Morphospace: Measurement, Modeling, Mathematics, and Meaning

N. Khiripet; R. Viruchpintu; J. Maneewattanapluk; J. Spangenberg; J.R. Jungck

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 6, Issue: 2, page 54-81
  • ISSN: 0973-5348

Abstract

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Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).

How to cite

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Khiripet, N., et al. "Morphospace: Measurement, Modeling, Mathematics, and Meaning." Mathematical Modelling of Natural Phenomena 6.2 (2010): 54-81. <http://eudml.org/doc/197624>.

@article{Khiripet2010,
abstract = {Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).},
author = {Khiripet, N., Viruchpintu, R., Maneewattanapluk, J., Spangenberg, J., Jungck, J.R.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {plant architecture; morphospace; multivariate-multidimensional modeling; 3D tree visualization, education; allometry; Lindenmayer systems; graph grammars; fractals; computational geometry; convex hulls; creative commons; 3D tree visualization; education},
language = {eng},
month = {10},
number = {2},
pages = {54-81},
publisher = {EDP Sciences},
title = {Morphospace: Measurement, Modeling, Mathematics, and Meaning},
url = {http://eudml.org/doc/197624},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Khiripet, N.
AU - Viruchpintu, R.
AU - Maneewattanapluk, J.
AU - Spangenberg, J.
AU - Jungck, J.R.
TI - Morphospace: Measurement, Modeling, Mathematics, and Meaning
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/10//
PB - EDP Sciences
VL - 6
IS - 2
SP - 54
EP - 81
AB - Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).
LA - eng
KW - plant architecture; morphospace; multivariate-multidimensional modeling; 3D tree visualization, education; allometry; Lindenmayer systems; graph grammars; fractals; computational geometry; convex hulls; creative commons; 3D tree visualization; education
UR - http://eudml.org/doc/197624
ER -

References

top
  1. M. T. Allen, P. Prusinkiewicz, T. M. DeJong. Using L-systems for modeling source-sink interactions, architecture and physiology of growing tree: The L-Peach model. New Phytologist., 166, 2005, No. 3, 869–880. 
  2. R. Aratsui. Leonardo was wise: Trees conserve cross-sectional area despite vessel structure. J. Young Investigators, 1, 1998, 1-23. 
  3. J. K. Bailey, R. K. Bangert, J. A. Schweitzer, R. T. Trote. III, S. M. Shuster, T. G. Whitman. Fractal geometry is heritable in trees. Evolution, 58, 2004, No. 9, 2100–2102. 
  4. D. Barthélémy, Y. Caraglio. Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny. Annals of Botany, 99, 2007, No. 3, 375–407. 
  5. F. S. Berezovskava, G. P. Karev, O. S. Kisliuk., R. G. Khlebopros, Y. L. Tsel’niker. A fractal approach to computer-analytical modeling of tree crowns. Trees, 11, 1997, 323–327. 
  6. N. Bessonov, V. Volpert. Dynamic models of plant growth. Mathematics and Mathematical Modelling. Editions Publibook Universite, Paris, France, 2008.  
  7. L. Breiman. Reflections after refereeing papers for NIPS. In David Wolpert, Ed., The Mathematics of Generalization. Santa Fe Institute Studies in the Sciences of Complexity Volume XX. Addison-Wesley: Reading, MA, 1995.  
  8. J. L. Casti. Would-Be worlds: How simulation is changing the frontiers of science. John Wiley and Sons: New York, NY, 1997.  
  9. J. E. Cohen. Mathematics is biology’s next microscope, only better; Biology is mathematics next physics, only better. Public Library of Science Biology, 2 (2004), No. 12, 2017–2023.  
  10. O. Deussen, B. Lintermann. Digital design of nature: Computer generated plants and organics. New York: Springer-Verlag, 2005.  
  11. A. E. Dimond. Pressure and flow relations in vascular bundles of the tomato plant. Plant Physiology, 41, 1966, 119–131. 
  12. J. Elkins. The object stares back: On the nature of seeing. Harcourt Brace and Company: San Diego, CA, 1996.  
  13. B. J. Enquist, K. J. Niklas. Global allocation rules for patterns of biomass partitioning in seed plants. Science, 295, 2002, No. 5559, 1517–1520. 
  14. B. Enquist, J. Sperry, P. Reich, V. Savage. Combining theories for plant architecture, allometry, and traits to develop the next generation of scaling theory. NSF Emerging Frontiers Award #0742800. Start Date: December 1, 2007 - Expires: November 30, 2010 (Estimated); NSF Program: THEORETICAL BIOLOGY, 2007.  
  15. FSPM07. Fifth international workshop on functional structural plant models. Napier, New Zealand, 4–9 November, 2007.  
  16. J. Fisher, H. Honda. Branch geometry and effective leaf area: A study of Terminalia-branching pattern: Theoretical ideas. American J. Botany, 66, 1979, 633–644. 
  17. C. Godin, O. Puech, F. Boudon, H. Sinoquet. Space occupation by tree crowns obey fractal laws: Evidence from 3D digitized plants. In Godin, C. Hanan, J. Kurth, W. Lacointe, A. Takenaka, A. Prusinkiewicz, P. DeJong, T. M. Beveridge, C. Andrieu, B., Eds. Proceedings of Proceedings of the 4th International Workshop on Functional-Structural Plant Models. Montpellier, 2004, 79–83. Proceedings of the 4th International Workshop on Functional-Structural Plant Models, 7-11/06/2004, Montpellier.  
  18. C. Godin, E. Costes, H. Sinoquet. A method for describing plant architecture which integrates topology and geometry. Annals of Botany, 84, 2004, 343–357. 
  19. Y. Grossman, T. De Jong, Peach. In Jungck , Eds., The BioQUEST Library, Academic Press: San Diego, 2003.  
  20. M. Hiratsuka, T. Toma, M. Yamada, I. Heriansyah, Y. Morikawa. A general allometric equation for estimating biomass in Acacia mangium plantations. In Proceedings of the 2003 International Conference on Tropical Forests and Climate Change, University of the Philippines Los Banos, Laguna, Philippines 2003, 212–218.  
  21. P. Hogeweg, B. Hesper. A model study on biomorphological description. Pattern Recognition, 6, 1974, 165–179. 
  22. H. Honda. Description of the form of trees by the parameters of a tree-like body: Effects of the branching angle and branch length on the shape of the tree-like body. J. Theoretical Biology, 31, 1971, 331–338. 
  23. H. S. Horn. Adaptive geometry of trees. Princeton University Press: Princeton, NY, 1971.  
  24. J. R. Jungck. Ten equations that changed biology. Bioscene: Journal of College Biology Teaching, 23, 1997, No. 1, 11–36. 
  25. J. R. Jungck, E. D. Stanley, S. J. Everse, V. Vaughan, Eds. The BioQUEST Library. Academic Press: North Reading, MA, 2003.  
  26. J. R. Jungck. The biological ESTEEM collection: Excel simulations and tools for exploratory, experiential mathematics. BioQUEST Notes, 14, 2005, No. 1, 1–2. 
  27. J. R. Jungck. Challenges, connections, complexities: Educating for collaboration. Chapter in Lynn Steen, Ed., Math & Bio2010: Linking Undergraduate Disciplines. The Mathematical Association of America: Washington, DC., 2005, 1–12.  
  28. J. R. Jungck. Fostering figuring and fascination: Engaging learners through alternative aesthetics. In J. R. Jungck, Er. Mazur, Wi. Schmidt, T. Schuller, Eds., Criticism of Contemporary Issues: Education. Serralves Museum of Art, Fundacao Serralves: Porto, Portugal, 2008, 19–54 and 93–126.  
  29. C. E. Knapp. Review of Last child in the woods: Saving our children from nature-deficit disorder  
  30. A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. J. Theoretical Biology, 30, 1971, 455–484. 
  31. A. Lindenmayer. Developmental systems and languages in their biological context. In G. T. Herman & G. Rozenberg, Eds., Developmental systems and languages. Amsterdam: North-Holland Publishing, 1975, 1–40. 0-7204-2806-8  
  32. A. Lindenmayer. Growing fractals and plants. In Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe, Eds. Chaos and Fractals, Second Edition. New Frontiers of Science. Springer: New York, 2004, 329–376. ISBN 978-0-387-20229-7  
  33. R. Louv. A brief history of the children &amp; nature movement. J. Sci. Educ. Technol, 17, 2008, 217–218. 
  34. B. Mandelbrot. Fractals: Form, chance and dimension. W. H. Freeman and Co.: San Francisco, 1977.  
  35. K. A. McCulloh, J. S. Sperry, R. F. Adler. Water transport in plants obey Murray’s law. Nature, 401, 2003, 939. 
  36. K. A. McCulloh, J. S. Sperry, R. F. Adler. Murray’s law and the hydraulic vs mechanical functioning of wood. Functional Ecology, 18, 2004, 931. 
  37. M. Mesterton-Gibbons, M. J. Childress. Constraints on reciprocity for non-sessile organisms. Bulletin of Mathematical Biology, 58, 1996, No. 5, 861–875. 
  38. National Research Council. Bio 2010: Transforming undergraduate education for future research biology. National Academies Press: Washington, DC, 2003.  
  39. K. J. Niklas. Computer simulations of branching-patterns and their implications on the evolution of plants. In L. J. Gross, R. M. Miura, Eds. Some Mathematical Questions in Biology: Plant Biology, Lectures on Mathematics in the Life Sciences, 18, 1986, 1–50 (American Mathematical Society).  
  40. K. J. Niklas. Evolutionary walks through a land plant morphoscape. J. Experimental Botany, 50, 1999, No. 330, 39–52. 
  41. K. J. Niklas, E. D. Cobb. Evidence for “diminishing returns” from the scaling of stem diameter and specific leaf area. Am. J. Botany, 95, 2008, 549–557. 
  42. K. Ogle, S. W. Pacala. A modeling framework for inferring tree growth and allocation from physiological, morphological and allometric traits. Tree Physiol., 29, 2009, 587–605. 
  43. R. W. Pearcy, H. Muraoka, F. Valladares. Crown architecture in sun and shade environments: Assessing function and trade-offs with a three-dimensional simulation model. New Phytologist, 166, 2005, No. 3, 791–800. 
  44. C. A. Price, B. J. Enquist, V. M. Savage. A general model for allometric covariation in botanical form and function. PNAS, 104, 2007, 13204–13209. 
  45. P. Przemyslaw, J. Hanan. Lindenmayer systems, fractals, and plants. Lecture Notes in Biomathematics, volume 79. Springer-Verlag, New York, 1989.  
  46. P. Prusinkiewicz, A. Lindenmayer. The algorithmic beauty of plants. New York: Springer-Verlag, 1990. (Available on-line in high quality pdf at ()).  URIhttp://algorithmicbotany.org/papers/#abop
  47. P. Prusinkiewicz, Y. Erasmus, B. Lane, L. D. Harder, E. Coen. Evolution and development of inflorescence architectures. Science, 316, 2007, No. 5830, 1452–1456. 
  48. D. M. Raup. Computer as aid in describing form in gastropod shells. Science, 138, 1962, 150–152. 
  49. P. de Reffye. Computer simulation of plant growth. In Clifford A. Pickover and Stuart K. Tewksbury, Eds. Frontiers of scientific visualization. John Wiley & Sons, Inc.: New York, NY, USA, 1994, 145–179.  
  50. D. Robinson. Implications of a large global root biomass for carbon sink estimates and for soil carbon dynamics. Proc. R. Soc. B, 274, 2007, 2753–2759. 
  51. A. Szalay, J. Gray. 2020 computing: Science in an exponential world. Nature, 440, 2006, 413–414. 
  52. R. Sattler, R. Rutishauser. The fundamental relevance of morphology and morphogenesis to plant research. Annals of Botany, 80, 1997, 571–582. 
  53. Seits, I.S. Antonova. Applying multidimensional statistics to tree architecture analysis. In Godin, C. Hanan, J. Kurth, W. Lacointe, A. Takenaka, A. Prusinkiewicz, P. DeJong, T. M.Beveridge, C. Andrieu, B., Eds. Proceedings of Proceedings of the 4th International Workshop on Functional-Structural Plant Models. Montpellier, 2004, 70–74. Proceedings of the 4th International Workshop on Functional-Structural Plant Models, 7–11/06/2004, Montpellier.  
  54. I. Shlyakhter, M. Rozenoer, J. Dorsey, S. Teller. Reconstructing 3D tree models from instrumented photographs. IEEE Computer Graphics and Animations, xx, 2001, 1–9.  
  55. S. Wegrzyn, J.-C. Gille, P. Vidal. Developmental systems: At the crossroads of system theory, computer science, and genetic engineering. Springer-Verlag: New York, 1990.  
  56. A. Weisstein, Ed. The Biological ESTEEM project (Excel simulations and tools for exploratory, experiential mathematics), 2009. (< >).  URIhttp//bioquest.org/esteem
  57. D. Willis, The sand dollar and the slide rule. Addison-Wesley Publishing Co.: North Reading, MA, 232 pp, 1995. ISBN 0-201-63275-6 23 ( 27 postpaid) from 
  58. W. Wimsatt. False models as means to truer theories. Chapter 6 in his book:Re-engineering philosophy for limited beings: Piecewise approximations to reality. Harvard University Press: Cambridge, MA, 2007.  
  59. M. S. Zens, C. O. Webb. Sizing up the shape of life. Science, 295, 2002, No. 5559, 1475. 

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