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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?
2012-01-19 18:13:44   来源:   点击:

    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:

      Whyis it that the number of petals in a flower is often one of thefollowing numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily hasthree petals, buttercups have five of them, the chicory has 21 of them,the daisy has often 34 or 55 petals, etc. Furthermore, when oneobserves the heads of sunflowers, one notices two series of curves, onewinding in one sense and one in another; the number of spirals notbeing the same in each sense. Why is the number of spirals in generaleither 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144?The same for pinecones : why do they have either 8 spirals from oneside and 13 from the other, or either 5 spirals from one side and 8from the other? Finally, why is the number of diagonals of a pineapplealso 8 in one direction and 13 in the other?

      Arethese numbers the product of chance? No! They all belong to theFibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (whereeach number is obtained from the sum of the two preceding). A more abstract way of puttingit is that the Fibonacci numbers fn are given by the formulaf1 = 1, f2 = 2, f3 = 3, f4 =5 and generally f n+2 = fn+1 + fn . For a longtime, it had been noticed that these numbers were important in nature,but only relatively recently that one understands why. It is a questionof efficiency during the growth process of plants (see below).

      Theexplanation is linked to another famous number, the golden mean, itselfintimately linked to the spiral form of certain types of shell. Let"smention also that in the case of the sunflower, the pineapple and ofthe pinecone, the correspondence with the Fibonacci numbers is veryexact, while in the case of the number of flower petals, it is onlyverified on average (and in certain cases, the number is doubled sincethe petals are arranged on two levels).

      Let"sunderline also that although Fibonacci historically introduced thesenumbers in 1202 in attempting to model the growth of populations ofrabbits, this does not at all correspond to reality! On the contrary,as we have just seen, his numbers play really a fundamental role in thecontext of the growth of plants

      THEEFFECTIVENESS OF THE GOLDEN MEAN

      Theexplanation which follows is very succinct. For a much more detailedexplanation, with very interesting animations, see the web site in thereference.

      Inmany cases, the head of a flower is made up of small seeds which areproduced at the centre, and then migrate towards the outside to filleventually all the space (as for the sunflower but on a much smallerlevel). Each new seed appears at a certain angle in relation to thepreceeding one. For example, if the angle is 90 degrees, that is 1/4 ofa turn, the result after several generations is that represented byfigure 1.

      Of course, thisis not the most efficient way of filling space. In fact, if the anglebetween the appearance of each seed is a portion of a turn whichcorresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that isa simple rational number), one always obtains a series of straightlines. If one wants to avoid this rectilinear pattern, it is necessaryto choose a portion of the circle which is an irrational number (or anonsimple fraction). If this latter is well approximated by a simplefraction, one obtains a series of curved lines (spiral arms) which eventhen do not fill out the space perfectly (figure 2).

      Inorder to optimize the filling, it is necessary to choose the mostirrational number there is, that is to say, the one the least wellapproximated by a fraction. This number is exactly the golden mean. Thecorresponding angle, the golden angle, is 137.5 degrees. (It isobtained by multiplying the non-whole part of the golden mean by 360degrees and, since one obtains an angle greater than 180 degrees, bytaking its complement). With this angle, one obtains the optimalfilling, that is, the same spacing between all the seeds (figure 3).

      Thisangle has to be chosen very precisely: variations of 1/10 of a degreedestroy completely the optimization. (In fig 2, the angle is 137.6degrees!) When the angle is exactly the golden mean, and only this one,two families of spirals (one in each direction) are then visible: theirnumbers correspond to the numerator and denominator of one of thefractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13,13/21, etc.

      Thesenumbers are precisely those of the Fibonacci sequence (the bigger thenumbers, the better the approximation) and the choice of the fractiondepends on the time laps between the appearance of each of the seeds atthe center of the flower.

      Thisis why the number of spirals in the centers of sunflowers, and in thecenters of flowers in general, correspond to a Fibonacci number.Moreover, generally the petals of flowers are formed at the extremityof one of the families of spiral. This then is also why the 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:

      百度了一下""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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