1 What it's about

Spirals with squares whose side lengths correspond to the Fibonacci numbers.

2 Fibonacci Squares

Fibonacci squares have side lengths corresponding to the Fibonacci sequence (Fig. 1)

1, 1, 2, 3, 5, 8, 13, 21, …

Fig. 1: Fibonacci Squares

3 The Classic Spiral

Figure 2 shows the classic spiral arrangement.

Fig. 2: Classic Arrangement

The spiral character can be made visible by drawing quarter circles (Fig. 3).

Fig. 3: Quarter Circles

The blue line is approximately a logarithmic spiral, but not exactly.

4 Sixfold Spiral

Figure 4 shows a different spiral arrangement of the Fibonacci squares.

Fig. 4: Different Arrangement

We can join four such spirals together, covering the entire area without overlapping (Fig. 5).

Fig. 5: Four Spirals

5 Animation

Figure 6 shows an animation with the classic arrangement (Fig. 2) as the starting point.

Fig. 6: From Landscape to Portrait Format

Figure 7 shows a special situation of the animation. The acute angles between two consecutive square sides measure 60°.

Fig. 7: Special Situation

6 Six Spirals

We can join six spirals together in the special situation (Fig. 7) (Fig. 8).

Fig. 8: Six Spirals

The spaces between the spirals are isosceles trapezoids whose sides correspond to three consecutive Fibonacci numbers.

Weblinks

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 (2022): Spiralen, Schraubenlinien und spiralartige Figuren. Mathematische Spielereien in zwei und drei Dimensionen. Springer Spektrum. ISBN 978-3-662-65131-5 und ISBN 978-3-662-65132-2 (eBook).

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.