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The bulletin of mathematical biophysics

, Volume 17, Issue 2, pp 111–126 | Cite as

Some theorems in topology and a possible biological implication

  • N. Rashevsky
Article

Abstract

In a previous paper (Bull. Math. Biophysics,16, 317–48, 1954) a transformationT of one graph into another was suggested, which may describe the relations between organisms of different complexity. In this paper some topological properties of the transformationT are studied. It is shown that the fundamental group of the transformed graph is homomorph to the fundamental group of the original graph. An expression is derived for the number of points in a point base of the transformed graph in terms of the number of points of the point base of the original when the point base of the latter consists only of residual points, and it is shown that the ratio of the number of points of the point base to the total number of points of the graph is in that case greater in the transformed graph than in the original. A combinatorial problem arising in connection with the transformationT is solved by deriving the number of possible ways in whichn-n i indistinguishable elements may be arranged inn i classes, permitting some of then i classes to be empty.

The possible biological meaning of the increased ratio of the number of points of the point base to the total number of points of the graph is discussed. It is suggested that it may be interpreted as a decrease of regenerating ability with increase of differentiation of the organism. Those considerations suggest the possibility of deriving some general biological laws from the consideration of the properties of the transformation only, regardless of the choice of the primordial graph.

Keywords

Point Base Fundamental Group Identity Element Representative Point Closed Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

  1. König, Denes. 1936.Theorie der Endlichen und Unendlichen Graphen. Leipzig: Akademische Verlagsgesellschaft.Google Scholar
  2. Netto, Eugen. 1901.Lehrbuch der Kombinatorik. Leipzig: B. G. Teubner.MATHGoogle Scholar
  3. Pontrjagin, Leon. 1946.Topological Groups. Princeton: Princeton University Press.Google Scholar
  4. Rashevsky, N. 1954. “Topology and Life: In Search of General Mathematical Principles in Biology and Sociology.”Bull. Math. Biophysics,16, 317–48.MathSciNetGoogle Scholar
  5. — 1955. “A combinatorial Problem in Biological Topology.”Ibid.,17, 45–50.MathSciNetGoogle Scholar
  6. Seifert, H. and W. Threlfall. 1934.Lehrbuch der Topologie. Leipzig-Berlin: B. G. Teubner.MATHGoogle Scholar

Copyright information

© The University of Chicago Press 1955

Authors and Affiliations

  • N. Rashevsky
    • 1
  1. 1.Committee on Mathematical BiologyThe University of ChicagoChicagoUSA

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