Self-similarity in Physics

Emmanuel  T. Rodulfo,

Self-similarity in Physics

Abstract:

A brief account of self-similarity manifest in certain physical phenomena is presented. In particular, the quantum harmonic oscillator is shown to be representable in terms of a self-similar matrix that contains the golden ratio as a special case.

References:

1. Euclid, Elements, Book 6, Definition 3.

2. Mandelbrot, Benoit B. (5 May 1967). "How long is the coast of Britain? Statistical self-similarity and fractional dimension". Science. New Series. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. PDF

3. Roger Penrose (2006). "Before the Big Bang: An Outrageous New Perspective and its Implications for Particle Physics" (PDF). Proceedings of the EPAC 2006, Edinburgh, Scotland: 2759–2762.

4. Gurzadyan, VG; Penrose, R (2013). "On CCC-predicted concentric low-variance circles in the CMB sky". Eur. Phys. J. Plus. 128 (2): 22. arXiv:1302.5162.Bibcode:2013EPJP..128...22G. doi:10.1140/epjp/i2013-13022-4.

5. Hassan M. K., Hassan M. Z., Pavel N. I. (2011). "Dynamic scaling, data-collapse and Self-similarity in Barabasi-Albert networks". J. Phys. A: Math. Theor. 44 (17):175101. arXiv:1101.4730. Bibcode:2011JPhA...44q5101K. doi:10.1088/1751-8113/44/17/175101.

6. Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-drivenaggregation". Phys. Rev. E. 79 (2):021406. arXiv:0901.2761. Bibcode:2009PhRvE..79b1406H. doi:10.1103/physreve.79.021406. PMID 19391746.

7. Dayeen F. R., Hassan M. K. (2016). "Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice". Chaos, Solitons & Fractals. 91:228. arXiv:1409.7928. Bibcode:2016CSF....91..228D. doi:10.1016/j.chaos.2016.06.006.

8. Koshy, T., 2001. Fibonacci and Lucas Numbers with Applications, (John Wiley and Sons). Binet representation of Fibonacci numbers.[Koshy,2001],[Oktay K. Pashaev, Sengul Nalci.arXiv:1107.4389 [math.QA].