HAPcryst

A GAP4 extension for HAP calculating resolutions for Bieberbach groups.

Marc Roeder
NUI Galway, Ireland

This package is an add-on for Graham Ellis’ HAP package. HAPcryst implements some functions for crystallographic groups, namely OrbitStabilizer-type methods. It is also capable of calculating free resolutions for Bieberbach groups.

HAPcryst is distributed under the terms of the GNU General Public License version 2.0 or later. See LICENSE for the text of version 2, or visit https://gnu.org/copyleft/gpl.html.

Please send bug reports to https://github.com/gap-packages/hapcryst/issues.

Installation

HAPcryst depends on the following packages:

hap (as HAPcryst is an extension of HAP)
polymaking (which in turn depends on external software)
Cryst

HAPcryst will not load if you do not have a recent version of those packages installed.

The following packages are not required but suggested:

Carat
CrystCat
GAPDoc

Download the package archive HAPcryst<ver>.tar.bz2, HAPcryst<ver>.tar.gz, or HAPcryst<ver>-win.zip (where <ver> is some version number) to the pkg/ directory of the GAP home directory. If you do not have permission to do so, create a directory called gap/pkg in your home directory.
Change directory to pkg/ and unpack the archive using the corresponding command:
```
tar -xjf HAPcryst<ver>.tar.bz2
tar -xzf HAPcryst<ver>.tar.gz
unzip HAPcryst<ver>-win.zip
```
Replace <ver> with the version number.
Start GAP. If you have created the directory gap/pkg in your home directory, use gap -l '<homedir>/gap;' where <homedir> is the path of your home directory. Use pwd to find out what it is if you do not know.
Call GAP and type LoadPackage("hapcryst"); to load HAPcryst. If GAP returns some error messages, load hap.
Run the test file by calling ReadPackage("hapcryst","tst/testall.g");.

Recommendation

Calculating resolutions of Bieberbach groups involves convex hull computations. polymake by default uses cdd to compute convex hulls. Experiments suggest that lrs is the more suitable algorithm for the convex hull computations done in HAPcryst than the standard cdd. You can change the behaviour of polymake by editing the file <homedir>/.polymake/prefer.pl. It should contain a section like this; just make sure lrs is before cdd. The position of beneath_beyond does not matter.

application polytope;

prefer "*.convex_hull  lrs, beneath_beyond, cdd";

Documentation

The documentation of the package can be found in the doc/ subdirectory.

A second part of the documentation describes datatypes used in HAPcryst. This can be found in the lib/datatypes/doc subdirectory of the HAPcryst main directory.

Examples

The examples directory contains code that generates pictures (JavaView applets) of Dirichlet domains of 3-dimensional Bieberbach groups with coloured orbits of the group acting on the faces of the tessellation of R^3.

gap> ReadPackage("hapcryst","examples/3dimBieberbachFD.gap");
gap> viewFundamentalDomain([0,0,0],1);

For instructions, see examples/3dimBieberbacFD.gap in the hapcryst package tree. The functions will also generate pictures of Dirichlet domains for non-Bieberbach groups, but the orientation of the shown edges does not necessarily make sense any more.