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

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.

16 pages, 1 figure

References in corpus (5)

• Evaluating the exact infinitesimal values of area of Sierpinski's carpet and volume of Menger's sponge
• Usage of infinitesimals in the Menger's Sponge model of porosity
• On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions
• An application of Grossone to the study of a family of tilings of the hyperbolic plane
• Cellular Automata Using Infinite Computations

Cited by in corpus (5)

• Independence of the grossone-based infinity methodology from non-standard analysis and comments upon logical fallacies in some texts asserting the opposite
• Some paradoxes of Infinity revisited
• The Mathematical Intelligencer flunks the Olympics
• Lower and upper estimates of the quantity of algebraic numbers
• Conjugate-symplecticity properties of Euler--Maclaurin methods and their implementation on the Infinity Computer