1 What it's about

A spiral figure with cuboids whose edges form a geometric sequence (Fig. 0).

Fig. 0: Cuboid as game material

2 The cuboid

A cuboid with edge lengths 1, a, a² (Fig. 1 and Fig. 2 for a = ½) has rectangular faces with dimensions 1×a, a×a², and 1×a². The first two rectangular faces are similar; the third is too long.

Fig. 1: Cuboid

Fig. 2: Cuboid net

3 Scaling and positioning

We scale a copy of the cuboid by the factor a and position the scaled copy on a matching rectangular face (orange in Fig. 3).

Fig. 3: Scaling and positioning

4 Iteration

We can iterate the process (dark yellow in Fig. 4).

Fig. 4: Iteration

In the next step (light yellow in Fig. 5), a cuboid is formed that is oriented the same as the initial cuboid but scaled by a factor of a³ relative to the initial cuboid.

Fig. 5: Next Step

Figure 6 shows further steps.

Fig. 6: Further Steps

Since a = ½ is small compared to 1, the sequence aⁿ rapidly converges to 0, and we cannot see any further steps.

5 Other Scaling Factors

In the trivial case a = 1, we obtain cubes (Fig. 7). The figure is a left-handed helix.

Fig. 7: Cubes

Figure 8 shows the situation for a = ⅘.

Fig. 8: a = ⅘

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