Authors: Kristoffer Jensen
Addresses: AD:MT, Aalborg University Esbjerg, Niels Bohrsvej 8, 6700 Esbjerg, Denmark
Abstract: The Farey sequences can be used to create the Euler totient function, by identifying the fractions for number n that did not occur in all Farey sequences up to n − 1. This function creates, when divided by n − 1, what is here called the primety measure, which is a measure of how close to being a prime number n is. The quality of a number sequence can be determined using the primety, which is further generalised to real numbers through the use of real numbered Farey sequences. Ambiguous and interesting results are obtained in this process; for instance it seems that all integers approach full primety when investigated in the real domain. The corresponding numerical sequences are furthermore shown to have interesting artistic properties. In particular the fractal behaviour is interesting. To further investigate fractal ambiguity, common fractals are extended with random elements, which render softer or more varied shapes.
Keywords: Farey sequences; Euler totient function; primety measure; self-similarity; fractals; aesthetics; quality of numbers; number sequences; fractal ambiguity.
International Journal of Arts and Technology, 2014 Vol.7 No.2/3, pp.247 - 260
Received: 02 Apr 2012
Accepted: 24 Sep 2012
Published online: 08 May 2014 *