Hans Walser, [20260520]

Fibonacci Spirals

Thanks to Shingo Nakanishi, OIT (Osaka Institute of Technology), Japan

1 What it's about

Spirals with isosceles trapezoids whose side lengths correspond to the Fibonacci numbers.

2 The Fibonacci Trapezoids

We draw the Fibonacci trapezoids (Fig. 1) on a regular triangular grid. The trapezoids are isosceles and have three consecutive Fibonacci numbers as their sides. We work with the following numbers from the Fibonacci sequence:

–1, 1, 0, 1, 1, 2, 3, 5, 8, 13, …

The isosceles trapezoid on the far right, for example, has a base length of 13, a leg length of 8, and a top side length of 5. The three consecutive Fibonacci numbers are therefore 5, 8, 13.

Fig. 1: Fibonacci Trapezoids

The "trapezoid" on the far left belongs to the triple –1, 1, 0. The base length is –1, the leg length is 1, and the top side length is 0.

The adjoining segment belongs to the triple 1, 0, 1.

The next triangle belongs to the triple 0, 1, 1.

The first true trapezoid belongs to the triple 1, 1, 2.

3 A Spiral

We construct a spiral from the trapezoids in Figure 1 (Fig. 2). It fits into the triangular grid.

Fig. 2: Spiral made of trapezoids

4 Six spirals

We can combine six spirals from Figure 2 so that they cover the entire plane but do not overlap (Fig. 3).

Fig. 3: Six spirals

In Figure 4, only the boundary lines are drawn. We can see two opposing spiral types.

Fig. 4: Boundary lines

Figures 5 and 6 show the spiral types individually.

Fig. 5: Line spirals

Fig. 6: Line spirals

Figure 7 shows the six area spirals without the background grid.

Fig. 7: Six area spirals without a grid

5 Another spiral

The spiral in Figure 8 contains right angles and therefore no longer fits into a triangular grid.

Fig. 8: Spiral with right angles

We can combine six such spirals (Fig. 9). The gaps between are squares with side lengths corresponding to the Fibonacci numbers.

Fig. 9: Six spirals with gaps between them

6 Alternating Coloring and Diagonals

We color the figures in Figure 1 alternately red and green and add diagonals to the trapezoids, running from the top left to the bottom right (Fig. 10).

Fig. 10: Alternating Coloring. Diagonals

We can stack all the red or all the green figures on top of each other to obtain an equilateral triangle (Fig. 11).

Fig. 11: Stack

We can also arrange the pieces so that the diagonals form the beginning of a spiral (Fig. 12).

Fig. 12: Spirals

Figure 13 shows the same spirals with three additional steps each.

Fig. 13: Three Additional Steps Each

We can also alternate between the colors red and green (Fig. 14).

Fig. 14: Red/green spiral

Weblinks

Hans Walser: Fibonacci Spirals

https://walser-h-m.ch/hans/Miniaturen/F/Fibonacci_Spirals2/Fibonacci_Spirals2.html

Hans Walser: Fibonacci-Spiralen

https://walser-h-m.ch/hans/Miniaturen/F/Fibonacci-Spiralen/Fibonacci-Spiralen.html

Hans Walser: Fibonacci-Star

https://walser-h-m.ch/hans/Miniaturen/F/Fibonacci-Star/Fibonacci_Star.htm

Hans Walser: Fibonacci-Trapeze

https://walser-h-m.ch/hans/Miniaturen/F/Fibonacci-Trapeze/Fibonacci-Trapeze.htm

References

Walser, Hans (2024): Spirals, Helical Lines, and Spiral-Like Figures. Mathematical Playfulness in Two and Three Dimensions. Springer.
ISBN 978-3-662-68930-1, ISBN 978-3-662-68931-8 (eBook)
https://doi.org/10.1007/978-3-662-68931-8

Walser, Hans (2012): Fibonacci. Zahlen und Figuren. Leipzig, EAGLE, Edition am Gutenbergplatz. ISBN 978-3-937219-60-8.