Reference: IUCr Computing Commission Newsletter No. 2, July 2003

Each cut plane of an asymmetric unit is defined by a condition of the form

h*x+k*y+l*z+c>=0

- exactly zero for points in the cut plane.
- greater than zero for points inside the asymmetric unit.
- less than zero for points outside the asymmetric unit.

To enhance readability the asymmetric unit conditions (shown under the pictures
in the gallery) are simplified by omitting terms with zeros (e.g.
`0*x`) and unit factors (e.g. `x` instead of
`1*x`). The constant term `c` is moved to the right-hand
side. For example:

x>=0 y<=1/4 z<1 x-y<=1/2A point

Often a face or edge on the surface of the asymmetric unit is only partially inside. The dividing lines are defined by face- or edge-specific sub-conditions. For example:

y<=1/4 [z<=1/2]The first condition defines the face as before. The second condition in square brackets only applies if

y<=1/4 [z<=1/2 [x<=1/4]]The third condition only applies if

Some asymmetric units require the combination of conditions with
the boolean operators *and* or *or*. For example:

y>=0 [x<=0 | x>=1/4] y<=1/4 [z>=1/8 & z<=5/8]In words:

- If
`y`is exactly zero, a point is inside the asymmetric unit only if`x`is less than or equal to zero or greater than or equal to`1/4`. - If
`y`is exactly`1/4`, a point is inside the asymmetric unit only if`z`is greater than or equal to`1/8`and at the same time less than or equal to`5/8`.

Gallery of direct-space asymmetric units