to initial conditions.

Findings

Exact infinite perimeter and finite area computed at infinity.

Different infinite steps yield distinguishable snowflakes with unique properties.

Variations in initial conditions affect the snowflake's characteristics at infinity.

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, $N_n$, of sides of the snowflake, the exact infinitesimal length, $L_n$, of each side and the exact infinite perimeter, $P_n$, of the Koch snowflake as the result of multiplication of the infinite $N_n$ by the infinitesimal $L_n$. It is established that for different infinite $n$ and $k$ the infinite perimeters $P_n$ and $P_k$ are also different and the difference can be infinite. It is shown that the finite areas $A_n$ and $A_k$ of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite $n$ and $k$ and the difference $A_n - A_k$ results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed.