This document continues the investigation of special categories (If / Then constants), and compares crystals with organisms.

Crystals and Organisms, Shape, Symmetry and Promorph.

Sequel to the investigation of some (intrinsic) **shapes** of two-dimensional crystals regarding their relationship to intrinsic **point symmetry** and **promorph.**

Square

We now consider a fully developed 2-D (i.e. two-dimensional) (imaginary) crystal having the **shape of a square :**

Any square, i.e. quadratic, crystal can be conceived as being built up by a periodic stacking of microscopic square units, as the next two Figures illustrate

The true, i.e. intrinsic, point symmetry of such a crystal can either be according to the Dihedral Group

That a **square crystal** can have these intrinsic symmetries can be explained succinctly as follows (and will be further evident in the sequel) **:**

A **Square** as such has the following symmetries, i.e. it will be superposed upon itself by the following transformations (which are then for that reason symmetry transformations) **:**

- 0
^{0}or 360^{0}rotation about any axis. - 180
^{0}rotation (half-turn) about the axis through its center. - Reflection in a line through its center and parallel to a side of it.
- Reflection in a line through its center and perpendicular to the reflection line just mentioned.
- Reflection in the NE diagonal line.
- Reflection in the SE diagonal line.
- 90
^{0}rotation (quarter-turn) about the axis through its center. - 270
^{0}rotation (three quarter-turn) about the axis through its center.

- When no symmetry is available (
**a**of the above list),**four Forms**are needed to produce a square. - When only a half-turn is available (
**b**of the above list),**two Forms**are needed to produce a square. - When only one diagonal reflection is available (
**e**__or__**f**of the above list),**two Forms**are needed to produce a square. - When two diagonal reflections are available (
**e**__and__**f**of the above list), only**one Form**is needed to produce a square. - When only a quarter-turn is available (
**g**of the above list), only**one Form**is needed to produce a square.

The next table shows the

Plane Group |
Point Symmetry |
Number of Forms |

P1 |
C_{1} ( 1 ) |
4 |

P2 |
C_{2} ( 2 ) |
2 |

PmCm Pg |
D_{1} ( m ) |
2 |

P2mmC2mm P2mg P2gg |
D_{2} ( 2mm ) |
1 |

P4 |
C_{4} ( 4 ) |
1 |

P4mmP4gm |
D_{4} ( 4mm ) |
1 |

For each of these plane groups mentioned in the above table, we will show how it supports a

A square, which has a symmetry according to the group D_{4} , will be supported by __those__ plane groups of which all the implied point symmetries (i.e. implied by the plane group) are also symmetries of the Square. This does not necessarily hold the other way around **:** A crystal can have an intrinsic shape that is a square, but nevertheless some, or even all, symmetries of this square can be absent in the crystal, resulting in the fact that the intrinsic symmetry of the crystal is lower than that of its intrinsic shape.

Figure above **:** Two-dimensional square (quadratic) crystal, point symmetry **D _{4}** , and supported by a quadratic point lattice (indicated by connection lines).

Figure above **:** Two-dimensional quadratic crystal, point symmetry **D _{4}** , and supported by a quadratic point lattice (indicated by connection lines).

Figure above **:** Two-dimensional square crystal, point symmetry **C _{4}** , and supported by a quadratic point lattice (indicated by connection lines).

While a

Let us, in the above Figure, denote the longer side of the unit rectangle **lo** and the shorter side **sh** . In the present case **lo / sh = 3.** Let us further denote the growth rate perpendicular to **lo : Rate(lo)** and perpendicular to **sh : Rate (sh)** . And let us express these rates in terms of the number of unit rectangles in a unit of time (and thus not in terms of units of distance). It is then clear that in the present case, in order for crystal growth to result in a **rectangular** crystal, **Rate(lo)** must be three times as big as **Rate(sh)** . And indeed generally we can say that if the stacking of rectangular building blocks should result in a square crystal, the relation

**Rate(lo) = (lo / sh) Rate(sh)** (measured in terms of numbers of building blocks)

must be satisfied.

Although such cases of the formation of square crystals out of rectangular building blocks can be imagined to occur, we will not pursue them further. Instead we consider the cases where the properties of the chemical motifs are such that, with respect to the plane groups P2mm, C2mm, P2mg, P2gg, Pm, Cm, Pg, P2 and P1, the lattice meshes become squares. See next Figures.

Figure above **:** Two-dimensional square crystal, point symmetry **D _{2}** , and supported by a primitive rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length.

The above Figure depicts a two-dimensional crystal with plane group symmetry P2mm and point group symmetry D

The next two Figures also show a square P2mm crystal (as in the Figure directly above), but now with its motifs rotated anticlockwise by 45

Figure above **:** Same as previous Figure. Some symmetry elements indicated **:** 2-fold rotation axes (small yellow solid circles) and mirror lines ( **m** ), demonstrating that the symmetry of the pattern is indeed according to the plane group P2mm.

The next Figures illustrate that a symmetry pattern having

Figure above **:** Construction (by means of auxiliary lines) of a two-dimensional square crystal with C2mm plane group symmetry, and (consequently) D_{2} point group symmetry. See next Figures.

Figure above **:** Two-dimensional square crystal, point symmetry **D _{2}** , and supported by a centered rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length. Compare with its

The next five Figures analyse the above square crystal.

Figure above **:** Two-dimensional square crystal of previous Figure. Some equivalent points indicated. Together they form the *centered rectangular point lattice* of the crystal. In the present case the meshes happen to be squares (as a special kind of rectangle).
Compare with its **rectangular analogue** of Part V .

Above Figure **:** Some points of an alternative set of equivalent points of the crystal of the previous Figures. Like the set depicted above, they also form a possible centered rectangular lattice of the crystal. The next Figure shows how, according to this lattice D_{2} motifs are repeated.

Above Figure **:** Two-dimensional square crystal of previous Figures. Some motifs highlighted. All this in fact shows D_{2} motifs being repeated according to a *centered rectangular point lattice,* which demonstrates that the pattern is indeed a C2mm pattern, despite its square meshes.

Above Figure **:** Two-dimensional square crystal of previous Figures. The repetition of the motifs can also be described with a *rhombic point lattice* (instead of with a centered rectangular lattice, as was done above). In our present case the rhombi of this lattice happen to be squares, which are a particular species of rhombus. Compare this square crystal with its **rectangular analogue** of Part V . Inspecting the present Figure, one would be tempted to see a D_{2} motif which is repeated by two independent translations perpendicular to each other, indicating that the plane group symmetry of the pattern is P2mm (instead of C2mm). But this is not so, because the reflection lines of the motif are not aligned with the edges of the square meshes, which should be so when the pattern's symmetry was according to P2mm, as can be seen in the **earlier Figure depicting a P2mm crystal** , where we see a D_{2} motif repeated according to a *primitive* rectangular point lattice (where in the present case the shape of the meshes has become a square). The next Figure shows this more clearly.

Figure above **:** Same as previous Figure (viz. 2-D square crystal with plane group symmetry C2mm and point group symmetry D_{2}), but rotated 45^{0} to clearly see the non-alignment of the motif's mirror lines with the edges of the square meshes. Compare with the **earlier Figure depicting a P2mm crystal** . The motif's mirror lines only aligns with the edges of the meshes when we have chosen to describe the repetition of the motifs by means of a *centered rectangular lattice.* See **above** , where such a repetition of the motifs, viz. a repetition according to the centered rectangular lattice (the meshes of which have in the present case become squares), is depicted, and where one can see that the mirror lines of the motifs do align with the edges of the (square) meshes.

To obtain the above pattern, i.e. a C2mm pattern allowing for s q u a r e lattice meshes, in particular that of the **above Figure** , **from** the corresponding pattern having *rectangular* lattice meshes as was depicted **earlier,** in Part V , it is __not__ enough simply to change the rectangle, that outlines the rectangular unit mesh, into a square. If we do this we get the following pattern (next Figure), and when we inspect this pattern we see that its plane group symmetry is not according to C2mm anymore but to Pm **:** We see a D_{1} motif repeated according to two translations perpendicular to each other, and because the two points indicated (red) are not equivalent, the lattice is primitiv (i.e. not centered) **:**

In order to get the correct pattern we must change the spacing between the motif units appropriately, as was done

After this analysis of our square C2mm crystal, we continue with the next plane group, viz.

Figure above **:** Construction (by means of auxiliary lines) of a two-dimensional square crystal with **P2mg** plane group symmetry, and (consequently) D_{2} point group symmetry. See next Figure.

Figure above **:** Two-dimensional square crystal, point group symmetry D_{2} , and supported by a primitive rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length. Compare with its **rectangular** counterpart, as depicted in Part V .

Now we're going to derive a

Figure above **:** Construction of a two-dimensional square crystal with plane group symmetry P2gg and (consequently) with point group symmetry D_{2} , based on a primitive rectangular point lattice, in which the meshes happen to be squares (as a particular species of rectangle). The *rectangular* mesh as it was so in the corresponding rectangular version, depicted **earlier,** in Part V , is transformed into a *square* mesh with the motif units at the appropriate locations. The next Figure completes the pattern.

Figure above **:** Two-dimensional square crystal, point symmetry D_{2} , and supported by a primitive rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length. Compare with its rectangular analogue as depicted **earlier,** in Part V .

The next Figure analyses the just obtained pattern. As one can see it has vertical and horizontal glide lines (

The next plane group to be discussed as to how it supports square crystals is

Figure above **:** Construction of a two-dimensional square crystal with plane group symmetry Pm and (consequently) with point group symmetry D_{1} , based on a primitive rectangular point lattice, in which the meshes happen to be squares (as a particular species of rectangle). The *rectangular* mesh as it was so in the corresponding rectangular version, depicted **earlier,** in Part V , is transformed into a *square* mesh with the motif units at the appropriate locations. The next Figure completes the pattern.

Figure above **:** Two-dimensional square crystal, point symmetry D_{1} , and supported by a primitive rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length. Compare with its rectangular analogue as depicted **earlier,** in Part V .

The next two Figures also depict a square Pm crystal, but with its motifs rotated anticlockwise by 45

The next plane group to be discussed as to how it supports square crystals is

Figure above **:** Construction of a two-dimensional square crystal with plane group symmetry Cm and (consequently) with point group symmetry D_{1} , based on a centered rectangular point lattice, in which the meshes happen to be squares (as a particular species of rectangle). The *rectangular* mesh as it was so in the corresponding rectangular version, depicted **earlier,** in Part V , is transformed into a *square* mesh with the motif units at the appropriate locations. The next Figure completes the pattern.

Figure above **:** Two-dimensional square crystal, point symmetry D_{1} , and supported by a centered rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length. Compare with its rectangular analogue as depicted **earlier,** in Part V .

The next Figure analyses the just constructed square crystal. One can see the mirror lines (

The next plane group to be discussed as to how it supports square crystals is

Figure above **:** Construction of a two-dimensional square crystal with plane group symmetry Pg and (consequently) with point group symmetry D_{1} , based on a centered rectangular point lattice, in which the meshes happen to be squares (as a particular species of rectangle). The *rectangular* mesh as it was so in the corresponding rectangular version, depicted **earlier,** in Part V , is transformed into a *square* mesh with the motif units at the appropriate locations. The next Figure completes the pattern.

Figure above **:** Two-dimensional square crystal, point symmetry D_{1} , and supported by a primitive rectangular point lattice (indicated by connection lines) where the two translations happen to be of equal length. Compare with its rectangular analogue as depicted **earlier,** in Part V .

The next Figure analyses the just constructed crystal (but a smaller version of it) as to its symmetry. One can see that the pattern's only symmetry elements are horizontal glide lines, demonstrating that the plane group symmetry of the pattern is indeed that of Pg.

The next Figure is about the plane group

Figure above **:** Two-dimensional square crystal, point symmetry C_{2} , and supported by an oblique point lattice (indicated by connection lines) where the angles happen to be 90^{0} and the two translations to be of equal length.

The next Figure, finally, is about the plane group

Figure above **:** Two-dimensional square crystal, point symmetry C_{1} , and supported by an oblique point lattice (indicated by connection lines) where the angles happen to be 90^{0} and the two translations to be of equal length.

This concludes a summary, showing when, and in what way,

The

**e-mail :**

To continue click HERE

**e-mail :**