1 What it's about

A kinematic experiment with ten congruent regular pentagons

2 A closing figure

We join ten congruent regular pentagons together (Fig. 1). The pentagons are overlapping, therefore they are colored transparently.

Fig. 1: Joining together

The figure closes after ten steps. The eleventh pentagon merges with the first.

Two consecutive pentagons share an edge (Fig. 2).

Consecutive pentagons alternate between having their vertices facing up and down.

Fig. 2: Ten pentagons

3 Spatial interpretations

The two-dimensional figure (Fig. 2) can be interpreted as a top-down view of the rhombic icosahedron (Fig. 3).

Fig. 3: Rhombic icosahedron. Top View

The two-dimensional figure (Fig. 2) can also be interpreted as a top view of the rhombic triacontahedron (Fig. 4).

Fig. 4: Rhombic triacontahedron. Top View

4 Relative Gliding

We now let two adjacent pentagons in Figure 2 glide along their common edge (Fig. 5). If you focus on a particular color, you can see that the corresponding pentagon completes a circuit without rotating.

Fig. 5: Relative Gliding

Figure 6 shows the situation at its maximum extent. In this situation, a change of glide edge occurs via the vertices of the pentagons.

Fig. 6: Maximum Extent

5 Background

The centers of the pentagons move on two regular pentagons rotated 36° relative to each other (blue in Fig. 7).

Fig. 7: Trajectory of the centers

The outer edges of the pentagons also move along two regular pentagons rotated 36° relative to each other (red in Fig. 8).

Fig. 8: Frame

6 Outlook

The author suggests that the same process can be carried out with other regular polygons with an odd number of sides.

Weblinks

Hans Walser: Bewegte Dreiecke

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Dreiecke/Bewegte_Dreiecke.html

Hans Walser: Moving Triangles

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Dreiecke/Moving_Triangles.html

Hans Walser: Bewegte Fünfecke

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Fuenfecke/Bewegte_Fuenfecke.html

Hans Walser: Moving Pentagons

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Fuenfecke/Moving_Pentagons.html

Hans Walser: Bewegte Quadrate

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Quadrate/Bewegte_Quadrate.html

Hans Walser: Moving Squares

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Quadrate/Moving_Squares.html

Hans Walser: Bewegte Siebenecke

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Siebenecke/Bewegte_Siebenecke.html

Hans Walser: Moving Heptagons

https://walser-h-m.ch/hans/Miniaturen/B/Bewegte_Siebenecke/Moving_Heptagons.html