Infinitesimal contact transformations and their applications to abstract growth patterns
Author:
Igor Frolow
Journal:
Quart. Appl. Math. 37 (1980), 389-399
MSC:
Primary 92A05; Secondary 51N05, 53A05, 58C28, 73P05
DOI:
https://doi.org/10.1090/qam/564730
MathSciNet review:
564730
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Abstract: Several abstract mathematical models of two-dimensional objects or “organisms” are introduced. The objects are abstractions, created by the combination of pattern-theoretic generators. They are deformed by what are shown to be infinitesimal contact transformations. Analytical methods are utilized to determine some of the resulting forms and a computer is programmed to display graphically the developing patterns.
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V. G. Boltianskii, Ogibaiushchaia, Gosudarstvennoe Izdatelstvo Fizico-Mathematicheskoy Literaturi, Moscow, 1961
C. Carathéodory, Calculus of variations and partial differential equations of the first order, Part I (translated by R. B. Dean and J. J. Brandstatter), Holden-Day, San Francisco, 1965
L. P. Eisenhart, Contact transformations, Annals Math. 2 series 30, 211–249 (1928)
I. Frolow, Abstract growth patterns, Ph.D. Thesis, Brown University, 1978
U. Grenander, Pattern synthesis—lectures in pattern theory, Vol. I, Springer-Verlag, New York, 1976
J. S. Huxley, Problems of relative growth, Methuen, London, 1932
A. Lindenmayer, Mathematical models for cellular interactions in development, Part I: filaments with one-sided inputs, J. Theor. Biol. 18, 280–299 (1968)
A. Lindenmayer, Mathematical models for cellular interactions in development, Part II: simple and branching filaments with two-sided inputs, J. Theor. Biol. 18, 300–315 (1968)
N. Rashevsky, Mathematical biophysics: physico-mathematical foundations of biology, Vol. I, 3rd ed., Dover, New York, 1960
R. Rosen, Morphogenesis, in Foundations of mathematical biology, Vol. II (R. Rosen, ed.), 1972, 1–77
M. S. Steinberg, Reconstruction of tissues by dissociated cells, in Mathematical models for cell rearrangement (G. D. Mostow, ed.), 1975, 82–99
R. Thom, Structural stability and morphogenesis (translated by D. H. Fowler), W. A. Benjamin, Reading, Massachusetts, 1975
D’Arcy Thompson, On growth and form, 2nd ed., Cambridge University Press, 1942
A. M. Turning, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London B237, 37–72 (1952)
L. Wolpert, Positional information and pattern formation, in Toward a theoretical biology, Vol. III (C. H. Waddington, ed.). 1970, 198–230
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Article copyright:
© Copyright 1980
American Mathematical Society
