Golden ratio and phyllotaxis, a clear mathematical link

被引：6

作者：

Bergeron, Francois ^{[1
]}

Reutenauer, Christophe ^{[1
]}

机构：

[1] Univ Quebec Montreal, Dept Math, Montreal, PQ, Canada

来源：

JOURNAL OF MATHEMATICAL BIOLOGY | 2019年 / 78卷 / 1-2期

基金：

加拿大自然科学与工程研究理事会;

关键词：

Modular group; Markoff approximation theory; Golden ratio; Phyllotaxis;

D O I：

10.1007/s00285-018-1265-3

中图分类号：

Q [生物科学];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

Exploiting Markoff's theory for rational approximations of real numbers, we explicitly link how hard it is to approximate a given number to an idealized notion of growth capacity for plants which we express as a modular invariant function depending on this number. Assuming that our growth capacity is biologically relevant, this allows us to explain in a satisfying mathematical way why the golden ratio occurs in nature.

引用

页码：1 / 19

页数：19

共 24 条

[1] MODEL OF CONTACT PRESSURE IN PHYLLOTAXIS [J].

ADLER, I .

JOURNAL OF THEORETICAL BIOLOGY, 1974, 45 (01) :1-79

[2]

Aigner M., 2013, Markov's Theorem and 100 Years of the Uniqueness Conjecture

[3] A dynamical system for plant pattern formation:: A rigorous analysis [J].

Atela, P ;

Golé, C ;

Hotton, S .

JOURNAL OF NONLINEAR SCIENCE, 2002, 12 (06) :641-676

[4]

Bacher R., 2014, CONFLU MATH, V6, P3, DOI [10.5802/cml.10, DOI 10.5802/CML.10]

[5] ROLE OF INTERMEDIATE CONVERGENTS IN TAITS EXPLANATION FOR PHYLLOTAXIS [J].

COXETER, HSM .

JOURNAL OF ALGEBRA, 1972, 20 (01) :167-&

[6] Phyllotaxis as a dynamical self organizing process .1. The spiral modes resulting from time-periodic iterations [J].

Douady, S ;

Couder, Y .

JOURNAL OF THEORETICAL BIOLOGY, 1996, 178 (03) :255-274

[7] Phyllotaxis as a dynamical self organizing process .3. The simulation of the transient regimes of ontogeny [J].

Douady, S ;

Couder, Y .

JOURNAL OF THEORETICAL BIOLOGY, 1996, 178 (03) :295-&

[8] Phyllotaxis as a dynamical self organizing process .2. The spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns [J].

Douady, S ;

Couder, Y .

JOURNAL OF THEORETICAL BIOLOGY, 1996, 178 (03) :275-294

[9]

Douady S., 1998, SYMMETRY PLANTS WORL, P335

[10]

Hermite C, 1916, J REINE ANGEW MATH, V41, P191

← 1 2 3 →