Hans Walser, [20260531]

Polygonal Spirals

Idea and inspiration: Shingo Nakanishi, OIT (Osaka Institute of Technology), Japan

1 The concept

Based on regular polygons, spirals are constructed that pave the plane.

2 Pentagons

The inner contour of the spiral with pentagons (Fig. 1) is congruent to the outer contour, but rotated by 72°.

Fig. 1: Spiral with pentagons

We can therefore pave the plane with five spirals (Fig. 2).

Fig. 2: Five spirals with pentagons

3 Squares

The inner contour of the spiral with squares (Fig. 3) is congruent to the outer contour, but rotated by 90°.

Fig. 3: Spiral with squares

We can pave the plane with four spirals (Fig. 4).

Fig. 4: Four spirals with squares

4 Hexagons

The inner contour of the spiral with hexagons (Fig. 5) is congruent to the outer contour, but rotated by 120°.

Fig. 5: Spiral with hexagons

We can pave the plane with three spirals (Fig. 6).

Fig. 6: Three spirals with hexagons

5 Outlook

I have not succeeded in creating an analogous construction based on heptagons, octagons, etc.

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