**Wurtzite 2H**

**Hexagonal 4H**

**Hexagonal 6H**

**Zincblende 3C**

**Rhombohedral 9R**

## Wurtzite 2H

Crystal system: Hexagonal

Space Group: P6_{3}mc

Stacking Sequence: *AB*

A: ^{1}⁄_{3} ^{2}⁄_{3} 0

B: ^{1}⁄_{3} ^{2}⁄_{3} ^{3}⁄_{8}

Example: GaN

## Hexagonal 4H

Crystal structure:^{1}⁄_{2} Hexagonal – ^{1}⁄_{2} Cubic

Space Group: P6_{3}mc

Stacking Sequence: *ABCB*

A_{1}: 0 0 0

A_{2}: ^{1}⁄_{3} ^{2}⁄_{3} ^{4}⁄_{16}

B_{1}: 0 0 ^{3}⁄_{16}

B_{2}: ^{1}⁄_{3} ^{2}⁄_{3} ^{7}⁄_{16}

Example: ZnS

Note:

Clearly the fractions of the z component can be simplified further but that would not show the coherent symmetry of the model. One can see clearly in this form that the component ascends by four steps and alternates between even and odd numbers for the numerator but with equal denominator.

If one is even more precise in their observation, by correlating the 2H-4H-6H denominators, one can see how the pattern just repeats with the denominator being respectively half and three times less the size from 2H, something that makes sense since 4H, 6H are two and three times the size of 2H. This is truly a wonderful depiction of the spatial symmetry that these structures have.

The same applies for 3C, being a diamond structure and needing only 2 components and 9R which exhibits the same translational and spatial symmetry by alternating its x and y components and following the principles of the common denominator, representing a part of a whole.

## Hexagonal 6H

Crystal structure: ^{2}⁄_{3} Hexagonal – ^{1}⁄_{3} Cubic

Space Group: P6_{3}mc

Stacking Sequence: *ABCACB*

A_{1}: 0 0 0

A_{2}: ^{1}⁄_{3} ^{2}⁄_{3} ^{4}⁄_{24}

A_{3}: ^{2}⁄_{3} ^{1}⁄_{3} ^{8}⁄_{24}

B_{1}: 0 0 ^{3}⁄_{24}

B_{2}: ^{1}⁄_{3} ^{2}⁄_{3} ^{7}⁄_{24}

B_{3}: ^{2}⁄_{3} ^{1}⁄_{3} ^{11}⁄_{24}

Example: SiC

Note:

Clearly the fractions of the z component can be simplified further but that would not show the coherent symmetry of the model. One can see clearly in this form that the component ascends by four steps and alternates between even and odd numbers for the numerator but with equal denominator.

If one is even more precise in their observation, by correlating the 2H-4H-6H denominators, one can see how the pattern just repeats with the denominator being respectively half and three times less the size from 2H, something that makes sense since 4H, 6H are two and three times the size of 2H. This is truly a wonderful depiction of the spatial symmetry that these structures have.

The same applies for 3C, being a diamond structure and needing only 2 components and 9R which exhibits the same translational and spatial symmetry by alternating its x and y components and following the principles of the common denominator, representing a part of a whole.

## Zincblende 3C

Crystal system: Cubic

Space group: F43m

Stacking Sequence: *ABC*

A: 0 0 0

B: ^{1}⁄_{4} ^{1}⁄_{4} ^{1}⁄_{4}

Example: GaN

## Rhombohedral 9R

Crystal structure: ^{2}⁄_{3} Cubic – ^{1}⁄_{3} Hexagonal

Space Group: R3m(Hexagonal axes)

Stacking Sequence: *CBABACACB*

A_{1}: 0 0 0

A_{2}: ^{2}⁄_{3} ^{1}⁄_{3} ^{4}⁄_{36}

A_{3}: ^{1}⁄_{3} ^{2}⁄_{3} ^{8}⁄_{36}

B_{1}: 0 0 ^{3}⁄_{60}

B_{2}: ^{2}⁄_{3} ^{1}⁄_{3} ^{7}⁄_{36}

B_{3}: ^{1}⁄_{3} ^{2}⁄_{3} ^{11}⁄_{36}

Example: SiC

Note:

Clearly the fractions of the z component can be simplified further but that would not show the coherent symmetry of the model. One can see clearly in this form that the component ascends by four steps and alternates between even and odd numbers for the numerator but with equal denominator.

If one is even more precise in their observation, by correlating the 2H-4H-6H denominators, one can see how the pattern just repeats with the denominator being respectively half and three times less the size from 2H, something that makes sense since 4H, 6H are two and three times the size of 2H. This is truly a wonderful depiction of the spatial symmetry that these structures have.

The same applies for 3C, being a diamond structure and needing only 2 components and 9R which exhibits the same translational and spatial symmetry by alternating its x and y components and following the principles of the common denominator, representing a part of a whole.

Blue and green images were produced using VESTA [1] by Almpes Kotzai under the supervision of Dr Carol Trager-Cowan. Yellow, orange and blue images made by Elena Pascal using Rayshade 4.0.9 software based on imput files developed by Marc De Graef [2].

[Bibtex]

```
@Article{Momma:db5098,
author = {Momma, Koichi and Izumi, Fujio},
title = {VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data},
journal = {Journal of Applied Crystallography},
year = {2011},
volume = {44},
number = {6},
pages = {1272--1276},
month = {Dec},
abstract = {VESTA is a three-dimensional visualization system for crystallographic studies and electronic state calculations. It has been upgraded to the latest version, VESTA3, implementing new features including drawing the external morphology of crystals; superimposing multiple structural models, volumetric data and crystal faces; calculation of electron and nuclear densities from structure parameters; calculation of Patterson functions from structure parameters or volumetric data; integration of electron and nuclear densities by Voronoi tessellation; visualization of isosurfaces with multiple levels; determination of the best plane for selected atoms; an extended bond-search algorithm to enable more sophisticated searches in complex molecules and cage-like structures; undo and redo in graphical user interface operations; and significant performance improvements in rendering isosurfaces and calculating slices.},
doi = {10.1107/S0021889811038970}
}
```

[Bibtex]

```
@Misc{teachingPointGroup,
author = {De Graef, M.},
title = {Teaching crystallographic and magnetic point group symmetry using three-dimensional rendered visualizations},
howpublished = {International Union of Crystallography},
year = {2008},
url = {https://www.iucr.org/education/pamphlets/23/full-text},
}
```