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►  The Infinite Fibonacci Number Sequence Table     ► the complete image :

     Abstract :

A Fibonacci-Number-Sequences-Table was developed, which contains infinite Fibonacci-Sequences. This was achieved with the help of research results from an extensive botanical study.

This study examined the phyllotactic patterns ( Fibonacci-Sequences ) which appear in the tree-species “Pinus mugo“ at different altitudes ( from 550m up to 2500m )

With the increase of altitude above around 2000m the phyllotactic patterns change considerably, the number of patterns ( different Fibonacci Sequences ) grows from 3 to 12,

and the relative frequency of the main Fibonacci Sequence decreases from 88 % to 38 %.

The appearance of more Fibonacci-Sequences in the plant clearly is linked to environmental ( physical ) factors changing with altitude.

Especially changes in temperature- / radiation- conditions seem to be the main cause which defines which Fibonnacci-Patterns appear in which frequency.

The developed ( natural ) Fibonacci-Sequence-Table shows interesting spatial dependencies between numbers of different Fibonacci-Sequences, which are connected to each other, by the golden ratio ( constant Phi ) > see Table

An interesting property of the numbers in the main Fibonacci-Sequence F1 seems to be, that these numbers contain all prime numbers as prime factors ! in all other Fibonacci-Sequences ≥ F2, which are not a multiple

of Sequence F1, certain prime factors seem to be missing in the factorized Fibbonacci-Numbers ( e.g. in Sequences F2, F6 & F8 ).

With the help of another study ( Title: "Phase spaces in Special Relativity : Towards eliminating gravitational singularities" ) a way was found to express (calculate) all natural numbers and their square roots only by using

constant Phi (golden ratio) and 1. An algebraic term found by Mr Peter Danenhower, in his study, made this possible.

With the formulas which I found, it seems to be possible to eliminate number systems and base mathematics only on Phi (ϕ) and 1 ( see my 12 conjectures )

Introduction :

In botany Phyllotaxis describes the arrangement of leaves on spiral paths on the stem of a plant. Phyllotactic spirals form a distinctive class of patterns in nature. But the true cause of these phyllotactic spirals , which appear

everywhere in nature, still isn’t found yet !

The current believe ist that the spiral patterns of leaves on the stem of a plant, which can be explained and described by Fibonacci Number Sequences, is controlled by plant hormones like Auxin.

However this can’t be the true cause for the precise Fibonacci-spiral-patterns seen on plants !

Because the extensive botanical study carried out by Dr. Iliya Vakarelov clearly shows that the Fibonacci-spiral formation is influenced by environmental conditions, especially temperature and radiation (light).

Therefore the Fibonacci-spiral formation seems to have a fundamental physical cause !   Dr. Vakarelov’s study also showed that the phyllotactic-patterns changed cyclic, with six year duration of the cycles. I

I have written an own hypothesis about the cause of phyllotactic ( Fibonacci ) patterns :

see study :   ►  Microscope Images indicate that Water Clusters are the cause of Phyllotaxis     and  :      ► Summary_Microscope Images indicate that Water Clusters are the cause of Phyllotaxis