In the previous document we studied crystals having the (total) symmetry according to the Plane Group **Pm.** Two types of possible motifs with D_{1} symmetry were investigated as to the Promorph they would imply.

In the present document we will consider still other motifs that have D_{1} symmetry and that can therefore also constitute a **Pm** pattern. Also these different motifs, although not changing any symmetry, imply different Promorphs (i.e. different species of the **Heterostaura allopola** or half amphitect pyramids).

The following motif, having D_{1} symmetry, will be placed in a primitive rectangular lattice, resulting in a **Pm** pattern **:**

Figure 1. *Newly chosen motif to constitute a *Pm* pattern.*

An enlargement of this motif is shown in the next Figure.

Figure 2. *Magnification of the newly chosen motif as depicted in the previous Figure.
It has *D

A possible 2-D crystal having this structure, i.e. consisting of a periodic repetition of the above motif according to a primitive rectangular lattice, is given in the next Figure.

Figure 3. *A possible 2-D crystal having the above described internal structure.
It consists of three Forms :
*

- A Form consisting of a vertical crystal 'face'.
- A Form consisting of a pair of oblique 'faces'.
- A Form consisting of a pair of 'faces' parallel to the mirror line.

An alternative unit cell still contains five antimers of the motif (2 + 2 + 1/2 + 1/2)

Figure 4. *Alternative unit cell (mesh) also contains five antimers, i.e. it contains a motif consisting of parts that are equivalent to the five antimers of the original motif.*

The five

Figure 5. *The five antimers of the 2-D crystal of Figure 3 are indicated. The boundaries separating these antimers are more or less arbitrary.*

The structure, symmetry and content of the motif determines the lattice according to which it will be repeated under given 'thermodynamic conditions. It also determines the 'chemical' atomic aspect a face can present to the growing environment. All this in turn determines the pattern of relative growth rates and thus the Growth Rate Vector Rosette, and finally the Vector Rosette of Actual Growth which expresses the shape of the crystal. So in fact the translation-free (microscopic) motif determines the intrinsic Shape of the corresponding crystal under given thermodynamic conditions. And this means that the shape of that motif is somehow related to the intrinsic shape the corresponding crystal will adopt. That's why we have chosen from the set of geometrically possible crystals a crystal of which the shape is somehow comparable with the shape of the translation-free motif.

The D_{1} symmetry of all the Pm crystals determines the *general* Promorph of them to be that of a 2-D analogue of the **Heterostaura allopola.** However, to what species of the latter it belongs is not revealed by the Shape or Symmetry of the crystal, but is determined by the promorphology of the translation-free motif. And indeed the latter indicates that the **Promorph** of the crystal belongs to the 2-D analogue of the **Pentamphipluera** (Heterostaura allopola). The solid representing the 3-D Pentamphipleura is *half a ten-fold amphitect pyramid* as the next Figure shows.

Figure 6. *Oblique top view of the Stereometric Basic Form of the Pentamphipleura. It is half a ten-fold amphitect pyramid. The five antimers are indicated by colors. The brown plane facing towards the beholder is the bisection plane.*

The antimers of a motif do not need -- in order for them to be true antimers -- to be equal, or exactly symmetric with respect to each other, as we have seen in the previous document. So our mirror symmetric motif can in other crystals (differeing slightly in 'chemical composition') be a motif where this mirror symmetry is lightly disturbed. Even then we will expect a crystal of a similar or equal shape as the one with the fully mirror symetric motif. And, as we saw, we can describe the point symmetry of this slightly asymmetric motif, and also the symmetry of the periodic pattern that can be obtained from such a motif, by the corresponding

Figure 7. *A two-dimensional crystal of which the classical symmetry of the internal structure is according to the Plane Group ***P1*** , but of which the antisymmetry is according to the antisymmetry group ***Pm / P1*** . In the latter interpretation the mirror reflection ***m*** is replaced by the antisymmetry transformation ***e _{1}m**

The two

Figure 8. *The two antimers of a two-dimensional crystal (as given in Figure 7) having ***P1*** classicla symmetry, but ***Pm / /P1*** antisymmetry, are indicated.*

The next Figure gives a 2-D periodic pattern according to the Plane group Pg.

Figure 9. *A pattern representing the Plane Group ***Pg*** .
The rectangular areas represent group elements (one such area is indicated as representing the identity element *

The next Figure depicts the same Pg pattern, but now the lines delineating the areas representing group elements deleted, and the above given coloration of the motifs undone.

Figure 10. *The *Pg* pattern of the above Figure. Each mesh contains a motif consisting of two commas related to each other by a glide reflection. A glide line is indicated (red dashed line).*

The point symmetry of any

According to all this the

And it also holds for all crystals having a symmetry of their internal structure according to the Plane Group **Cm,** because also here the point symmetry is that of D_{1} (crystallographically denoted by m)

The next Figure gives a periodic pattern according to the Plane Group **P2mm.**

Figure 11. *A two-dimensional periodic pattern according to the Plane Group ***P2mm*** . The yellow and green areas represent group elements (they do not represent symmetry features). A unit mesh is indicated (blue).*

The next Figure gives this same pattern, without indicating areas representing group elements.

Figure 12. *The 2-D periodic pattern of Figure 11. The lattice underlying this pattern is a primitive rectangular net. Each mesh is provided with a motif consisting of four commas. The symmetry of the motif is according to the group ***D _{2}**

There are two alternative ways to conceive of the motif (s.str.). One as the content of a mesh, the other as

Figure 13. *Two alternative ways to see the motifs of the *P2mm* pattern of the previous Figures. In both cases the motif has ***D _{2}**

Figure 14. *The *D_{2}* motif of the above *P2mm* pattern has its directional axes in interradial positions. The commas represent antimers, and these in turn represent the radii of the motif.*

From Figure 14 it is clear that the Promorph of this motif is a 2-D analogue of the

The stereometric solid, representing the

Figure 15. *Slightly oblique top view of a rectangular pyramid. In case of there being four antimers it is the basic form of the Tetraphragma interradialia, or, (it could be interpreted as just) a modification of the Tetraphragma.*

The directional planes of the

See next Figure.

Figure 16. *Equatorial plane (or base for that matter) of a rectangular pyramid with four antimers (indicated by coloration). The radial cross axes are indicated with red, the interradial cross axes (which here are at the same time the directional axes) are indicated with green. *

So in virtue of the promorphology of the translation-free motif of the pattern of Figure 12, all 2-D crystals having this structure (i.e. having an internal structure with plane group P2mm, and possessing the motifs as given) have a

The next Figure gives a possible crystal having the above structure.

Figure 17. *A possible 2-D crystal having the structure as depicted in Figure 12. It consists of two Forms :
*

- A Form consisting of two horizontal 'faces' (i.e. when one horizontal face parallel, but not coincident, to a mirror line, is initially given, then a second face parallel to it is automatically generated, resulting in a Form consisting of two faces). It is an open Form, and thus needs another Form to be able to exist as faces of a crystal.
- A Form consisting of two vertical 'faces' (i.e. when one vertical face parallel, but not coincident, to a mirror line, is initially given, then a second face parallel to it is automatically generated, resulting in a Form consisting of two faces). It is an open Form, and thus needs another Form to be able to exist as faces of a crystal.

The

Figure 18. *Point symmetry -- *D_{2}* -- of the 2-D crystal of the previous Figure. There are two mirror lines perpendicular to each other. In the point of intersection of these mirror lines there is a 2-fold rotation axis (perpendicular to the plane of the drawing).*

Although other shapes of possible crystals having this particular structure are in principle possible, we assume that the structure of the translation-free motif has much influence on the shape adopted by the crystal, because, as has been said, the structure of the motif determines the atomic aspects presented to the environment by possible faces, that in turn determine the pattern of relative growth rates, and thus finally determine the intrinsic shape of the crystal.

On the basis of the motif as present in the above crystal, the latter must have four **antimers** **:**

Figure 19. *The four antimers of the 2-D crystal of Figure 17.
The two directional (directive) axes are posited interradially. The Promorph of the crystal therefore belongs to the 2-D analogue of the *

The

The structure of the

Figure 20. *A possible crystal with (the symmetry of its internal structure according to the) Plane Group ***P2mm*** (as in Figure 12), but with different motifs. *

It is however more likely that the shape of the crystal relates to the shape of the translation-free motif. This is expressed in the next Figure.

Figure 21. *A possible crystal with ( the symmetry of its internal structure according to the) Plane Group ***P2mm*** (as in Figure 12), having motifs as present in the previous Figure. While the symmetry of these motifs is still according to the symmetry group ***D _{2}**

The two

Figure 22. *The two antimers of the crystal of the previous Figure.
The Promorph is of the crystal is then that of the 2-D analogue of the *

Itis perhaps convenient to add some explanation as to the difference between the

Figure 23. *Perspectivic view of a Rhombic pyramid.
Left image : Rhombic pyramid with t w o antimers, representing the basic form of the D i p h r a g m a.
Right image : Rhombic Pyramid with four antimers (basic form of the Tetraphragma). *

The stereometric basic form of the

Figure 24. *Base (or a plane parallel to it and containing the mid-point of the main axis) of a four-fold amphitect pyramid as the basic form of the Autopola orthostaura diphragma. This configuration represents organismic forms with only t w o antimers. These a n t i m e r s are indicated by colors. There is only one radial cross axis (red) and one interradial cross axis (green). These two axes coincide with the directional axes. The axis, indicated with green is indeed interradial because it lies between antimers.*

The two antimers must of course be congruent, because the dorsal side differs as little from the ventral side as the right side from the left.

The next Figure gives a possible 2-D crystal also having an internal symmetry according to the Plane Group P2mm, but with a different translation-free motif. This motif, however, has the same point symmetry -- D

Figure 25. *A possible 2-D crystal, still having internal *P2mm* symmetry, but with a different translation-free motif. Although this motif still has a point symmetry according to the group *D_{2} , it has **four*** antimers, like the motif in Figure 12, but they are differently arranged. The Promorph of this motif is belonging to the 2-D analogue of the ***Tetraphragma radialia** (Heterostaura autopola orthostaura). *See next Figure.*

Figure 26. *The translation-free motif of the crystal of the previous Figure has four antimers, arranged in such a way that the directive axes go through these antimers, meaning that these axes are radial. We could think of the motif as consisting of four equally sized, but qualitatively different atoms or atomic complexes.*

The four

Figure 27. *The four antimers of the above crystal are indicated.*

The

The next Figures elucidate the *Tetraphragma radialia*.

Figure 28. *Slightly oblique top view of a Rhombic Pyramid as the basic form of the Tetraphragma radialia.
The main axis, the directional axes and the interradial cross axes are indicated.*

These

Figure 29. *Equatorial plane (or base for that matter) of a regular 4-fold pyramid. This pyramid is the basic form of the Homostaura isopola tetractinota. Radial cross axes are indicated as ***R,*** interradial cross axes as ***I.*** The four antimers are indicated by coloration.From this form can be derived the Rhombic Pyramid of the Autopola tetraphragma radialia by differentiating between the two directional axes (See next Figure). *

Figure 30. *(colored image within the rectangle) Equatorial plane (or base for that matter) of an amphitect 4-fold pyramid (with four antimers). This pyramid is the basic form of the Autopola tetraphragma radialia (Heterostaura). Radial cross axes are indicated as ***R***,** interradial cross axes as ***I***.** The four antimers are indicated by coloration. From this form can be derived the basic form of the Eutetrapleura radialia by making one of the directional axes heteropolar (See next Figure). *

Figure 31. *Equatorial plane (or base for that matter) of an ideally halved (i.e. without thereby loosing antimers) 4-fold amphitect pyramid (with four antimers). This ideally halved pyramid is the basic form of the Allopola eutetrapleura radialia (Heterostaura). Radial cross axes are indicated as *

The next Figure gives a possible crystal, again with an internal symmetry according to the Plane group P2mm, but with a different translation-free motif, although still having D

Figure 32. *A possible crystal with (the symmetry of its internal structure according to the) Plane Group ***P2mm*** (as in Figure 12), but with different motifs. Such a (different) motif still has *D_{2}* symmetry but possesses six antimers, arranged in such a way that the promorph of the motif belongs to the 2-D analogue of the *

- A Form consisting of four oblique 'crystal faces'.
- A Form consisting of two horizontal 'crystal faces'.

The next Figure indicates the six

Figure 33. *The six antimers of the crystal of Figure 32 are indicated.*

The

The next Figures elucidate the *Oxystaura hexaphragma*.

Figure 34. *Slightly oblique top view of a six-fold amphitect pyramid as the basic form of the Oxystaura hexaphragma.*

Figure 35. *Slightly oblique top view of a six-fold amphitect pyramid as the basic form of the Oxystaura hexaphragma. The six a n t i m e r s are indicated by colors.*

In the previous document we discussed the Plane Group

Figure 36. *A two-dimensional crystal (consisting of three Forms) having ***Pm*** plane group symmetry and ***D _{1}**

But the symmetry of the crystal of Figure 36 can also be interpreted as the result of a desymmetrization of the Plane Group P2mm (and of the Point Group D

Here the reflection

If the motif is found to represent D_{2} symmetry, and has four equal antimers arranged in such a way as to result in a promorph belonging to the 2-D analogue of the Tetraphragma interradialia (Heterostaura autopola), then the right half must be relflected in the line **a** (see next Figure) going through the two antimers residing in this right half, and at the same time be reduced in size. If, on the other hand, the motif is found to be in the latter state, it must reverse to the first state.

In short we can describe **e _{1}** as the permutation

(cycle notation)

Graphically this looks as follows **:**

Figure 37. *Illustration of the action of the antiidentity transformation ***e _{1}**

The full antisymmetry transformation is then

(or, equivalently , perform e

Figure 38. *The action of the antisymmetry transformation ***e _{1}m**

So if we replace, in the Plane Group P2mm, the vertical reflection

So indeed we can describe the plane group symmetry -- which classically is Pm -- of the crystal of

This

In the

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