Hans Walser, [20260611]

Kaleidoscope

1 What it's about

Modeling kaleidoscopes

2 The classic kaleidoscope

The classic kaleidoscope consists of an internally mirrored triangular prism. For two-dimensional modeling, we work with an equilateral triangle. Inside the equilateral triangle, we draw an arbitrary figure (Fig. 1).

Fig. 1: Figure in the triangle

Now we reflect this figure across each side of the triangle (Fig. 2).

Fig. 2: First step

Now we reflect the entire figure, that is, the original figure plus its three reflections, across each side of the triangle (more precisely: across the supporting line of each side of the triangle) (Fig. 3).

Fig. 3: Second step

In this step, for example, the red dot is reflected back in from the outside; it is now four layers high inside.

Again, we reflect the entire figure across each side of the triangle (Fig. 4).

Fig. 4: Third Step

Figure 5 shows the situation after six steps. The outline is a regular hexagon.

Fig. 5: Situation after six steps

Figure 6 shows an animation of the first six steps. In a real kaleidoscope, the number of steps is unlimited.

Fig. 6: Animation

Figure 7 shows how the central triangle expands. The color is transparent; overlays appear darker.

Fig. 7: Expansion of the central triangle

If the figure in the central triangle (Fig. 1) is changed, the overall figure in the kaleidoscope changes accordingly (Fig. 8).

Fig. 8: Change in the figure in the central triangle

3 General Triangle

The equilateral triangle cannot be replaced by a general triangle (Fig. 9).

Fig. 9: General Triangle

While it looks quite decent in the first few steps, we then lose all symmetry (Fig. 10 after six steps).

Fig. 10: Loss of Symmetry

In the general triangle, we replace the line reflections with point reflections across the midpoints of the sides (Figs. 11 and 12).

Fig. 11: Point Reflections

Fig. 12: Situation after six steps

The outline is an affine regular hexagon.

How can such a kaleidoscope be constructed?

4 Square Kaleidoscope

Figure 13 shows the kaleidoscope with a square as its basis.

Fig. 13: Square Kaleidoscope

Fig. 14: Situation after six steps

The outline is a square standing on one vertex.

Animation 15 shows the expansion and superposition of the kaleidoscope.

Fig. 15: Expansion and Superposition

5 Regular Pentagon

For the regular pentagon, I had to simplify the figure in the central pentagon graphically because (in the third step) superpositions occur.

Fig. 16: Regular pentagon

After six steps, a pattern emerges (Fig. 17). Lines can be seen that run orthogonally to the blue sides of the pentagon. I don't understand this.

Fig. 17: Situation after six steps

In animation 18 and figure 19, the circle has been moved to the center of the pentagon.

Fig. 18: Circle at the center

The sides of the blue pentagon appear to curve inwards, but this is an optical illusion.

Fig. 19: Situation after six steps

Animation 20 shows the expansion and superposition of the reflected pentagons. We don't seem to have a closing figure.

Fig. 20: Expansion and superposition

6 Regular hexagon

In the regular hexagon, overlaps also occur in the central hexagon (after the third step), but after the fourth step, the figure remains stable in the hexagon (Fig. 21).

Fig. 21: Inside the hexagon

Fig. 22 shows the situation after five steps.

Fig. 22: Situation after five steps

Animation 23 shows the expansion and overlap of the hexagons.

Fig. 23: Expansion and overlap

Weblinks

Wikipedia: Kaleidoskop

https://de.wikipedia.org/wiki/Kaleidoskop

Hans Walser: Kaleidoskop

https://walser-h-m.ch/hans/Miniaturen/K/Kaleidoskop/Kaleidoskop.htm