1 Preliminaries

For our construction, we need the Golden Ratio:

Φ = (1 + √5)/2 ≈ 1.618

2 The Base Trapezoid

We work with an isosceles trapezoid with base Φ, leg length 1, and top edge 1/Φ (Fig. 1). The base angles measure 60°.

Fig. 1: Base Trapezoid

3 Construction

We reduce the size of the base trapezoid by a factor of 1/Φ ≈ 0.618. We rotate the reduced trapezoid once by 120° and once by –120° and join the two smaller trapezoids at one vertex as shown in Figure 2.

Fig. 2: Reduce, rotate, and join

The convex hull of the figure (Fig. 3.1) is congruent to the base trapezoid.

Fig. 3.1: First step

Now we repeat the procedure of reducing, rotating, and joining (Fig. 3.2).

Fig. 3.2: Second step: Repeating the procedure

Figures 3.3 to 3.12 show the further steps. The number of red trapezoids doubles with each step.

Fig. 3.2 Fig. 3.3: Third Step

Fig. 3.4: Fourth Step

Fig. 3.5: Fifth Step

Fig. 3.6: Sixth Step

Fig. 3.7: Seventh Step

Fig. 3.8: Eighth Step

Fig. 3.9: Ninth Step

Fig. 3.10: Tenth Step

Fig. 3.11: Eleventh Step

Fig. 3.12: Twelfth Step

4 Animation

Fig. 4: Animation

Weblink

Hans Walser: Goldenes Fraktal

https://walser-h-m.ch/hans/Miniaturen/G/Goldenes_Fraktal/Goldenes_Fraktal.html

References

Walser, Hans (2024): Der Goldene Schnitt. Geometrische und zahlentheoretische Betrachtungen. 7. Auflage. Springer Spektrum.
Print-ISBN 978-3-662-68556-3. E-Book_ISBN 978-3-662-68557-0.
https://doi.org/10.1007/978-3-662-68557-0

Walser, Hans (2024): The Golden Ratio. Geometric and Number Theoretical Considerations. Springer. ISBN 978-3-662-69889-1, ISBN 978-3-662-69890-7 (eBook)
https://doi.org/10.1007/978-3-662-69890-7