Promorphology of Crystals Preparation XIV

Stereometric Basic forms of Inorganic Beings

Promorphology of Crystals
preparation
14

e-mail :

Sequel to the Plane Group P31m

First we reproduce Figure 28 of the previous document :

Figure 1. The P31m pattern of Figure 27 of the previous document furnished with its appropriate hexagonal point lattice. A unit cell is indicated.

The pattern is adapted from the ionic pattern of Figure 14 of the previous document. This adaptation -- indicated by the green star-shaped patches in Figure 1 -- has qualitatively changed the pattern from a P3m1 pattern to a P31m pattern.
The next Figures will demonstrate that the pattern of Figure 1 indeed belongs the Plane Group P31m. Before we do so we give the symmetry diagram for that Plane Group :

Figure 2. Symmetry content of 2-D patterns according to the Plane Group P31m. Glide lines are indicated by dashed lines. Mirror lines are indicated by red and by black solid lines, while the 3-fold rotation axes are indicated by small triangles.

Figure 3. The 3-fold rotation axes are present in accordance with the symmetry diagram for the Plane Group P31m (Figure 2). Only some axes indicated.

Figure 4. The mirror lines (purple) are present in accordance with the symmetry diagram for the Plane Group P31m (Figure 2). Only some of them are indicated.

Figure 5. A first set of glide lines is present in accordance with the symmetry diagram for the Plane Group P31m (Figure 2). Only one glide line is indicated (purple).

Figure 6. A second set of glide lines is present in accordance with the symmetry diagram for the Plane Group P31m (Figure 2). Only one glide line is indicated (purple).

Figure 7. A third set of glide lines is present in accordance with the symmetry diagram for the Plane Group P31m (Figure 2). Only one glide line is indicated (purple).

We have now demonstrated that the pattern of Figure 1 and of Figure 27 of the previous document belongs to the Plane Group P31m.
The next Figure indicates a suitable repeat unit that can serve to represent the Complex Motif of the present pattern :

Figure 8. A hexagonal repeat unit of the present P31m pattern is indicated.

When we isolate this unit we automatically eliminate the simple translations, and obtain the Complex Motif of the pattern. See next Figure.

Figure 9. Complex Motif of the P31m pattern of Figure 1 (and also of Figure Figure 27 of the previous document).
The hexagonal boundary line (dark blue) does not as such belong to the Complex Motif. It belongs to the motif's background (in fact half its thickness).

The next Figure clearly shows the point symmetry of this Complex Motif :
Three mirror lines intersecting at angles of 120⁰ and a 3-fold rotation axis. So the symmetry is 3m. The motif has three congruent antimers.

Figure 10. 3m point symmetry of the Complex Motif of the P31m pattern of Figure 1 (and also of Figure Figure 27 of the previous document).
The hexagonal boundary line (dark blue) does not as such belong to the Complex Motif. It belongs to the motif's background (in fact half its thickness).
The Complex Motif has three congruent antimers.

The simplest planimetric figure expressing this symmetry is the Equilateral Triangle (regular trigon), which accordingly is the promorph or planimetric basic form of all crystals of the Plane Group P31m, and with it of the Class 3m. See next Figure.

Figure 11. The Equilateral Triangle (regular trigon) is the promorph or planimetric basic form of all crystals belonging to the Plane Group P31m and with it to the Class 3m.

In the next document we are going to c o m p a r e the Plane Groups P3m1 and P31m, to detect their similarities and their differences.

e-mail :

To continue click HERE to further investigate the Plane Groups with respect to their Complex Motifs.

e-mail :

back to homepage

back to retrospect and continuation page

back to Internal Structure of 3-D Crystals

back to The Shapes of 3-D Crystals

back to The Thermodynamics of Crystals

back to Tectology

back to Introduction to Promorphology

back to Anaxonia, Homaxonia, Polyaxonia

back to Protaxonia : Monaxonia