Research output: Contribution to journal › Article › peer-review
Multidimensional generalization of phyllotaxis. / Lodkin, Andrei .
In: Cybernetics and Physics, Vol. 8, No. 3, 28.11.2019, p. 153-160.Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - Multidimensional generalization of phyllotaxis
AU - Lodkin, Andrei
N1 - Publisher Copyright: © Publication Year, publisher Name. All rights reserved.
PY - 2019/11/28
Y1 - 2019/11/28
N2 - 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.
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 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.
KW - филлотаксис, золотое сечение, диофантовы приближения, парус Клейна
KW - Diophantine approximation
KW - Golden number
KW - Klein sail
KW - Phyllotaxis
UR - http://www.scopus.com/inward/record.url?scp=85076189779&partnerID=8YFLogxK
UR - http://www.mendeley.com/research/multidimensional-generalization-phyllotaxis
U2 - https://doi.org/10.35470/2226-4116-2019-8-3-153-160
DO - https://doi.org/10.35470/2226-4116-2019-8-3-153-160
M3 - Article
VL - 8
SP - 153
EP - 160
JO - Cybernetics and Physics
JF - Cybernetics and Physics
SN - 2223-7038
IS - 3
ER -
ID: 49394343