Space group diagram

If you give me the name of a certain space group, for example, the P222 (orthorhombic system). How could I construct the following diagram? 


[![enter image description here](https://i.stack.imgur.com/uSo9p.png)](https://i.stack.imgur.com/uSo9p.png)


Image source: <http://img.chem.ucl.ac.uk/sgp/large/sgp.htm>


I know it's probably an ambiguous question. I, in the subject of Chemical Crystallography, have given the concepts of sliding plane, helical axis, among others. But, at the time of making this type of diagrams and giving the coordinates of the special positions or those of Wyckoff.




Here is a rough outline how you could proceed:


**Symmetry elements along principal, secondary and tertiary direction**


The space group symbol summarizes the symmetry operations in the crystal. In your example, there are two-fold axis in three directions (222). Which directions depends on the crystal system; for orthorhombic, they are mutually perpendicular. 


For a tetragonal space group, the principal direction is along the $c$-axis, the secondary direction is perpendicular to it along the cell axes, and the tertiary direction is along the diagonal. So for ${P4\_3 2\_1 2}$, there is a ${4\_3}$ axis along $c$, a $\mathrm{2\_1}$ axis along $a$ and $b$, and a 2-fold along the diagonals ($a+b$ and $a-b$).


For a cubic space group, the principal direction is along the cell axes ($a$, $b$, and $c$), the secondary along the body diagonal ($a+b+c$) and the tertiary along the face diagonals ($a+b$, $a+c$, etc).


For the complete set of crystal systems, see [here](http://www.globalsino.com/EM/page1623.html) or [here](http://img.chem.ucl.ac.uk/sgp/misc/notation.htm) for example.


**Additional symmetry elements by combining rotational and translational symmetry**


In your example, there are additional symmetry elements that don't go through the origin, but cut axes at 1/2. These are from combining translation perpendicular to a 2-fold with the 2-fold operation.


**Positions**


If you start with coordinates $(x,y,z)$ and apply the 2-fold axes running along $a$, $b$, and $c$, you get the four general positions of the space group $P222$. The coordinates are given as fractional coordinates. For example, if you apply a two-fold along a to the coordinates $(x,y,z)$, you get $(x,-y,-z)$, the second general position.


Now, you can show these in the diagram. The coordinates in $a$ and in $b$ are shown by positioning, while the coordinate in $c$ is shown by $(+)$ or $(-)$ in this case. As you can see, there are 4 circles inside the cell, corresponding to four positions.


**Centering**


Your example space group is primitive (the $P$ of $P222$), so there is no centering. For space groups with symbols starting with $C$, $I$ or other letters, you would have to add the centering to the general positions, and to the circles in the cell. Combining the centering operation with the symmetry elements given in the space group symbol might give you additional symmetry elements in the cell.


**International tables of crystallography**


There are a finite number of space groups, and they are all known and tabulated in the international tables. So as an exercise, you could go through each one of them, correlating the space groups symbol with the general positions and the arrangement of symmetry elements. 


The International Tables are not available for free online, but are available in many chemistry libraries. Online, you can find a list of space groups at <http://img.chem.ucl.ac.uk/sgp/mainmenu.htm>, and also a [guide to the symbols used](http://img.chem.ucl.ac.uk/sgp/misc/symbols.htm) for the different symmetry elements. [Wikipedia](https://en.wikipedia.org/wiki/List_of_space_groups) also lists the space groups with short names and full names by space group number and crystal system.


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