Skip to main content
SpringerLink
Log in

Multifractal Characterization of Butterfly Wings Scales

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

A lot of insect families have physical structures created by evolution for coloration. These structures are a source of ideas for new bio-inspired materials. The aim of this study was to quantitatively characterize the micromorphology of butterfly wings scales using atomic force microscopy and multifractal analysis. Two types of butterflies, Euploea mulciber (“striped blue crow”) and Morpho didius (“giant blue morpho”), were studied. The three-dimensional (3D) surface texture of the butterfly wings scales was investigated focusing on two areas: where the perceived colors strongly depend on and where they do not depend on the viewing angle. The results highlight a correlation between the surface coloration and 3D surface microtexture of butterfly wings scales.

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

  • Adam JA (2003) Mathematics in nature: modeling patterns in the natural world. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Arman A, Ţălu Ş, Luna C, Ahmadpourian A, Naseri M, Molamohammadi M (2015) Micromorphology characterization of copper thin films by AFM and fractal analysis. J Mater Sci: Mater Electron 26:9630–9639. https://doi.org/10.1007/s10854-015-3628-5

    Article  Google Scholar 

  • Biro LP, Balint Z, Kertesz K, Vertesy Z, Mark GI, Horvath ZE, Balazs J, Mehn D, Kiricsi I, Lousse V, Vigneron J-P (2003) Role of photonic-crystal-type structures in the thermal regulation of a Lycaenid butterfly sister species pair. Phys Rev E 67(2):021907-1–021907-7

    Article  Google Scholar 

  • Castrejón-Pita AA, Sarmiento-Galán A, Castrejón-Pita JR, Castrejón-García R (2005) Fractal Dimension in Butterflies’ Wings: a novel approach to understanding wing patterns? J Math Biol 50(5):584–594

    Article  MathSciNet  Google Scholar 

  • Cloudsley-Thompson JL (1996) Biotic interactions in arid lands. Springer, Berlin

    Book  Google Scholar 

  • Dallaeva D, Tomanek P (2012) AFM study of structure influence on butterfly wings coloration. Adv Electr Electron Eng 10(2):120–124

    Google Scholar 

  • Dallaeva D, Ţălu Ş, Stach S, Škarvada P, Tománek P, Grmela L (2014) AFM imaging and fractal analysis of surface roughness of AlN epilayers on sapphire substrates. Appl Surf Sci 312:81–86. https://doi.org/10.1016/j.apsusc.2014.05.086

    Article  Google Scholar 

  • GraphPad InStat software, version 3.20 (GraphPad, San Diego, CA). http://www.graphpad.com/instat/instat.htm. Accessed 25 June 2018

  • Howse P, Thomas J (2014) Seeing butterflies: new perspectives on colour, patterns and mimicry. Publisher Papadakis, London

    Google Scholar 

  • Jovani R, Perez-Rodrıguez L, Mougeot F (2013) Fractal geometry for animal biometrics: a response to Kuhl and Burghardt. Trends Ecol Evol 28:499–500

    Article  Google Scholar 

  • Kemp DJ, Rutowski RL (2007) Condition dependence, quantitative genetics, and the potential signal content of iridescent ultraviolet butterfly coloration. Evolution 61:168–183

    Article  Google Scholar 

  • Kingsolver JG (1985) Butterfly engineering. Sci Am 253(2):106–113

    Article  Google Scholar 

  • Kishimoto S, Wang Q, Xie H, Zhao Y (2007) Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method. Appl Opt 46(28):7026–7034

    Article  Google Scholar 

  • Krushnamegh JK (2008) Ph.D. thesis: evolution of sex-limited mimicry in swallowtail butterflies, The University of Texas at Austin, USA. http://hdl.handle.net/2152/18078. Accessed 15 Jan 2018

  • Kwaśny W, Mikuła J (2012) Fractal and multifractal characteristics of the PVD and CVD coatings deposited onto compound tool ceramics. Arch Mater Sci Eng 56(1):37–44

    Google Scholar 

  • Liu F, Dong B, Zhao F, Hu X, Liu X, Zi J (2011) Ultranegative angular dispersion of diffraction in quasiordered biophotonic structures. Opt Express 19:7750–7755

    Article  Google Scholar 

  • Mandelbrot BB (1982) The fractal geometry of nature. Freeman W. H., San Francisco

    MATH  Google Scholar 

  • Martin S, Sami M (eds) (2011) Animal camouflage: mechanisms and function. Cambridge University Press, Cambridge

    Google Scholar 

  • Mika F, Matějková-Plšková J, Jiwajinda S, Dechkrong P, Shiojiri M (2012) Photonic crystal structure and coloration of wing scales of butterflies exhibiting selective wavelength iridescence. Materials 5:754–771

    Article  Google Scholar 

  • Nijhout HF (2010) Molecular and physiological basis of colour pattern formation. In: Casas J, Simpson SJ (eds) Advances in insect physiology: insect integument and colour, vol 38. Elsevier, Amsterdam, pp 219–265

    Chapter  Google Scholar 

  • Pauly M, Mitra NJ, Wallner J, Pottmann H, Guibas LJ (2008) Discovering structural regularity in 3D geometry. J ACM Trans Graph 27(3):43. https://doi.org/10.1145/1360612.1360642

    Article  Google Scholar 

  • Perez-Rodriguez L, Jovani R, Mougeot F (2013) Fractal geometry of a complex plumage trait reveals bird’s quality. Proc R Soc B Biol Sci 280:1755

    Article  Google Scholar 

  • Punnett RC (1915) Mimicry in butterflies. Cambridge University Press, Cambridge

    Google Scholar 

  • Ramazanov S, Ţălu Ş, Sobola D, Stach S, Ramazanov G (2015) Epitaxy of silicon carbide on silicon: micromorphological analysis of growth surface evolution. Superlattices Microstruct 86:395–402. https://doi.org/10.1016/j.spmi.2015.08.007

    Article  Google Scholar 

  • Reed RD, Serfas MS (2004) Butterfly wing pattern evolution is associated with changes in a notch/distal-less temporal pattern formation process. Curr Biol 14:1159–1166. https://doi.org/10.1016/j.cub.2004.06.046

    Article  Google Scholar 

  • Seckbach J (ed) (2002) Symbiosis: mechanisms and model systems. Springer, Dordrecht

    Google Scholar 

  • Sobola D, Țălu Ș, Sadovský P, Tománek P (2015) Application of AFM measurement and fractal analysis to study the surface of natural optical structures. Progress in applied surface, interface and thin film science 2015 (SURFINT), 23–26 November 2015, Florence, Italy. Comenus University, Bratislava. ISBN: 978-80-223-3975-9. Book: Extended abstract book of conference progress in applied surface, interface and thin film science 2015, pp 127–128

  • Ţălu Ş (2012a) Mathematical methods used in monofractal and multifractal analysis for the processing of biological and medical data and images. Anim Biol Anim Husb 4(1):1–4

    Google Scholar 

  • Ţălu Ş (2012b) Texture analysis methods for the characterisation of biological and medical images. Extreme Life Biospeology Astrobiol 4(1):8–12

    Google Scholar 

  • Ţălu Ş (2013) Characterization of surface roughness of unworn hydrogel contact lenses at a nanometric scale using methods of modern metrology. Polym Eng Sci 53(10):2141–2150. https://doi.org/10.1002/pen.23481

    Article  Google Scholar 

  • Ţălu Ş (2015) Micro and nanoscale characterization of three dimensional surfaces: basics and applications. Napoca Star Publishing House, Cluj-Napoca

    Google Scholar 

  • Ţălu Ş, Giovanzana S (2012) Image analysis of the normal human retinal vasculature using fractal geometry. Hum Vet Med Bioflux 4(1):14–18

    MATH  Google Scholar 

  • Ţălu Ş, Stach S (2014) Multifractal characterization of unworn hydrogel contact lens surfaces. Polym Eng Sci 54(5):1066–1080. https://doi.org/10.1002/pen.23650

    Article  Google Scholar 

  • Ţălu Ş, Stach S, Mendez A, Trejo G, Ţălu M (2013) Multifractal characterization of nanostructure surfaces of electrodeposited Ni–P coatings. J Electrochem Soc 161(1):D44–D47. https://doi.org/10.1149/2.039401jes

    Article  Google Scholar 

  • Ţălu Ş, Ghazai AJ, Stach S, Hassan A, Hassan Z, Ţălu M (2014a) Characterization of surface roughness of Pt Schottky contacts on quaternary n-Al0.08In0.08Ga0.84N thin film assessed by atomic force microscopy and fractal analysis. J Mater Sci: Mater Electron 25(1):466–477

    Google Scholar 

  • Ţălu Ş, Stach S, Zaharieva J, Milanova M, Todorovsky D, Giovanzana S (2014b) Surface roughness characterization of poly(methylmethacrylate) films with immobilized Eu(III) β-diketonates by fractal analysis. Int J Polym Anal Charact 19(5):404–421. https://doi.org/10.1080/1023666X.2014.904149

    Article  Google Scholar 

  • Ţălu Ş, Stach S, Mahajan A, Pathak D, Wagner T, Kumar A, Bedi RK (2014c) Multifractal analysis of drop-casted copper (II) tetrasulfophthalocyanine film surfaces on the indium tin oxide substrates. Surf Interface Anal 46(4):393–398. https://doi.org/10.1002/sia.5492

    Article  Google Scholar 

  • Ţălu Ş, Stach S, Mahajan A, Pathak D, Wagner T, Kumar A, Bedi RK, Ţălu M (2014d) Multifractal characterization of water soluble copper phthalocyanine based films surfaces. Electron Mater Lett 10(4):719–730

    Article  Google Scholar 

  • Ţălu Ş, Stach S, Valedbagi S, Bavadi R, Elahi SM, Ţălu M (2015) Multifractal characteristics of titanium nitride thin films. Mater Sci Pol 33(3):541–548. https://doi.org/10.1515/msp-2015-0086

    Article  Google Scholar 

  • Ţălu Ş, Bramowicz M, Kulesza S, Ghaderi A, Dalouji V, Solaymani S, Fathi Kenari M, Ghoranneviss M (2016) Fractal features and surface micromorphology of diamond nanocrystals. J Microsc 264:143–152. https://doi.org/10.1111/jmi.12422

    Article  Google Scholar 

  • Turner JRG (1977) Butterfly mimicry: the genetical evolution of an adaptation. Evol Biol 10:163–206

    Google Scholar 

  • Xie Yubing (2012) The nanobiotechnology handbook. CRC Press, Boca Raton

    Book  Google Scholar 

Download references

Acknowledgements

Research described in the paper was financially supported by the Ministry of Education, Youth and Sports of the Czech Republic under the Project CEITEC 2020 (LQ1601), by the National Sustainability Program under Grant LO1401. For the research, infrastructure of the SIX Center was used. Part of the work was carried out with the support of CEITEC Nano Research Infrastructure (ID LM2015041, MEYS CR, 2016–2019), CEITEC Brno University of Technology.

Author information

Authors and Affiliations

  1. The Technical University of Cluj-Napoca, The Directorate of Research, Development and Innovation Management (DMCDI), Constantin Daicoviciu Street, No. 15, 400020, Cluj-Napoca, Cluj County, Romania

    Ştefan Ţălu

  2. Institute of Continuous Media Mechanics UB RAS, 1 Korolev St., Perm, Russia, 614013

    Ilya A. Morozov

  3. Faculty of Electrical Engineering and Communication, Physics Department, Brno University of Technology, Technická 8, 616 00, Brno, Czech Republic

    Dinara Sobola & Pavel Škarvada

Authors
  1. Ştefan Ţălu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilya A. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dinara Sobola
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pavel Škarvada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dinara Sobola.

Ethics declarations

Conflict of interest

The authors claim to have no financial interest, either directly or indirectly, in the products or information listed in the article. The authors alone are responsible for the content and writing of the paper.

Appendix

Appendix

The values of multifractal singularity spectrum f(α) for all groups are listed in Tables 1, 2, 3, 4 and 6.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ţălu, Ş., Morozov, I.A., Sobola, D. et al. Multifractal Characterization of Butterfly Wings Scales. Bull Math Biol 80, 2856–2870 (2018). https://doi.org/10.1007/s11538-018-0490-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-018-0490-7

Keywords

Search

Navigation