1 Basic Shape

The basic shape is a non-convex quadrilateral with three acute angles of 45° (Fig. 1).

Fig. 1: Basic Shape

We work with a yellow and a red quadrilateral.

2 Reduction Factor

We reduce the size of the red quadrilateral by the factor √2 – 1 ≈ 0.414. This is the ratio of the shorter sides of the quadrilateral to the longer sides.

We create three copies of the reduced red quadrilateral (Fig. 2).

Fig. 2: Reduction and Copying

3 Construction

We place the reduced red copies into the 45° vertices of a copy of the yellow quadrilateral (Fig. 3). A regular octagon appears in the center.

Fig. 3: Insertion into the Corners

Now we create three more reduced copies (Fig. 4).

Fig. 4: Scaling and Copying

We place the scaled-down copies again into the 45° vertices of a copy of the yellow quadrilateral (Fig. 5).

Fig. 5: Inserting into the vertices

And so it continues (Fig. 6). The red portion gets smaller and smaller.

Fig. 6: The Next Steps

Figure 7 shows an animation of the construction process.

Fig. 7: Construction Steps

4 Assembling into a Star

We can assemble eight copies of any step into an eight-pointed star (Figs. 8, 9, and 10).

Fig. 8: Third-Generation Star

Fig. 9: Which Generation is This?

Fig. 10: Animation

5 Remarks

The fractal has the fractal dimension D

D = log(3)/log(√2 + 1) ≈ 1.246

Figures 2 to 7 have the same structure as the Sierpiński triangle.

Weblinks

Hans Walser: Achteck-Fraktal

https://walser-h-m.ch/hans/Miniaturen/A/Achteck1/Achteck1.htm

Hans Walser: Miniaturen: Fraktale

https://walser-h-m.ch/hans/Miniaturen_Uebersicht/Fraktale/index.html