Lissajous curves with a finite sum of prime number frequencies
LLissajous curves with a finite sum of prime number frequencies
Imre F. Barna and L. M´aty´as Wigner Research Center for Physics,Konkoly-Thege Mikl´os ´ut 29 - 33,1121 Budapest, HungaryEmail: [email protected] Department of Bioengineering,Faculty of Economics,Socio-Human Sciences and Engineering,Sapientia Hungarian University of Transylvania Libert˘atii sq. 1,530104 Miercurea Ciuc, Romania (Dated: September 4, 2020)
Abstract
The Ulam spiral inspired us to calculate and present Lissajous curves where the orthogonallyadded functions are a finite sum of sinus and cosines functions with consecutive prime numberfrequencies.
PACS numbers: a r X i v : . [ m a t h . G M ] S e p We may say that prime numbers fascinates mankind more than two thousand years. Thescientific literature of number theory - which in great part deals with prime - is enormousand fills libraries. Number theory is not our field of interest at all, so it is not our duty togive any kind of overview of the field, therefore we just mention two works about primes[1, 2]. (Our scientific interest is laser-matter interaction [3] and self-similar solutions of non-linear partial differential equations of flow systems [4].) We just would like to show a tinyidea about primes which might be interesting to the experts. There are two starting pointswhich gave as the idea. The first is the Ulam spiral which was found by Stanislaw Ulam in1967 [5]. The left figure in Fig. (1) basically shows how the spiral is defined, the middlefigure shows how the prime numbers are distributed among the first 400 natural numbers,and the last right figure presents the prime distribution on a much lager scales. The primesare represented with dark dots, spot and with short lines. It is evident, that there is anon-trivial correlation of primes even on large scales in this representation.Our second starting point is the definition of Lissajous (or Bowditch) curves [6] whichare a well-known object for physicists. The parametric formula of the curve reads x ( t ) = sin ( a · t + δ ) ,y ( t ) = cos ( b · t ) , (1)where ’a’, ’b’ are the relative frequencies and δ is the phase between the two oscillations.Figure 2 presents three classical Lissajous curves with various relative frequencies and phasesbetween the two trigonometric functions. The length of the corresponding parametric curveis defined as L = (cid:90) π (cid:112) ˙ x ( t ) + ˙ y ( t ) dt, (2)where prime means derivation with respect to the parameter t . With this formula it istrivial to get back the circumference of a unit circle. It is also clear that the length ofvarious Lissajous curves are proportional to π . Differential geometrical analysis helps toderive and study additional parameters of the curve. Let’s try to image the distribution ofprimes somehow with the help of the Lissajous curves.2 IG. 1: The Ulam spiral. The left figure is just the definition of the spiral, the middle figure showsthe prime distributions among the first 400 natural numbers, the right figure presents the largescale distribution of primes in the Ulam spiral.FIG. 2: Three classical Lissajous curves. The three parameter sets (from left to right) are ( a =1 , δ = π , b = 2), ( a = 3 , δ = π , b = 2) , and ( a = 3 , δ = π , b = 4) We applied the next parametric formula for the curve x ( t ) = N (cid:88) i =1 sin ( a i · t ) a i ,y ( t ) = N (cid:88) i =1 cos ( a i · t ) a i , (3)where a i s are the first N prime numbers. Figure 3 shows the Lissajous curves for N =100 , , - - - - - - - - - FIG. 3: Our Lissajous curves with different kind of finite Fourier sums with prime number fre-quencies. From left to right, the sum of the first 100, 1000 and 5000 primes were taken.
As second case, Figure 4 presents two curves where only the second neighbor primes areconsidered to the ’x’ and ’y’ coordinates such as 2 , ,
11 and 3 , ,
13. Note, the much quickerconvergence, it is not possible to see the differences between the two figures with naked eyes.We tried to modify Eq. (3) with additional logarithmic, square root or power law functionsof the argument of the sinus and cosines function to create much more internal structure ofthe curves at larger number of primes. Unfortunately in vain. This is the present endpointof our idea and analysis. (The presented calculations and figures were evaluated with Maple12.) It can happen that our toy model might give idea for such kind of further investigations. [1] H. Maier and W.P. Schleich,
Prime Numbers 101 , Wiley Interscience 2009.[2] T. Estermann,
Introduction to Modern Prime Number Theory , Cambridge the University Press,1952.[3] I.F. Barna, M.A. Pocsai and S. Varr´o, Eur. Phys. J. Appl. Phys. , 20101 (2018).[4] I.F. Barna, M.A. Pocsai and L. M´aty´as, Fluid. Dyn. Res. , 015515 (2020).[5] M. Stein and S.M. Ulam, ”An Observation on the Distribution of Primes.” American Mathe-matical Monthly , 43 (1967). - - - - - - - FIG. 4: Our Lissajous curves with different kind of finite Fourier sums with prime number fre-quencies. From left to right, the sum of the first 100 and 1000 second neighboring primes weretaken.[6] D. Lawrence,
A Cataloge of Special Plane Curves , (Page 178.) Dover Publication 1972., (Page 178.) Dover Publication 1972.