Dear Boaz,
I think you are the one who is finally asking the essential question.
The classification we all know about, which goes back to the 19th
century, is not into 230 space groups, but 230 space-group *types*, i.e.
classes where every form of equivalencing (esp. by choice of setting) has
been applied to the enumeration of the classes and the choice of a unique
representative for each of them. This process of maximum reduction leaves
very little room for the introducing "conventions" like a certain ordering
of the lengths of cell parameters. This seems to me to be a major mess-up in
the field - a sort of "second-hand mathematics by (IUCr) committee" which
has remained so ill-understood as to generate all these confusions. The work
on the derivation of the classes of 4-dimensional space groups explained the
steps of this classification beautifully (arithmetic classes -> extension by
non-primitive translations -> equivalencing under the action of the
normaliser), the last step being the choice of a privileged setting *in
termns of the group itself* in choosing the representative of each class.
The extra "convention" a<b<c leads to choosing that representative in a way
that depends on the metric properties of the sample instead of once and for
all (how about that for a brilliant step backward!). Software providers then
have to de-standardise the set of 230 space group *types* (where each
representative is uniquely defined once you give the space group (*type*)
number) to accommodate all alternative choices of settings that might be
randomly thrown at them by the metric properties of e.g. everyone's
orthorhombic crystals. Mathematically, what one then needs to return to is
the step before taking out the action of the normaliser, but this picture
gets drowned in clerical disputes about low-level software issues.
My own take on this (when I was writing symmetry-reduction routines for
my NCS-averaging programs, along with space-group specific FFT routines in
the dark ages) was: once you have a complete mathematical classification
that is engraved in stone (i.e. in the old International Tables and in
crystallographic software as we knew it), then stick to it and re-index back
and forth to/from the unique representative listed under the IT number, as
needed - don't try and extend group-theoretic Tables to re-introduce
incidental metrical properties that had been so neatly factored out from the
final symmetry picture. Otherwise you get a dog's dinner.
So much for my 0.02 Euro.
With best wishes,
Gerard.
--
On Fri, Apr 01, 2011 at 11:30:12AM +0000, Boaz Shaanan wrote:
> Excuse my naive (perhaps ignorant) question: when was the
> a<b<c rule/convention/standard/whatever introduced? None of the
> textbooks I came across mentions it as far as I could see (not that this is
> reason for or against this rule of course).
>
> Thanks,
>
> Boaz
>
>
> Boaz Shaanan, Ph.D.
> Dept. of Life Sciences
> Ben-Gurion University of the Negev
> Beer-Sheva 84105
> Israel
> Phone: 972-8-647-2220 ; Fax: 646-1710
> Skype: boaz.shaanan
--
===============================================================
* *
* Gerard Bricogne g...@globalphasing.com *
* *
* Global Phasing Ltd. *
* Sheraton House, Castle Park Tel: +44-(0)1223-353033 *
* Cambridge CB3 0AX, UK Fax: +44-(0)1223-366889 *
* *
===============================================================
