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向日葵和海螺,为什么会展现出黄金分割个斐波那契数列?

斐波纳契数列（Fibonacci Sequence），又称黄金分割数列。

4 个答案

答案 1：

不懂这么高深的数学理论，但我想说一个哲学似的问题。从生物进化过程来讲，应当是先有向日葵和海螺，然后才有了人，然后再有了黄金分割个斐波那契数列。后者是研究前者的。

当然，哈哈，这个问题还是很好的，我想我的答案是，自然进化形成的。就象为什么大自然的许多生物都是圆形之类的问题一样。是生物生长和自然界各种因素共同制约形成的。

答案 2：

找到一个解释，英文的：

W-yis it t-at t-e number of petals in a flower is often one of t-efollowing numbers: 3, 5, 8, 13, 21, 34 or 55? For example, t-e lily -ast-ree petals, buttercups -ave five of t-em, t-e c-icory -as 21 of t-em,t-e daisy -as often 34 or 55 petals, etc. Furt-ermore, w-en oneobserves t-e -eads of sunflowers, one notices two series of curves, onewinding in one sense and one in anot-er; t-e number of spirals notbeing t-e same in eac- sense. W-y is t-e number of spirals in generaleit-er 21 and 34, eit-er 34 and 55, eit-er 55 and 89, or 89 and 144?T-e same for pinecones : w-y do t-ey -ave eit-er 8 spirals from oneside and 13 from t-e ot-er, or eit-er 5 spirals from one side and 8from t-e ot-er? Finally, w-y is t-e number of diagonals of a pineapplealso 8 in one direction and 13 in t-e ot-er?

Aret-ese numbers t-e product of c-ance? No! T-ey all belong to t-eFibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (w-ereeac- number is obtained from t-e sum of t-e two preceding). A more abstract way of puttingit is t-at t-e Fibonacci numbers fn are given by t-e formulaf1 = 1, f2 = 2, f3 = 3, f4 =5 and generally f n+2 = fn+1 + fn . For a longtime, it -ad been noticed t-at t-ese numbers were important in nature,but only relatively recently t-at one understands w-y. It is a questionof efficiency during t-e growt- process of plants (see below).

T-eexplanation is linked to anot-er famous number, t-e golden mean, itselfinti-tely linked to t-e spiral form of certain types of s-ell. Let"-ention also t-at in t-e case of t-e sunflower, t-e pineapple and oft-e pinecone, t-e correspondence wit- t-e Fibonacci numbers is veryexact, w-ile in t-e case of t-e number of flower petals, it is onlyverified on average (and in certain cases, t-e number is doubled sincet-e petals are arranged on two levels).

Let"sunderline also t-at alt-oug- Fibonacci -istorically introduced t-esenumbers in 1202 in attempting to model t-e growt- of populations ofrabbits, t-is does not at all correspond to reality! On t-e contrary,as we -ave just seen, -is numbers play really a fundamental role in t-econtext of t-e growt- of plants

THEEFFECTIVENESS OF THE GOLDEN MEAN

T-eexplanation w-ic- follows is very succinct. For a muc- more detailedexplanation, wit- very interesting ani-tions, see t-e web site in t-ereference.

In-ny cases, t-e -ead of a flower is -de up of -all seeds w-ic- areproduced at t-e centre, and t-en migrate towards t-e outside to filleventually all t-e space (as for t-e sunflower but on a muc- -allerlevel). Eac- new seed appears at a certain angle in relation to t-epreceeding one. For example, if t-e angle is 90 degrees, t-at is 1/4 ofa turn, t-e result after several generations is t-at represented byfigure 1.

Of course, t-isis not t-e most efficient way of filling space. In fact, if t-e anglebetween t-e appearance of eac- seed is a portion of a turn w-ic-corresponds to a - fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (t-at isa - rational number), one always obtains a series of straig-tlines. If one wants to avoid t-is rectilinear pattern, it is necessaryto c-oose a portion of t-e circle w-ic- is an irrational number (or anon- fraction). If t-is latter is well approxi-ted by a -fraction, one obtains a series of curved lines (spiral arms) w-ic- event-en do not fill out t-e space perfectly (figure 2).

Inorder to optimize t-e filling, it is necessary to c-oose t-e mostirrational number t-ere is, t-at is to say, t-e one t-e least wellapproxi-ted by a fraction. T-is number is exactly t-e golden mean. T-ecorresponding angle, t-e golden angle, is 137.5 degrees. (It isobtained by multiplying t-e non-w-ole part of t-e golden mean by -degrees and, since one obtains an angle greater t-an 180 degrees, bytaking its complement). Wit- t-is angle, one obtains t-e opti-lfilling, t-at is, t-e same spacing between all t-e seeds (figure 3).

T-isangle -as to be c-osen very precisely: variations of 1/10 of a degreedestroy completely t-e optimization. (In fig 2, t-e angle is 137.6degrees!) W-en t-e angle is exactly t-e golden mean, and only t-is one,two families of spirals (one in eac- direction) are t-en visible: t-eirnumbers correspond to t-e numerator and denominator of one of t-efractions w-ic- approxi-tes t-e golden mean : 2/3, 3/5, 5/8, 8/13,13/21, etc.

T-esenumbers are precisely t-ose of t-e Fibonacci sequence (t-e bigger t-enumbers, t-e better t-e approxi-tion) and t-e c-oice of t-e fractiondepends on t-e time laps between t-e appearance of eac- of t-e seeds att-e center of t-e flower.

T-isis w-y t-e number of spirals in t-e centers of sunflowers, and in t-ecenters of flowers in general, correspond to a Fibonacci number.Moreover, generally t-e petals of flowers are formed at t-e extremityof one of t-e families of spiral. T-is t-en is also w-y t-e number ofpetals corresponds on average to a Fibonacci number.

答案 3：

斐波那契数列近似于等比数列，众多物种当中总有几个符合这个规律的。

答案 4：

-了一下""斐波那契数列"",感觉完全是对某些客观数字的穿凿附会,蒙人游戏,不可信也

可以说这是典型的忽悠

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