Dear gap-forum,
I am sorry, there were two wrong lines in my first answer
to Giancarlo Bassi's letter. Here is a new attempt.
Giancarlo Bassi wrote:
I read from some books about the 17 wall-paper groups.
According to Coxeter's notation these groups are represented by these
symbols with the following meaning:
p1 two translations
p2 three half turns
pg two parallel glide reflections
pm two reflections and a translation
cm a reflection and a parallel glide reflection
pmm reflection in the four sides of a rectangle
pmg A reflection and two half-turns
pgg Two perpendicular glide reflections
cmm two perpendicular reflection and a half-turn
pgg two perpendicular reflection and a half-turn
p4 a half turn and a quarter turn
p4m Reflections in three sides of a (45,45,90) triangle
p4g A reflection and a quarter-turn
p3 Two rotations through 120
p3m1 A reflection and a rotation through 120
p31m Reflection in the three sides of an equilateral triangle
p6 A half-turn and a rotation through 120
p6m Reflections in the three sides of a (30,60,90) triangle
According Grossman Magnus's book the wall-paper figures are
present in the graphs which can completely cover the plane
by a fundamental region.
I have little experience with GAP too.
I know there's a GAP-package for crystallographic groups.
Now my question:
How can I identify or build the 17 groups by GAP?
---------------------------------------------------------------------
Dear Giancarlo Bassi,
the two-dimensional wall-paper groups are available in GAP under the
following Hermann-Mauguin symbols:
"p1", "p2", "pm", "pg", "cm", "p2mm", "p2mg", "p2gg", "c2mm", "p4", "p4mm", "p4gm", "p3", "p3m1", "p31m", "p6", "p6mm".
Note that some of these symbols differ from those in your list.
GAP provides display commands like
gap> DisplaySpaceGroupType( "p2gg" ); #I Space-group type (2,2,2,1,3); IT(8) = p2gg; orbit size 1
or
gap> DisplaySpaceGroupGenerators( "p2gg" );
#I Non-translation generators of SpaceGroupOnLeftBBNWZ( 2, 2, 2, 1, 3 )[ [ 1, 0, 1/2 ], [ 0, -1, 1/2 ], [ 0, 0, 1 ] ] [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ] ]
or commands to actually construct the groups like
gap> s := SpaceGroupBBNWZ( "p2gg" ); SpaceGroupOnRightBBNWZ( 2, 2, 2, 1, 3 )
You should perhaps look through the CrystCat manual to get the full
list of the GAP functions that accept an Hermann-Mauguin symbol as
argument.
With kind regards, Volkmar Felsch (Aachen)
Miles-Receive-Header: reply