For Fortran programs, the original set of subroutines (held in symlib.f) has been replaced by an CSYM library interface to the C library. From the point of view of an application programmer, this interface should be identical to the original set of subroutines. This document originates from the original Fortran library, but should be applicable to the new library.
The available Fortran calls have been arranged as much as possible into groups by function. There are often several versions of calls apparently performing the same or very similar tasks, reflecting the policy of never removing existing functionality in order to maintain compatibility with programs written using older versions of the library.
syminfo.lib holds information for all the standard spacegroups in the International Tables. For each spacegroup, several alternative settings are included (e.g. "P 1 2 1", "P 1 1 2" (a.k.a. 1003) and "P 2 1 1" for spacegroup 3).
number = standard spacegroup number
basisop = change of basis operator
symbol ccp4 = CCP4 spacegroup number e.g. 1003
(0 if not a CCP4 group)
symbol Hall = Hall symbol
symbol xHM = extended Hermann Mauguin symbol
symbol old = CCP4 spacegroup name
(blank if not a CCP4 group)
symbol laue = Laue group symbol
symbol patt = Patterson group symbol
symbol pgrp = Point group symbol
hklasu ccp4 = reciprocal space asymmetric unit
(with respect to standard setting)
mapasu ccp4 = CCP4 real space asymmetric unit
(with respect to standard setting)
(negative ranges if not a CCP4 group)
mapasu zero = origin based real space asymmetric unit
(with respect to current setting)
mapasu nonz = non-origin based real space asymmetric unit
(with respect to current setting)
cheshire = Cheshire cell
(with respect to standard setting)
symop = list of primitive symmetry operators
cenop = list of centering operators
For example:
begin_spacegroup number 3 basisop z,x,y symbol ccp4 1003 symbol Hall ' P 2y (z,x,y)' symbol xHM 'P 1 1 2' symbol old 'P 1 1 2' symbol laue '-P 2' '2/m' symbol patt '-P 2' '2/m' symbol pgrp ' P 2' '2' hklasu ccp4 'k>=0 and (l>0 or (l=0 and h>=0))' mapasu ccp4 0\<=x\<=1/2; 0\<=y\<1; 0\<=z\<1 mapasu zero 0\<=x\<1; 0\<=y\<=1/2; 0\<=z\<1 mapasu nonz 0\<=x\<1; 0\<=y\<=1/2; 0\<=z\<1 cheshire 0\<=x\<=1/2; 0\<=y\<=0; 0\<=z\<=1/2 symop x,y,z symop -x,-y,z cenop x,y,z end_spacegroup
A call to Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RlSymmMatrx) MSYMLB3 should be used to retrieve information from the symmetry library. Note that not all the data items are compulsory for MSYMLB3, although older versions of the routine (MSYMLB2, MSYMLB, MSYGET) still need them.
Typically a symmetry operation RSym will consist of a rotation operation R and a translation operation T (basically a vector). These are applied to a vector x to obtain x':
It is convenient to represent the rotation by a 3*3 matrix:
( R11 R12 R13 )
[R] = ( R21 R22 R23 )
( R31 R32 R33 )
and the translation by a column vector with 3 elements:
( T1 )
[T] = ( T2 )
( T3 ).
CCP4 uses 4x4 arrays to store these symmetry operations as follows:
RSym = ( R11 R12 R13 T1 )
( R21 R22 R23 T2 )
( R31 R32 R33 T3 )
( 0 0 0 1 )
or
RSym = ( [R] | [T] )
( 0 0 0 | 1 )
Essentially this is a 4x4 matrix holding 3x3 transformation matrix in the "top-left-hand corner", the 3-element column (translation) vector in the "top-right-hand corner", and then (0 0 0 1) in the bottom row.
The subroutine Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RlSymmMatrx) MSYMLB3 will obtain the set of symmetry matrices in this representation for a given spacegroup, whilst Subroutine SYMFR2(ICOL,I1,NS,ROT) SYMFR2 or Subroutine SYMFR3(ICOL,I1,NS,ROT,EFLAG) SYMFR3 will obtain individual matrices from the string representation mentioned above. (Subroutine SYMTR4(NSYM,RSM,SYMCHS) SYMTR4 will perform the inverse operation, converting matrices to string representation.)
RSym . [xf]
[yf]
[zf]
[1 ]
will apply both the symmetry and the translation operations to real space coordinates with a single matrix multiplication. The CCP4 modlib MODLIB library provides matrix_vector_ops matrix-vector routines MATVEC4 and TRANSFRM which can be used to perform this operation.
Note that only the operations in the appropriate Laue group are applied to reflection indices, so there are no translational components (i.e. the vector part of the operation, [T], is zero). The subroutine Subroutine INVSYM(S,ST) INVSYM will invert a 4x4 matrix stored in this representation, for the purpose of applying symmetry operations to reflection indices.
while reciprocal space axis vectors are transformed like real space coordinates, i.e.
(See also the REINDEX documentation.)
x+z,z,-y), reciprocal space operations (e.g. h,l-h,-k) and reciprocal- and real-space axis vector operations (e.g. a*+c*,c*,-b* and a,c-a,-b respectively). SYMFR3 only translates real space coordinate operations. Use MSYMLB3 to obtain the set of symmetry operator matrices given the spacegroup name or number. SYMFR2/3 will generate individual symmetry operator matrices from their string representation (useful if the operators are a subset of a spacegroup). SYMTR4 performs the opposite action, and generates string representations of individual symmetry operations from the matrices.
INVSYM will generate the inverse matrix of a real space symmetry operation, to be applied to reflection indices as described in HOW SYMMETRY OPERATIONS ARE STORED AND APPLIED section 2.
These routines all derive additional information from the symmetry operators or the spacegroup name. The subroutine HKLRANGE returns the information stored in the common block which it shares with PGNLAU
Nb: routines Logical Function CENTRC(KHKL,ICENT) CENTRC and CENPHS both appear to be unused.
Call CENTRIC once to set up the symmetry elements in common blocks shared with CENTR. This defines the set of centric reflections. Then for each reflection, a call to CENTR will return whether it is centric.
Logical Function HKLEQ(IH,KH) HKLEQ - used in SCALA to test if two reflections have equal indices.
Both these routines are called from within other routines, although they can also be called independently. ASUSET must be called before INASU can be used.
Neither of the routines XSPECIALS or KROT appear to be used in supported CCP4 programs.
These routines are not directly related to symmetry operations. Hashing is a method of storing data value pairs in such a way that they can be be efficiently retreived later on; the hash table is the resulting data structure.
These three routines share the common block RECPLT. SETRSL and STHLSQ are used only in CAD, whilst STS3R4 does not appear in any supported program.
This is how the routines are used in CAD. A call to SETRSL with the cell dimensions and angles sets up coefficients in RECPLT, which are then used by the function STHLSQ to calculate the quantity "(sin(theta)/lamba)**2" for any given set of h, k, l indices. From Bragg's Law, this quantity is equal to 1/(4*d**2), that is, one-quarter of the resolution. Within CAD, multiplication by 4 yields the resolution 1/d**2.
The exact function of this routine is unclear and it does not appear in any supported program.
On input:
On input:
On input:
On input:
ARGUMENTS
Taken from Alexei Vagin
When ccp4_hash_lookup(nser) is later evaluated it will return nfind
This function will allow the efficient retrieval of an identifier for a large range variable (such as a crystal number). The values of the function ccp4_hash_lookup(nser) are stored in the array it(2, kpri) where kpri is the prime number used to generate the function.
The array 'it' lives in the common block which is shared by ccp4_hash_setup and the function ccp4_hash_lookup
NOTES: A hash table is a way of storing information so that it easily be retrieved without the need for indexing or long searches. NSER is referred to as the "key", which is "hashed" (computer- science speak for "messed up") by the hashing function (in this case MOD(NSER4,KPRI) + 1) to determine where the value pair will be stored. The function LOOKUP can then search on the same basis when supplied with the key, to retreive the pair in (at most) 3 calculations. Note that KPRI (the table size) MUST BE A PRIME in order for this method to work.
Returns IC
Determine whether a reflection is centric (return IC=1) or not (IC=0). If none of the zone tests is satisfied, the reflection is non-centric.
In this case the required tests are controlled by 7 flags in icent for
(the last is needed in pg312)
It decides how many centric zones there are, and flags them.
set up tests for 0kl h0l hk0 hhl hkh hkk h,-hl hk-h hk-k -h 2h l 2h -h l hkl
Zones are encoded using an idea from a program by Bricogne. If h*zone(1) + k*zone(2) + l*zone(3) is equal to 0.0, that reflection is in that zone. All that is needed is the most general conditions - a reflection is either centric or not.
This is Randys program description:
Returns EPSI ( epsilon zone) , and ISYSAB flag. Systematic absences flagged with ISYSAB = 1
Find the zone a reflection falls into, and return the appropriate value for the reflection multiplicity factor. Each reflection must have a zone.
On entry:
Returns true if indices ih = kh
Arguments: (INTEGER) HRNG0,HRNG1,KRNG0,KRNG1,LRNG0,LRNG1
INASU = +1 if h k l chosen
INASU = -1 if -h-k-l chosen
INASU = 0 if reflection is out-of-bounds
Arguments:
Returns KROT=0 correct operation, =1 if not.
Arguments:
Arguments:
NAMSPG_CIF should be as in _symmetry.space_group_name_H-M
It is satisfied by the first fit it finds:
eg: 47 8 8 Pmmm PGmmm ORTHORHOMBIC 'P 2/m 2/m 2/m' 'P m m m'
You could call the subroutine with NAMSPG_CIF = 'Pmmm' or 'P 2/m 2/m 2/m' or 'P m m m'
But it will always return the LONGEST possible name. ie 'P 2/m 2/m 2/m'
LSPGRP NLINS and contain either NAMSPG or NAMSPG_CIF
LSPGRP NLINS NLINP NAMSPG NAMPG CRYSTAL NAMSPG_CIF
where:
On entry:
Returns:
On entry:
This subroutine chooses the primitive set of symmetry operators.
If necessary it re-orders the symmetry operators to give the primitive ones first.
This subroutine works out the point group name NAMPG. That is ; it checks rotation axes, etc etc and recognises these point groups. (It DOES NOT cope with mirror planes etc)
Gronigen MDF stuff: It now sets up the common block MDFPAR for MDF file mods and fills in the symmetry info. See subroutine for details.
Use this subroutine to transfer information to and from MDFPAR.
If JLASS eq 0 then fill JLASS JCENTR JSCREW from common block.
If JLASS gt 0 then fill KLASS ICENTR ISCREW in common block.
On entry:
This subroutine returns a laue code number used to choose the unique region of reciprocal space for each point group.
The number nlaue is the same as the one set in CAD for this purpose.
Pointgroup Laue group Limits
3 pg1 1bar hkl:l>=0 hk0:h>=0 0k0:k>=0 1,2
pg1bar
4 pg2 (b) 2/m hkl:k>=0, l>=0 hk0:h>=0 3/b,4/b....
pgm pg2/m
5 pg2 (c) 2/m hkl:k>=0, l>=0 h0l:h>=0 1003,1004
6 pg222 mmm hkl:h>=0, k>=0, l>=0 16 ...
pgmm2 pgmmm
7 pg4 4/m hkl:h>=0, l>=0 with k>=0 if h=0 and
pg4bar pg4/m k>0 if h>0
8 pg422 4/mmm hkl:h>=0, k>=0, l>=0 89..
pg4mm pg4bar2m pg4barm2 pg4/mmm
9 pg3 3bar hkl:h>=0, k>0 00l:l>0 143..
pg3bar
10 pg312 3/m hkl:h>=0, k>=0 with k\<=h for all l.
pg32 pg3m pg3m1 pg3barm1 if k = 0 l>=0
Space group numbers : 149-151-153 157 159 162 163
11 pg321 3bar1m hkl:h>=0, k>=0 with k\<=h for all l.
pg31m pg3bar1m if h = k l>=0
Space group numbers : 150-152-154
12 pg6 6/m hkl:h>=0, k>=0, l>=0 with k>=0 if h=0
pg6bar 6/m and k> 0 if h>0
13 pg622 6/mmm hkl:h>=0, k>=0, l>=0 with h>=k 177..
pg6mm pg6barm2 pg6bar2m pg 6/mmm
14 pg23 m3 hkl:h>=0, k>=0, l>=0 with l>=h, k>=h
pgm3bar
15 pg432 m3m hkl:h>=0, k>=0, l>=0 with k>=l
pg4bar3m pgm3barm
This has been modified by permuting the Nth column by matrix PERM.
C---- Permute C C DO 10 I = 1,3 C BV(I) = PERM(I,1)*JV(N,1) + PERM(I,2)*JV(N,2) + C + PERM(I,3)*JV(N,3) C 10 CONTINUE C C---- Copy back C C DO 20 I = 1,3 C JV(N,I) = NINT(BV(I)) C 20 CONTINUE
See PRMVCI - this routine is its real equivalent.
The real-space symmetry matrices are applied by premultiplying them by a row vector hkl, ie (h'k'l') = (hkl)R
All angles in radians
Matrices passed in Common block ATSYM
Input:
The sampling intervals must satisfy the following conditions:
PG1 PG1bar PG2 PGm PG2/m PG222 PGmm2 PGmmm PG4 PG4bar PG4/m PG422 PG4mm PG4bar2m PG4/mmm PG3 PG3bar PG32 PG3m PG3barm PG6 PG6bar PG6/m PG622 PG6mm PG6bar2m PG6/mmm PG23 PGm/3bar PG432 PG4bar3m PGm3bar m
We use:
PG1 PG1bar PG2 PG2/m PG222 PGmmm PG4 PG4/m PG422 PG4/mmm PG3 PG3bar PG32 PG3bar/m PG6 PG6/m PG622 PG6/mmm PG23 PGm/3bar PG432 PGm3barm
For grid restrictions we only need to know the laue number. Here is the table:
3 pg1 1bar hkl:l>=0 hk0:h>=0 0k0:k>=0 1,2
4 pg2 2/m hkl:k>=0, l>=0 hk0:h>=0 3/b,4/b....
5 pg2(c) 2/m hkl:k>=0, l>=0 h0l:h>=0 1003,1004
6 pg222 mmm hkl:h>=0, k>=0, l>=0 16 ...
7 pg4 4/m hkl:h>=0, l>=0 with k>=0 if h=0 and
8 pg422 4/mmm hkl:h>=0, k>=0, l>=0 89..
9 pg3 3bar hkl:h>=0, k>0 00l:l>0 143..
10 pg312 3/m hkl:h>=0, k>=0 with k\<=h for all l.
if k = 0 l>=0
Space group numbers : 149-151-153
11 pg321 3/m hkl:h>=0, k>=0 with k\<=h for all l.
if h = k l>=0
Space group numbers : 150-152-154
12 pg6 6/m hkl:h>=0, k>=0, l>=0 with k=0 if h=0
13 pg622 6/mmm
14 pg23 m3
15 pg432 m3m
On entry:
NB This s/r differs from SETLIM in that the limits are taken from cctbx via CCP4's syminfo.lib file.
On entry:
First calculates the components of input axes in an orthonormal basis, then calculate components of reciprocal axes in same basis.
The input angles are in degrees.
SYMFR2 recognises the following types of input:
real space symmetry operations, e.g. X+1/2,Y-X,Z
reciprocal space operations, e.g. h,l-h,-k
reciprocal axis vectors, e.g. a*+c*,c*,-b*
real space axis vectors, e.g. a,c-a,-b
The subroutine returns the appropriate 4x4 transformation matrix for each operation. The calling program must interpret the resulting matrix(ces) correctly.
On entry I1 is the first character of ICOL to look at (say after keyword 'SYMM')
NS is the number of the first symmetry operation to be read, & returns with the number of the last one read.
On exit, ROT(4,4,NS) contains the real-space symmetry matrices, in standard convention, i.e.
Input:
Arguments:
This translates the symmetry matrices RSM(4,4,NSM) into INT TAB character strings
It gives the real and reciprocal space operations.
eg X,Y,Z H , K, L
eg -Y,X-Y, Z -H-K, H, L etc
This translates the symmetry matrices RSM(4,4,NSM) into INT TAB character strings
It gives the real and reciprocal space operations.
eg X,Y,Z H , K, L
eg -Y,X-Y, Z -H-K, H, L etc
Arguments :
This translates the symmetry matrices RSM(4,4,NSM) into INT TAB character strings
It gives the real and reciprocal space operations.
eg X,Y,Z H , K, L
eg -Y,X-Y, Z -H-K, H, L etc
Arguments :
Returns ISYSAB flag. Systematic absences flagged with ISYSAB = 1 Only reflns with EPSI > 1 need be considered.
This subroutine finds what coordinates occupy special positions i.e. have occupancies less than 1.0 from consideration of the symmetry operations.
Below are the definitions of the a. Reciprocal Space Asymmetric Unit Definitions reciprocal space and the b. Real Space Asymmetric Unit Definitions real space asymmetric units under the CCP4 convention.
| Pointgroup | Laue group | Limits | Spacegroup Nos | |
|---|---|---|---|---|
| 3 | pg1 pg1bar | 1bar | hkl:l>=0 hk0:h>=0 0k0:k>=0 | 1,2 |
| 4 | pg2 (b) pgm pg2/m | 2/m | hkl:k>=0, l>=0 hk0:h>=0 | 3,4.... |
| 5 | pg2 (c) | 2/m | hkl:k>=0, l>=0 h0l:h>=0 | 1003, 1004 |
| 6 | pg222 pgmm2 pgmmm | mmm | hkl:h>=0, k>=0, l>=0 | 16 ... |
| 7 | pg4 pg4bar pg4/m | 4/m | hkl:h>=0, l>=0 with k>=0 if h=0 and k>0 if h>0 | 75,.. |
| 8 | pg422 pg4mm pg4bar2m pg4barm2 pg4/mm | 4/mmm | hkl:h>=0, k>=0, l>=0 | 89,.. |
| 9 | pg3 pg3bar | 3bar | hkl:h>=0, k>0 00l:l>0 | 143,.. |
| 10 | pg312 pg32 pg3m pg3m1 pg3barm1 | 3/m | hkl:h>=0, k>=0 with k<=h for all l. if k=0 l>=0 | 149 151 153 157 159 162 163 |
| 11 | pg321 pg31m pg3bar1m | 3bar1m | hkl:h>=0, k>=0 with k<=h for all l. if k=h l>=0 | 150 152 154 |
| 12 | pg6 pg6bar | 6/m | hkl:h>=0, k>=0, l>=0 with k>=0 if h=0 and k>0 if h>0 | 168.. |
| 13 | pg622 pg6mm pg6barm2 pg6bar2m pg6/mmm | 6/mmm | hkl:h>=0, k>=0, l>=0 with h>=k | 177.. |
| 14 | pg23 pgm3bar | m3 | hkl:h>=0, k>=0, l>=0 with l>=h, k>=h | 195.. |
| 15 | pg432 pg4bar3m pgm3barm | m3m | hkl:h>=0, k>=0, l>=0 with k\=1 | 209.. |
| No. | Spacegroup | Upper limits on x, y, z asterisk (*) |
|---|---|---|
| 1 | P 1 | x < 1, y < 1, z < 1, |
| 2 | P -1 | x < 1, y <= 1/2, z < 1, |
| 3 | P 1 2 1 | x <= 1/2, y < 1, z < 1, |
| 4 | P 1 21 1 | x < 1, y < 1/2, z < 1, |
| 5 | C 1 2 1 | x <= 1/2, y < 1/2, z < 1, |
| 10 | P 1 2/M 1 | x <= 1/2, y <= 1/2, z < 1, |
| 16 | P 2 2 2 | x <= 1/2, y <= 1/2, z < 1, |
| 17 | P 2 2 21 | x <= 1/2, y <= 1/2, z < 1, |
| 18 | P 21 21 2 | x < 1, y <= 1/4, z < 1, |
| 19 | P 21 21 21 | x < 1, y < 1, z <= 1/4, |
| 20 | C 2 2 21 | x <= 1/2, y <= 1/4, z < 1, |
| 21 | C 2 2 2 | x <= 1/2, y <= 1/4, z < 1, |
| 22 | F 2 2 2 | x <= 1/4, y <= 1/4, z < 1, |
| 23 | I 2 2 2 | x <= 1/2, y <= 1/4, z <= 1, |
| 24 | I 21 21 21 | x <= 1/2, y <= 1/4, z < 1, |
| 47 | P 2/M 2/M 2/M | x <= 1/2, y <= 1/2, z <= 1/2, |
| 65 | C 2/M 2/M 2/M | x <= 1/2, y <= 1/4, z <= 1/2, |
| 69 | F 2/M 2/M 2/M | x <= 1/4, y <= 1/4, z <= 1/2, |
| 71 | I 2/M 2/M 2/M | x <= 1/2, y <= 1/4, z <= 1/2, |
| 75 | P 4 | x <= 1/2, y <= 1/2, z < 1, |
| 76 | P 41 | x < 1, y < 1, z < 1/4, |
| 77 | P 42 | x <= 1/2, y < 1, z < 1/2, |
| 78 | P 43 | x < 1, y < 1, z < 1/4, |
| 79 | I 4 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 80 | I 41 | x <= 1/2, y < 1, z < 1/4, |
| 83 | P 4/M | x <= 1/2, y <= 1/2, z <= 1/2, |
| 87 | I 4/M | x <= 1/2, y <= 1/2, z <= 1/4, |
| 89 | P 4 2 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 90 | P 4 21 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 91 | P 41 2 2 | x < 1, y < 1, z <= 1/8, |
| 92 | P 41 21 2 | x < 1, y < 1, z <= 1/8, |
| 93 | P 42 2 2 | x <= 1/2, y < 1, z <= 1/4, |
| 94 | P 42 21 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 95 | P 43 2 2 | x < 1, y < 1, z <= 1/8, |
| 96 | P 43 21 2 | x < 1, y < 1, z <= 1/8, |
| 97 | I 4 2 2 | x <= 1/2, y <= 1/2, z <= 1/4, |
| 98 | I 41 2 2 | x <= 1/2, y < 1, z <= 1/8, |
| 123 | P 4/M 2/M 2/M | x <= 1/2, y <= 1/2, z <= 1/2, |
| 139 | I 4/M 2/M 2/M | x <= 1/2, y <= 1/2, z <= 1/4, |
| 143 | P 3 | x <= 2/3, y <= 2/3, z < 1, |
| 144 | P 31 | x < 1, y < 1, z < 1/3, |
| 145 | P 32 | x < 1, y < 1, z < 1/3, |
| 146 | H 3 | x <= 2/3, y <= 2/3, z < 1/3, |
| 147 | P -3 | x <= 2/3, y <= 2/3, z <= 1/2, |
| 148 | R -3 | x <= 2/3, y <= 2/3, z <= 1/6, |
| 149 | P 3 1 2 | x <= 2/3, y <= 2/3, z <= 1/2, |
| 150 | P 3 2 1 | x <= 2/3, y <= 2/3, z <= 1/2, |
| 151 | P 31 1 2 | x < 1, y < 1, z <= 1/6, |
| 152 | P 31 2 1 | x < 1, y < 1, z <= 1/6, |
| 153 | P 32 1 2 | x < 1, y < 1, z <= 1/6, |
| 154 | P 32 2 1 | x < 1, y < 1, z <= 1/6, |
| 155 | H 3 2 | x <= 2/3, y <= 2/3, z <= 1/6, |
| 162 | P -31 2/M | x <= 2/3, y <= 1/2, z <= 1/2, |
| 164 | P -3 2/M 1 | x <= 2/3, y <= 1/3, z <= 1, |
| 166 | R -3 2/M | x <= 2/3, y <= 2/3, z <= 1/6, |
| 168 | P 6 | x <= 2/3, y <= 1/2, z < 1, |
| 169 | P 61 | x < 1, y < 1, z < 1/6, |
| 170 | P 65 | x < 1, y < 1, z < 1/6, |
| 171 | P 62 | x < 1, y < 1, z < 1/3, |
| 172 | P 64 | x < 1, y < 1, z < 1/3, |
| 173 | P 63 | x <= 2/3, y <= 2/3, z < 1/2, |
| 175 | P 6/M | x <= 2/3, y <= 2/3, z <= 1/2, |
| 177 | P 6 2 2 | x <= 2/3, y <= 1/2, z <= 1/2, |
| 178 | P 61 2 2 | x < 1, y < 1, z <= 1/12, |
| 179 | P 65 2 2 | x < 1, y < 1, z <= 1/12, |
| 180 | P 62 2 2 | x < 1, y < 1, z <= 1/6, |
| 181 | P 64 2 2 | x < 1, y < 1, z <= 1/6, |
| 182 | P 63 2 2 | x <= 2/3, y <= 2/3, z <= 1/4, |
| 191 | P 6/M 2/M 2/M | x <= 2/3, y <= 1/3, z <= 1/2, |
| 195 | P 2 3 | x < 1, y < 1, z <= 1/2, |
| 196 | F 2 3 | x <= 1/4, y <= 1/4, z < 1, |
| 197 | I 2 3 | x < 1, y < 1, z <= 1/2, |
| 198 | P 21 3 | x <= 1/2, y <= 1/2, z < 1, |
| 199 | I 21 3 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 200 | P 2/M -3 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 202 | F 2/M -3 | x <= 1/2, y <= 1/2, z <= 1/4, |
| 204 | I 2/M -3 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 207 | P 4 3 2 | x < 1, y <= 1/2, z <= 1/2, |
| 208 | P 42 3 2 | x <= 1/2, y < 1, z <= 1/4, |
| 209 | F 4 3 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
| 210 | F 41 3 2 | x <= 1/2, y < 1, z <= 1/8, |
| 211 | I 4 3 2 | x <= 1/2, y <= 1/2, z <= 1/4, |
| 212 | P 43 3 2 | x < 1, y < 1, z <= 1/8, |
| 213 | P 41 3 2 | x < 1, y < 1, z <= 1/8, |
| 214 | I 41 3 2 | x <= 1/2, y < 1, z <= 1/8, |
| 221 | P 4/M -3 2/M | x <= 1/2, y <= 1/2, z <= 1/2, |
| 225 | F 4/M -3 2/M | x <= 1/2, y <= 1/4, z <= 1/4, |
| 229 | I 4/M -3 2/M | x <= 1/2, y <= 1/2, z <= 1/4, |
(*) The limits are in fractional coordinates, and the lower limits are always x=0, y=0, z=0.
1.5.6