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# Orientation relationship

Dave Schuller schuller at indigo1.biomol.uci.edu
Tue Feb 13 17:27:05 EST 1996

```Naiming Chu <zu04117 at uabdpo.dpo.uab.edu> wrote:
>I am working on a dimer structure which has 1,2 molecules in asymmetry
>unit. Its space group is P21.
>...
> My question is why the rotation angles are not same before and after
> symmetry operation?

clearly, 1' is going to be rotated by 180 d from 1.  The axis of this
rotation is the crystallographic 2-fold symmetry axis (|| to the Y axis).
The reason the rotation angle from 1 to 2 is not the same as that from
1' to 2 is that there is no special relationship between the crystallographic
axis and the noncrystallographic axis, which is specified by direction vector
(0.4555, 0.0080, 0.8902).  These rotation axes are not parallel, nor are they
perpendicular.  (Hey, maybe thats why they call it NONCRYSTALLOGRAPHIC
SYMMETRY!)
So, if you do a 180 d rotation about one axis followed by a
180 d rotation about the other, you should not expect the resultant combined
operation to be a 180 d rotation about any particular single axis.  To convince
yourself of this, sit down for a while with some asymmetrical objects such
as toy soldiers.  Sure, people will look at you funny, but you want to learn
this, right?

If Mc is the crystallographic operator 1 -> 1'
we can specify the inverse operator 1' -> 1 as Md.
[Md = Mc in this case; except the translation vector is  (0, -0.5, 0) in
fractional coordinates instead of (0, 0.5, 0) but these are equivalent]

if M1 is the operator 1 -> 2

then M1', the operator for 1' -> 2, can be described by
M1' = (Md)(M1)

If you want to calculate some of these things for your case,
I happen to have some utilities for inverting matrices and multiplying them.
ftp indigo4.biomol.uci.edu
cd pub/schullersoft
binary
get ncs.tar.Z
check out sqmatinv and sqmatmul.

I cannot work out the correct answer for you since you do not include complete
transformations in your message.  If you work out a transformation and want to
check if it is reasonable, the most straightforward way to do this is to
actually apply the transformation to a set of coordinates and see if they end
up where you expected them to.  I have a utility to do this, pdb_rotm, and
there are certainly others available; I think it can even be done in X-plor.

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