70 research outputs found
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area
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
Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems
In this survey, a recent computational methodology paying a special attention to the separation
of mathematical objects from numeral systems involved in their representation is described.
It has been introduced with the intention to allow one to work with infinities and infinitesimals
numerically in a unique computational framework in all the situations requiring these notions. The
methodology does not contradict Cantor’s and non-standard analysis views and is based on the
Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite,
infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a
computational device called the Infinity Computer (patented in USA and EU) working numerically
(recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite
and infinitesimal numbers that can be written in a positional numeral system with an infinite radix.
It is argued that numeral systems involved in computations limit our capabilities to compute and lead
to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility
to use the same numeral system for measuring infinite sets, working with divergent series, probability,
fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally
different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical
objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher
accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not
related to their nature but is a consequence of the weakness of traditional numeral systems used to
study them. It is shown that the introduced methodology and numeral system change our perception
of the mathematical objects studied in the two problems
The Olympic medals ranks, lexicographic ordering and numerical infinities
Several ways used to rank countries with respect to medals won during
Olympic Games are discussed. In particular, it is shown that the unofficial
rank used by the Olympic Committee is the only rank that does not allow
one to use a numerical counter for ranking – this rank uses the lexicographic
ordering to rank countries: one gold medal is more precious than any number
of silver medals and one silver medal is more precious than any number of
bronze medals. How can we quantify what do these words, more precious,
mean? Can we introduce a counter that for any possible number of medals
would allow us to compute a numerical rank of a country using the number
of gold, silver, and bronze medals in such a way that the higher resulting
number would put the country in the higher position in the rank? Here we
show that it is impossible to solve this problem using the positional numeral
system with any finite base. Then we demonstrate that this problem can be
easily solved by applying numerical computations with recently developed
actual infinite numbers. These computations can be done on a new kind of
a computer – the recently patented Infinity Computer. Its working software
prototype is described briefly and examples of computations are given. It is
shown that the new way of counting can be used in all situations where the
lexicographic ordering is required
Numerical methods for solving initial value problems on the Infinity Computer
New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity
Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations
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