Skip to main content
SpringerLink
Log in

Are periodicity and symmetry the properties of a discrete space? (On one paradox of cellular automata)

  • Original Research
  • Published:
Structural Chemistry Aims and scope Submit manuscript

Is The God a mathematician?

M. Livio, The Golden Ratio

Mathematics is the language that all exact sciences use.

N.I. Lobachevsky

Abstract

We consider a three-color cellular automaton (CA) that acts on a square lattice. In ternary numeral system, the CA has the number 111010011111110111011102010, which corresponds to 3704707887996 in the decimal system. With the finite size of the input row, CA-3704707887996 generates aperiodic structures with repeating self-similar patterns. In order to investigate the formation of periodic patterns by the CA-3704707887996, the input row was used that represents a periodic structure containing variable number of cells. The behavior of the CA-3704707887996 work for the input rows with periods up to 21 cells was studied. All generated patterns were analyzed from the viewpoint of their periodicity along the vertical direction. It was found out that the behavior of the system depending upon the length (l input) and structure of the periodic input row is strongly nonlinear. The complexity C of a pattern considered as a size of a unit cell increases exponentially with the increasing l input value and can be described by the function C = 1.8986 exp [0.3334 l input], with the correlation coefficient R 2 = 0.8967. As a rule, initially the CA generates a metastable aperiodic structure and then adopts a stable regime of generation of a stable periodic pattern. The abstract properties of the CA-3704707887996 have several important consequences for structural chemistry: (1) relatively simple schemes of local interactions of particles may result in the formation of very complex structures depending upon the initial conditions and abstract properties of the interactions; (2) symbolic complexity of the generated patterns increases exponentially depending upon the structure of initial conditions; (3) under certain initial conditions, explosive fluctuations of complexity are possible that lead to the formation of giant superstructures with extremely large periods; and (4) under certain conditions, the border between aperiodic and periodic structures virtually disappears.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Pauling L (1939) The nature of the chemical bond. Cornell University Press, Ithaka, New York

    Google Scholar 

  2. Mackay AL (1981) De nive quinquangula - on the pentagonal snowflake. Sov Phys Crystallogr 26:910–919

    Google Scholar 

  3. Weinstein BK (1979) Modern Crystallography, vol 1. Moscow, Nauka (in Russian)

    Google Scholar 

  4. Eliseev EN, Sachkov YuV, Belov NV (1982) The flows of ideas and the regularities in the development of natural sciences. Leningrad, Nauka (in Russian)

    Google Scholar 

  5. Borisov SV, Magarill SA, Pervukhina NV (2012) Crystallographic analysis of atomic structures and the phenomenological mechanism of crystallization. J Struct Chem 53(Suppl 1):55–62

    Article  CAS  Google Scholar 

  6. Shevchenko VY, Krivovichev SV (2008) Where are genes in paulingite? Mathematical principles of formation of inorganic materials on the atomic level. Struct Chem 19:571–577

    Article  CAS  Google Scholar 

  7. Shevchenko VY, Krivovichev SV, Mackay AL (2010) Cellular automata and local order in the structural chemistry of the lovozerite group minerals. Glass Phys Chem 36:1–9

    Article  CAS  Google Scholar 

  8. Shevchenko VY, Krivovichev SV, Tananaev IG, Myasoedov BF (2013) Cellular automata as models of inorganic structures self-assembly. Glass Phys Chem 39:1–10

    Article  CAS  Google Scholar 

  9. Mackay AL (1976) Crystal symmetry. Phys Bull 1976:495–497

    Article  Google Scholar 

  10. Mackay AL (1984) Descriptors for complex inorganic structures. Croat Chem Acta 57:725–736

    CAS  Google Scholar 

  11. Mackay AL (1986) Generalised crystallography. Comp Math Appl 12B:21–37

    Article  Google Scholar 

  12. Mackay AL (1995) Generalised crystallography. J Mol Struct (Theochem) 336:293–303

    Article  CAS  Google Scholar 

  13. Mackay AL (2002) Generalized crystallography. Struct Chem 13:217–222

    Google Scholar 

  14. Weber T, Dshemuchadse J, Kobas M, Conrad M, Harbrecht B, Steurer W (2009) Large, larger, largest—a family of cluster-based tantalum copper aluminides with giant unit cells. I. Structure solution and refinement. Acta Crystallogr B65:308–317

    Article  Google Scholar 

  15. Conrad M, Harbrecht B, Weber T, Jung DY, Steurer W (2009) Large, larger, largest—a family of cluster-based tantalum copper aluminides with giant unit cells. II. The cluster structure. Acta Crystallogr B65:318–325

    Article  Google Scholar 

  16. Krivovichev SV (2014) Which inorganic structures are the most complex? Angew Chem Int Ed 53:654–661

    Article  CAS  Google Scholar 

  17. Krivovichev SV (2004) Crystal structures and cellular automata. Acta Crystallogr A 60:257–262

    Article  Google Scholar 

  18. Toffoli T, Margolus N (1987) Cellular automata machines. MIT Press, Boston

    Google Scholar 

  19. Ilachinski A (2001) Cellular automata: a discrete universe. World Scientific, Singapore

    Book  Google Scholar 

  20. Wolfram S (2002) A new kind of science. Wolfram Media Inc, Urbana

    Google Scholar 

  21. Steurer W (2011) Measures of complexity. Acta Crystallogr A67:C184

    Article  Google Scholar 

  22. Krivovichev SV (2013) Structural complexity of minerals: information storage and processing in the mineral world. Miner Mag 77:275–326

    Article  CAS  Google Scholar 

Download references

Acknowledgments

We are grateful to anonymous reviewer for the insightful and detailed comments on the manuscript and to Alan L. Mackay for many years of inspirational scientific discussions.

Author information

Authors and Affiliations

  1. Institute of Silicate Chemistry, Russian Academy of Sciences, Makarova Emb. 6, St. Petersburg, Russia, 199034

    Vladimir Ya Shevchenko & Sergey V. Krivovichev

  2. Department of Crystallography, Institute of Earth Sciences, Saint-Petersburg State University, University Emb. 7/9, St. Petersburg, Russia, 199034

    Sergey V. Krivovichev

Authors
  1. Vladimir Ya Shevchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey V. Krivovichev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey V. Krivovichev.

Additional information

Dedicated to A.L. Mackay on the occasion of his 90th birthday.

Parts of this article are based upon the ideas expressed by Alan L. Mackay in private correspondence to VYaS.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shevchenko, V.Y., Krivovichev, S.V. Are periodicity and symmetry the properties of a discrete space? (On one paradox of cellular automata). Struct Chem 28, 45–50 (2017). https://doi.org/10.1007/s11224-016-0844-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11224-016-0844-4

Keywords

Search

Navigation