Fibonacci Tornado Phyllotaxy spirals consisting of all similar triangles

Year: 2008 Authors: Akio Hizume

Core claim

Only a restricted set of Fibonacci numbers yields phyllotaxy spirals made of all similar triangles, and their vertices lie on a logarithmic spiral.

Topics

phyllotaxy spirals, Fibonacci numbers, public sculpture, computer graphic design

Domains

number sequences, similar triangles, logarithmic spirals, golden ratio, generative design, public sculpture, computer graphics, visual pattern design

Methods

recurrence diagram, trigonometric formulas, spiral fitting, mod 1 to mod 3 visualization

Media

computer graphic images, public sculpture, diagram figures, logarithmic spiral

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Akio Hizume Star Cage Institute of Geometry Chiba, Japan E-mail: akio@starcage.org http://www.starcage.org

Abstract

“Fibonacci Tornado” is a generalization of the classical phyllotaxy spirals, which consist of all similar triangles. The result is a computer graphic design for a public sculpture. These spirals based on phyllotaxy are only possible for a restricted set of numbers — the Fibonacci numbers!

Fibonacci Tornado

Figure 1: Fibonacci Tornado mod 2 as a computer graphic design for a public sculpture

You can choose any Fibonacci Number $F_{(j)}$ .

where $F_{(0)}=0$ , $F_{(1)}=1$ , $F_{(j)}=F_{(j-1)}+F_{(j-2)}$ , integer $j\ge 2$ .

On the recurrence diagram as shown Figure 2, it should require the following formula.

$\delta =F_{(j+1)}\ast 2\pi /\tau$

$\varphi =F_{(j+2)}\ast 2\pi /\tau$

where $\tau =(1+\sqrt{5})/2$ , that is the golden ratio.

The ratio of similarity $s$ which is represented as $ak+1/ak=bk+1/bk=ck+1/ck$ must accord the following formula.

$b{s}^{F(j+1)}+c{s}^{F(j+1)}=a$

in case of $j=3$

$a=1$

$b=\sin \delta /\sin (\delta +\varphi )=0.7966510959\cdots$

$c=\sin \varphi /\sin (\delta +\varphi )=0.5387910616\cdots$

Figure 2: Recurrence Diagram

We can decide one logarithmic spiral which contains all vertexes of triangles.

It should be represented as;

The logarithmic spiral can be clockwise or counterclockwise.

We should get value of the $G$ which makes $r=s$ under each case of $\omega =2\pi (\tau -1)$ or $\omega =2\pi (2-\tau )$

in case of $\omega =2\pi (\tau -1)$ and $j=3$ $G=0.9822502411\dots$

in case of $\omega =2\pi (2-\tau )$ and $j=3$ $G^{\prime }=0.9714381720\dots$

Then we get the figure right.

Figure 3: mod 2 $(j=3)$ spiral

Let me show the solution from mod 1 $(j=2)$ to mod 3 $(j=4)$ on Figure 4.

Figure 4: The Fibonacci Tornado from mod1 to mod3

References

[1] This article was published in Japanese on MANIFOLD #11, pp. 7-8. 2005. [2] Akio Hizume, inter-native architecture OF music, ISBN978-4-9902966, pp. 117-118. 2006. [3] See more images in http://www.starcage.org/dragon/tornado..