Multidimensional generalization of phyllotaxis

Research output

Abstract

The regular spiral arrangement of various parts of biological objects (leaves, florets, etc.), known as phyllotaxis, could not find an explanation during several centuries. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar patterns were discovered in inorganic objects. After a series of computer models, it was only in the XXI century that the rigorous explanation of the appearance of
the golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parameter. The variation of this parameter leads to an interesting bifurcation diagram where the limiting object is the SL(2;Z)-orbit of the golden number on the segment [0,1].
We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns. A conjecture about the object that plays the role of the golden number is given.
Original languageEnglish
Pages (from-to)153-160
JournalCybernetics and Physics
Volume8
Issue number3
DOIs
Publication statusPublished - 28 Nov 2019

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Phyllotaxis
Orbits
Mathematical models
divergence
Divergence
leaves
mathematical models
Angle
diagrams
analogs
orbits
Computer Model
operators
Bifurcation Diagram
Arrangement
Leaves
Limiting
Orbit
Fixed point
Generalization

Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Multidimensional generalization of phyllotaxis",
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Multidimensional generalization of phyllotaxis. / Lodkin, Andrei .

In: Cybernetics and Physics, Vol. 8, No. 3, 28.11.2019, p. 153-160.

Research output

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AB - The regular spiral arrangement of various parts of biological objects (leaves, florets, etc.), known as phyllotaxis, could not find an explanation during several centuries. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar patterns were discovered in inorganic objects. After a series of computer models, it was only in the XXI century that the rigorous explanation of the appearance ofthe golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parameter. The variation of this parameter leads to an interesting bifurcation diagram where the limiting object is the SL(2;Z)-orbit of the golden number on the segment [0,1].We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns. A conjecture about the object that plays the role of the golden number is given.

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