CARAT is a computer package which handles enumeration, construction, recognition,
and comparison problems for crystallographic groups up to dimension 6.
The name CARAT itself is an acronym for Crystallographic AlgoRithms And
CARAT is a compilation of various programs written in C developed under
HP-UX and Linux, and should be portable to most Unices.
In particular CARAT does not come together with an environment, but
relies on the ordinary unixes shell and files for input and output.
This is one of the points which distiguishes CARAT from most other packages
for computer algebra, like GAP
. If you would like such a user interface, the current version of GAP comes
with an interface to CARAT, which enables one to use the most important
functions of CARAT, but not all.
Some of the features of CARAT
CARAT can construct any space group up to degree 6 from its build in table
CARAT can name a space group R given by generators unily, ie. generate
a name which only depends on the isomorphism of R and determines it uniquely.
Decide Q-, Z- and affine equivalence of finite unimodular groups and space
Calculate the integral normalizer of a finite unimodular group.
CARAT-introduction and Page of low dimensional Bieberbach groups
Below we listes a few papers dealing with
CARAT or have been written with the help of it:
J. Opgenorth, W. Plesken, T. Schulz: Crystallographic
algorithms and tables, Acta. Cryst (1998), 517-531.
deals with the basic algorithms behind CARAT, and can be viewed as an introduction
to the underlying structures to crystallographic groups from an algorithmic
point of view.
W. Plesken, T. Schulz: Counting crystallographic
groups in low dimensions, to appear in Exp. Math. Issue 3 (2000), 407-411.
The paper describes the results obtained by CARAT in counting all space
groups up to dimension 6.
C. Cid, T. Schulz: Computation of Five and Six Dimensional Bieberbach
Groups Exp. Math. Issue 1 (2001), 109-115.
This paper is concerned with the computation and classification of 5- and 6-dimensional torsion-free
crystallographic groups, also known as Bieberbach groups.
As a result, we have the following table for Crystallographic and Bieberbach groups up
to dimension 6:
|No of Q-classes
|No of Z-classes
|No of affine classes
|No of Bieberbach groups
You can download the
newest version of CARAT
(version 2.1b1 from 2008-07-19). For
installation, please extract the file downloaded and refer to the file README.md
Changes of the new version
- Major rewrite of the code...
- New build system.
- CARAT has been moved to GitHub.
If you have any question concerning CARAT (ie. installation etc.),
or find a bug in the package, please do not hesitate to contact us:
Lehrstuhl B für Mathematik
Prof. Dr. i.R. W. Plesken
Tel: +49 241 80 97079