An application of Grossone to the study of a family of tilings of the hyperbolic plane

Abstract

In this paper, we look at the improvement of our knowledge on a family of tilings of the hyperbolic plane which is brought in by the use of Sergeyev's numeral system based on grossone, see [17,18,19]. It appears that the information we can get by using this new numeral system depends on the way we look at the tilings. The ways are significantly different but they confirm some results which were obtained in the traditional but constructive frame and allow us to obtain an additional precision with respect to this information.presented in this paper are slightly different from those briefly presented at the International Workshop Infinite and Infinitesimal in Mathematics, Computing and Natural Sciences held at Cetraro, Italy, in May 2010, see [13]. They are in some sense a more precise version of what was given in [13]. In Section 6 we indicate a few possible continuations. The new numeral systemIn papers [17,18,19], Yaroslav Sergeyev gives the main arguments in favour of the new numeral system he founded, allowing to obtain more precise results on infinite sets that what was obtained previously.We can sum up the properties of the system as follows.We distinguish the objects of our study from the tools we use to observe them. These three parts of the knowledge process have to be more clearly distinguished as they were traditionally in mathematics, contrarily to other domains of science, as physics and natural sciences where this distinction is clearly observed. This is the content of Postulate 2 in the quoted papers. It is an important issue for mathematics where the distinction between an observer and what is observed is very often forgotten. In particular, not enough attention is paid to subjectivity of the observer and the relative validity of his/her observations. The latter are very dependent of cultural elements, especially the language used by the observer to describe what he/she sees.We are interested in the properties of the objects, some of them being possibly infinite or infinitesimal, but operations on the objects, performed by a human being or a machine, necessarily deal with finitely many of them and only finitely many operations can be applied within the frame of an argument. This is the content of Postulate 1 in the quoted papers.At last and not the least, we consider that the principle The part is less than the whole has to be applied to all numbers, finite, infinite or infinitesimal, and also to all sets and processes, whether finite or infinite. This is the content of Postulate 3 of [17,18,19].On the basis of these principles, Yaroslav Sergeyev introduced a new numeral system in order to be able to write down infinite numbers. To this aim, an infinite natural number is introduced, grossone, denote by 1 , which is the number of elements of the set of positive integers. This number satisfies the following three properties which are axioms of the system: − for any finite natural number n, n < 1 .− we have 0. 1 = 1 .0 = 0, 1 − 1 = 0, 1 1 = 1, 1 0 = 1, 1 1 = 1 and 0 1 = 0. − let IN k,n be the set of p...

Reprint of Infinity computations in cellular automaton forest-fire model

“…A number of applications of the new approach can be found in [7,8,16,17,22,23,25,[27][28][29][30][31][32]34,35]. We start by introducing three postulates that will fix our methodological positions (having a strong applied character) with respect to infinite and infinitesimal quantities and Mathematics, in general.…”

Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

“…(see [8,9,14,22,30,32,34,35,36,37,39,40,41,44,48,49]). It is important to emphasize that the new numeral system avoids situations of the type (5)- (7) providing results ensuring that if a is a numeral written in this system then for any a (i.e., a can be finite, infinite, or infinitesimal) it follows a + 1 > a.…”

“…The new methodology has been successfully applied for studying a number of applications: percolation (see [14,44]), Euclidean and hyperbolic geometry (see [22,30]), fractals (see [32,34,41,44]), numerical differentiation and optimization (see [8,35,39,49]), infinite series (see [36,40,48]), the first Hilbert problem (see [37]), and cellular automata (see [9]). …”

Infinigons of the hyperbolic plane and grossone

“…However, using a trick we explained in [5], we can handle the situation no matter which the parity of λ is. The idea consists in replacing the regions we considered when q is even by new regions to which we now turn.…”

“…Let R 0 be the mid-point of the side ω 0 which abuts τ 0 at W 0 and which makes an angle ϕ 0 = h π q with τ 0 , the angles ϑ 0 and ϕ 0 being on different sides of τ 0 . From the construction, the isosceles triangles M 0 V 1 0 N 0 and N 0 W 0 R 0 are equal so that the points M 0 , N 0 and R 0 lie on a same ray u issued from M 0 whose supporting line is called a h-midpoint line in [5]. A similar h-mid-point line v can be drawn from the mid-point of s 1 which is symmetric to u with respect to the bisector β of the angle (ℓ, m).…”