Hans
Walser, Basel University, Switzerland

*hwalser@bluewin.ch*

Abstract: Cutting away a rhombus from a regular
pentagon, the leftover will be a semi-regular pentagon. Using this semi-regular
pentagon as tile, we can build various ornaments, tilings, crystallographic
patterns, and spirals. We will find also a semi-regular not convex
dodecahedron. The regular and the semi-regular dodecahedron together can fill
the space without gaps or overlapping.

The regular pentagon is a bad guy.

We canŐt use it for a tiling of the plane. There
occurs a gap of 36ˇ (Fig. 1c) (Grźnbaum and Shephard 1987, Frontispiece).

Fig. 1: No tiling with regular pentagons

Now we modify the regular pentagon (Fig. 2a). We flip
down a vertex (Fig. 2b). The remaining part will be our semi-regular pentagon
(Fig. 2c). The semi-regular pentagon has equal sides, but not equal angles.

Fig. 2: The semi-regular pentagon

Though our semi-regular pentagon has some defects
compared to the regular pentagon, it has also some advantages.

We find a tiling with a combination of regular and
semi-regular pentagons (Fig. 3a).

Fig. 3: Tiling with pentagons and semi-regular pentagons

The tiling contains translational symmetries,
indicated by blue arrows in the figure 3b, axial symmetries (red) and glide-reflection
symmetries (purple).

The figure 4 depicts an example with a fivefold
rotational symmetry.

Fig. 4: Fivefold rotational symmetry

Nevertheless we may build frieze patterns and tilings
just out of our semi-regular pentagon.

The figure 5 depicts a frieze pattern. With respect to
the colors it has translational symmetry only. Without respect to the colors we
have additionally a glide-reflection symmetry.

Fig. 5: Frieze pattern

The tiling of the figure 6a is just an overlapping of
the frieze pattern of the figure 5. In this figure we may get the victim of an
optical illusion. Are the horizontal lines parallel?

The figure 6b depicts a more elegant tiling.

WhatŐs the difference between the figures 6c and 6d?

Fig. 6: Tilings

The figure 7a gives a constellation with concentric
rings. The figure 7b is basically the same, but the colors are exchanged such
that there are always two different colors at an edge.

Fig. 7: Concentric rings

First we see nothing special in the figure 8a. But a
closer look reveals a spiral (Fig. 8b). The width of the spiral is constant, so
we have an Archimedean spiral.

Fig. 8: Spirals

The figure 9 depicts two and ten Archimedean spirals
respectively.

Fig. 9: More spirals

With twelve regular pentagons as faces we can build
the regular dodecahedron (Fig. 10a). Together with the regular tetrahedron, the
cube, the octahedron, and the icosahedron it belongs to the five regular
Platonic solids.

Fig. 10: Regular and semi-regular dodecahedron

With twelve semi-regular pentagons we can build the
semi-regular dodecahedron (Fig. 10b).

The semi-regular dodecahedron fits obviously in a cube
(Fig. 11b). But we can also draw a cube on the faces of the regular
dodecahedron (Fig. 11a).

Fig. 11: Relation to the cube

The regular dodecahedron may be seen as a compound of
a cube with a hip roof on every face. And surprisingly the semi-regular
dodecahedron is just the leftover when we cut away a hip roof at every side of
the cube. So we have related construction principles. The regular dodecahedron
is positive, the semi-regular dodecahedron negative. The hip roof excess of the
regular dodecahedron corresponds to the hip roof deficit of the semi-regular
dodecahedron.

The figure 12 depicts the situation between the two
dodecahedra.

Fig. 12: Hip roofs

The hip roof cut away from the cube at the right (Fig.
12b) goes to the cube at the left (Fig. 12a). We have a situation similar to
the situation of anions and cations in chemistry.

A singular hip roof has the dimensions indicated in
the figure 13.

Fig. 13: Dimensions of the hip roof

The notation means the
Golden Section (Walser 2001 and 2013) . The height of the roof is .

The semi-regular dodecahedron is an eight-pointed star
(Fig. 14b). But it is different from the similar looking *Kepler star* or *stella** octangula* (Fig.
14a).

Fig. 14: Kepler star and semi-regular dodecahedron

The symmetry groups of the regular and the
semi-regular dodecahedra are different. The semi-regular dodecahedron has just
the symmetry group S4 of the regular tetrahedron.

The regular and the semi-regular dodecahedra have the
same topology (Fig. 15 and 16). Both have 20 vertices, but in the semi-regular
case 12 of the vertices are hyperbolic.

Fig. 15: Vertices, edges, and faces

Both have 30 edges, but in the semi-regular case six
of the vertices are like the bottom of a valley.
The regular dodecahedron
has 12 regular pentagons as faces, in the case of the semi-regular dodecahedron
the 12 faces are semi-regular pentagons.

In the diagrams of the figure 16 we see how the
vertices and edges are connected. There is no difference between the two
dodecahedra.

Fig. 16: Diagrams for regular and semi-regular dodecahedron

We can build a paper model of the semi-regular
dodecahedron (Fig. 17a).

Fig. 17: Paper model

We need 6 parts according to figure 17b. We have to
cut along the black lines, fold in the sense of a valley fold along the red
line and in the sense of a mountain fold along the blue lines.

The red area remains visible on the outside, and the
gray areas can be tucked in or glued. Hint: The model will not be very stable.
Therefore I built first a model with reduced size (98%) and the visible model
as second layer on it.

Because of the relative situation between the two
dodecahedra (Fig. 12) the regular dodecahedron can sit on the semi-regular
dodecahedron like the egg on an eggcup (Fig. 18).

Fig. 18: Eggcup

Neither the regular nor the semi-regular dodecahedra
are space fillers (Coxeter 1973, p. 68f). But we can fill the space with
regular dodecahedra combined with semi-regular dodecahedra (Fig. 19).

Fig. 19: Filling the space

The figure 19 is somehow the spatial analogue to the
figure 3.

The proof of the space-filling property is easy. First, the cube is a space filler. So we fill the space with cubes and color the cubes in black and white like a three-dimensional chessboard. Now we cut away the hip roofs from the black cubes and add them to the adjacent white cubes.

Coxeter, H. S. M. (1973): Regular Polytopes. Third Edition. New York: Dover 1973. ISBN 0-486-61480-8.

Grźnbaum, B. and Shephard, G. C. (1987): Tilings and Patterns. New York: Freeman. ISBN 0-7167-1193-1.

Walser, H. (2001): The Golden Section. Translated by Peter Hilton and Jean Pedersen. The Mathematical Association of America 2001. ISBN 0-88385-534-8.

Walser, H. (2013): Der Goldene Schnitt. 6., bearbeitete und erweiterte Auflage. Edition am Gutenbergplatz, Leipzig. ISBN 978-3-937219-85-1.