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.**

Parallelogram / Rhombus

We shall start with the Parallelogram

Let us then consider a fully developed two-dimensional crystal having a **parallelogram** as its intrinsic shape **:**

Any parallelogram can be conceived as being built up by a periodic stacking of microscopic parallelogrammatic units, as the next Figure illustrates.

The true (i.e. intrinsic) point symmetry of such a parallelogrammatic crystal is either according to the Cyclic Group

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

A **Parallelogram** as such has the following symmetries, i.e. it will be superposed upon itself by the following transformations (which are 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.

- When no symmetry is available (
**a**of the above list),**four Forms**are needed to produce a parallelogram. - When only a half-turn is available (
**b**of the above list),**two Forms**are needed to produce a parallelogram.

Plane Group |
Point Symmetry |
Number of Forms |

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

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

The

P2, which has a point symmetry according to C_{2} .

P1, which has a point symmetry according to C_{1} .

For each one of these plane groups we will show how it supports a **parallelogrammatic** crystal shape. The **intrinsic shape** of the crystal, in the present case its parallelogrammatic shape, depends on the Growth Rate Vector Rosette, which in turn depends on the atomic aspects (chemical nature of motifs and geometry of lattice) presented to the growing environment by the possible crystal faces. The parallelogrammatic crystal can be formed by the stacking of parallelogrammatic building blocks (unit meshes) containing a motif s.str. compatible with the point symmetry implied by the given plane group. The point symmetry is indicated in each case. Recall that the point symmetry of a crystal is the translation-free residue of its plane group symmetry.

A parallelogram, which has a symmetry according to the group C

Figure above **:** A parallelogrammatic two-dimensional C_{2} crystal supported by an oblique point lattice (indicated by connection lines).

Figure above **:** A parallelogrammatic two-dimensional C_{1} crystal supported by an oblique point lattice (indicated by connection lines).

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

In the

Now we shall deal with the

Rhombus

So let us consider a fully developed two-dimensional crystal having a **rhombus** as its intrinsic shape (a *rhombus* is a parallelogram with all sides of equal length) **:**

Any rhombus-shaped crystal can be conceived as being built up by a periodic stacking of microscopic rhombus-shaped units, as the next Figure illustrates.

The true point symmetry of such an intrinsically rhombic crystal is either according to the Dihedral Group

That a **rhombic crystal** (i.e. a two-dimensional crystal having as its intrinsic shape the rhombus) can possess either of these symmetries can be explained succinctly as follows (and will be further evident in the sequel) **:**

A **Rhombus** 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 connecting two opposite corners.
- Reflection in a line connecting two opposite corners and perpendicular to the one just mentioned (c).

- When no symmetry is available (
**a**of the above list),**four Forms**are needed to produce a rhombus. - When only a half-turn is available (
**b**of the above list),**two Forms**are needed to produce a rhombus. - When only one reflection is available (
**c**__or__**d**of the above list),**two Forms**are needed to produce a rhombus. - When two reflections are available (
**c**__and__**d**of the above list), only**one Form**is needed to produce a rhombus.

The next table shows which

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 |

So the Rhombus is supported by the following plane groups

P2mm, C2mm, P2mg, P2gg, which have a point symmetry according to D_{2} .

Pm, Cm, Pg, which have a point symmetry according to D_{1} .

P2, which has a point symmetry according to C_{2} .

P1, which has a point symmetry according to C_{1} (asymmetric group).

For each of these plane groups we will show how it supports a **rhombic** crystal shape. The **intrinsic shape** of the crystal, in the present case its rhombic shape, depends on the Growth Rate Vector Rosette, which in turn depends on the atomic aspects (chemical nature of motifs and geometry of lattice) presented to the growing environment by the possible crystal faces. The rhombic crystal can be formed by the stacking of rectangular or rhombic building blocks (unit meshes) containing a motif s.str. compatible with the point symmetry implied by the given plane group. The point symmetry is indicated in each case. Recall that the point symmetry of a crystal is the translation-free residue of its plane group symmetry.

A rhombus, which has a symmetry according to the group D_{2} , 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 Rhombus. This does not necessarily hold the other way around **:** A crystal can have an intrinsic shape that is a rhombus, but nevertheless some, or even all, symmetries of this rhombus 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 **:** A rhombus-shaped two-dimensional D_{2} crystal supported by a primitive rectangular point lattice (indicated by connection lines).

In order to have a better overview of such a crystal, we depict a somewhat different and smaller version of it

Figure above **:** A rhombus-shaped two-dimensional D_{2} crystal supported by a primitive rectangular point lattice (indicated by connection lines).

Figure above **:** A rhombus-shaped two-dimensional D_{2} crystal supported by a rhombic point lattice (indicated by connection lines).

Figure above **:** A rhombus-shaped two-dimensional D_{2} crystal supported by a primitive rectangular point lattice (indicated by connection lines).

**:** A rhombus-shaped two-dimensional D_{2} crystal supported by a primitive rectangular point lattice (indicated by connection lines).

Figure above **:** A rhombus-shaped two-dimensional D_{1} crystal supported by a primitive rectangular point lattice (indicated by connection lines).

Figure above **:** A rhombus-shaped two-dimensional D_{1} crystal supported by a rhombic point lattice (indicated by connection lines).

Figure above **:** A rhombus-shaped two-dimensional D_{1} crystal supported by a primitive rectangular point lattice (indicated by connection lines).

Figure above **:** A rhombus-shaped two-dimensional C_{2} crystal supported by an oblique point lattice (indicated by connection lines) where the angles happen to be 90^{0}.

Figure above **:** Same as previous Figure, but smaller version **:** A rhombus-shaped two-dimensional C_{2} crystal supported by an oblique point lattice (indicated by connection lines) where the angles happen to be 90^{0}.

Figure above **:** A rhombus-shaped two-dimensional C_{1} crystal supported by an oblique point lattice (indicated by connection lines) where the angles happen to be 90^{0}.

Figure above **:** Same as previous Figure, but smaller version **:** A rhombus-shaped two-dimensional C_{1} crystal supported by an oblique point lattice (indicated by connection lines) where the angles happen to be 90^{0}.

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

In the

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