, Volume 132, Issue 1, pp 67–80 | Cite as

Birkhoff’s aesthetic ratio as a morphometric tool in the analysis of anatomical development of biological tree-like structures

  • Julio Gil
  • Miguel Gimeno
  • Jesús Laborda
  • Javier Nuviala
  • Rafael Lahoz-Beltra
Original Paper


Morphometric analyses are commonly performed on organisms detecting evolutionary changes in the shape of organisms. Tree-like structures are very common in Nature and have been attracted a lot of attention, for example, blood vessel system, plants and trees, neurone tissues, bronchial systems, etc. In 1928, the mathematician George David Birkhoff introduced a ratio for aesthetic measurement of an object. Birkhoff’s ratio has been successfully applied in painting, music, poetry, architecture or other fine arts. In the present paper, we analyse the morphology of tree-like structures by reducing shape to a numerical aesthetic value. Such value has been calculated as Birkhoff’s ratio between entropy and fractal dimension, showing how this aesthetic ratio is a better index than others traditional tree-like shape descriptors. An important novelty of our approach is that we represent a tree-like structure as a topological map emphasizing connections among branches instead the internodes length, node order, etc. The utility of this approach is tested studying samples of trees obtained with Monte Carlo method and the mammalian vascular system of eight representative animals, resulting new insights into the organization of the animal vascular system. We found that Birkhoff’s ratio has a better statistical performance and properties (robustness to isometry and the possibility to be transformed to a normal variable) than other morphometric indexes designed for tree-like structures classically used. Our approach could be useful in the analysis of anatomical development of biological tree-like structures and in consequence it could be applied in structural, functional and comparative biology as well as a tool for comparing such structures in evolutionary biology.


Birkhoff’s aesthetic ratio Morphology Shape Morphometric analysis 



We wish to thank Cormac de Brun for reading the manuscript and perfecting the English. R. Lahoz-Beltra was supported by the grant ‘Profesores UCM en el Extranjero 2008’ and by the Laboratorio de Bioinformatica (UCM).

Supplementary material

435_2012_171_MOESM1_ESM.doc (172 kb)
Supplementary material 1 (DOC 172 kb)


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Julio Gil
    • 1
  • Miguel Gimeno
    • 1
  • Jesús Laborda
    • 1
  • Javier Nuviala
    • 1
  • Rafael Lahoz-Beltra
    • 2
  1. 1.Department of Anatomy, Embryology and Animal Genetics, Faculty of Veterinary MedicineUniversity of ZaragozaZaragozaSpain
  2. 2.Department of Applied Mathematics (Biomathematics), Faculty of Biological SciencesComplutense University of MadridMadridSpain

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