# ASU Gallery - Guide to notation

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
```
h,k,l are Miller indices and define the normal vector of the cut plane, c is a constant which determines the distance from the origin. x,y,z are fractional coordinates in direct space. The expression h*x+k*y+l*z+c is
• 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.
If all points that are exactly in a cut plane are not inside the asymmetric unit, the condition changes from h*x+k*y+l*z+c>=0 to h*x+k*y+l*z+c>0.

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<=1/2
```
A point x,y,z is inside the asymmetric unit only if all conditions are simultaneously true.

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. This notation is recursive. For example:
```  y<=1/4 [z<=1/2 [x<=1/4]]
```
The third condition only applies if y=1/4 and z=1/2.

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.
The boolean operators may occur at any level in the recursive hierarchy defined by nested square brackets. An example is the asymmetric unit of space group I 41/a (No. 88).
Gallery of direct-space asymmetric units