How to use cosymlib¶
Cosymlib is a a python library for computing continuous symmetry & shape measures (CSMs & CShMs). Besides using the APIs contained in cosymlib to build your own custom-made python programs we have also written some general scripts to perform standard tasks such as calculating a continuous shape measure for a given structure without the need of writing a python script. All this general task scripts are called using a similar syntax which includes the name of the script, the name of the file containing the structural data, and optional arguments specifying the tasks we want to perform:
$ script filename -task1 -task2 ... -taskn
For instance, consider a struct.xyz
file containing the following structural information
for a H4 molecule in an approximately square geometry:
4
H4 Quadrangle
H 1.1 0.9 0.0
H -1.0 1.1 0.0
H -0.9 -1.2 0.0
H 1.1 -1.0 0.0
If we would like to compute the square shape measure S(SP-4) for this 4-vertex
polygon we simply can call the shape script indicating the name of our .xyz file
containing the coordinates and use the -m
flag (m stands for measure) with the
SP-4 label to indicate that we want to compute a shape measure using a perfect square
(SP-4 stands for square planar structure with 4 vertices) as the reference shape:
$ shape struct.xyz -m SP-4
and the shape script wil call the APIs in cosymlib to read first our input file, generate a molecule object, calculate the S(SP-4) continuous shape measure for it, and print the result of the calculation on the screen:
----------------------------------------------------------------------
COSYMLIB v0.10.5
Electronic Structure & Symmetry Group
Institut de Quimica Teorica i Computacional (IQTC)
Universitat de Barcelona
----------------------------------------------------------------------
Structure SP-4
H4, 0.520,
----------------------------------------------------------------------
End of calculation
----------------------------------------------------------------------
If, for instance, we also want the coordinates for the square with the optimal overlap
with our problem structure, we just need to include the -s
flag (where s stands
for structure) in our call:
$ shape struct.xyz -m SP-4 -s
A longer, explicit version for some flags is also available using a double - sign. With these explicit flags the previous command becomes:
$ shape struct.xyz --measure SP-4 --structure
The general task scripts include also gsym
and cchir
for
calculating continuous symmetry and chirality measures for polyhedral structures, shape_map
for plotting
shape maps, as well as the esym
and mosym
scripts for the continuous symmetry analysis of electron
densities and the pseudosymmetry analysis of molecular orbitals, respectively.
Besides these six basic scripts, we have also developed cosym
, a general script that allows to perform
any of the basic calculations above. We could, for instance, use directly cosym
to calculate the previous
shape measure using the following command:
$ cosym struct.xyz -shp_m SP-4 -s
Note that when using cosym
some of the optional flags in shape
change to indicate which type
of calculation we would like to perform. For instance, -m
becomes -shp_m
to distinguish it from a
symmetry measure ( -m
flag in sym
) that becomes -sym_m
when called from cosym
.
On the other hand, other arguments such as -s
, which have the same meaning when calculating
shape, symmetry or chirality measures, remain unchanged when used in combination with the
general cosym
script.
Taking into account the users of our previous programs, we have also written a stand-alone script
shape_classic
which is able to read an old_shape.dat
input file containing both
the structural information and the necessary keywords to run a full CShM calculation as it was done
in our previous SHAPE
program.
In the sections below you can find a detailed description of all stand-alone scripts as well as all APIs included in the present distribution of cosymlib.
General task scripts¶
The cosymlib library includes several scripts to perform basic tasks that can be run in a terminal as command line instructions, without the need of writing a full python script. The following subsections describe the general usage of all of them.
shape¶
shape
can be used for computing continuous shape measures (CShMs) for geometrical structures
defined by the cartesian coordinates for a set of vertices, for instance, a molecular structure
defined by the positions of the atomic nuclei.
The minimal information needed to run shape
is an input file containing the coordinates of a set
of vertices. Since shape
is mainly intended to be used in the context of structural chemistry, the main
source of structural information will be a fname.xyz
file containing a molecular geometry in xyz
format (http://en.wikipedia.org/wiki/XYZ_file_format).
An example of a cocl6.xyz
file with the structure for a perfect octahedral CoCl6 fragment
with 2.4Å Co-Cl interatomic distances is:
7
CoCl6
Co 0.0 0.0 0.0
Cl -2.4 0.0 0.0
Cl 2.4 0.0 0.0
Cl 0.0 2.4 0.0
Cl 0.0 -2.4 0.0
Cl 0.0 0.0 2.4
Cl 0.0 0.0 -2.4
The first line in the file indicates the number of atoms (vertices in the geometric structure), the second line contains a free-format descriptive title, and the following lines (as many as indicated in the first line) contain a label (usually the atomic symbol) and the cartesian coordinates x, y, z for each atom (vertex) in the structure.
fname.xyz
files read by shape may contain a single structure as in the previous example or
multiple structures (all with the same number N of atoms). In this case you must append a
N
Structure_name
label_1 x1 y1 z1
...
label_N xN yN zN
block for each structure, without leaving any blank lines between them. Note that even if the number of vertices N must be the same for all structures, it is mandatory to include it explicitly for each block.
Shape is also able to read input structures from files in other formats used in structural chemistry. A detailed description of the structural files read by shape can be found in the information on input formats section.
The basic call to the shape script must provide the the file containing the input structure and the reference shape with respect to which the shape measure is calculated.
$ shape input_file -m SH
where input_file
is a file containing the structural information in a valid format, for instance
a .xyz file, -m
requests a shape measure calculation, and is SH
a label indicating a given
reference structure, for instance SP-4
for a square or OC-6
for an octahedron. Note that
the reference shape must be compatible with the problem structure, i. e., they must both contain the
same number of atoms (vertices). To obtain a list of the labels for the reference structures compatible
with a given input structure you may use:
$ shape input_file -l
If input_file
contains, for instance, a structure with 6 atoms (vertices) your will get the
following output on screen:
Available reference structures with 6 Vertices:
Label Sym Info
HP-6 D6h Hexagon
PPY-6 C5v Pentagonal pyramid
OC-6 Oh Octahedron
TPR-6 D3h Trigonal prism
JPPY-6 C5v Johnson pentagonal pyramid J2
We can then use this information to compute the desired continuous shape measure
$ shape input_file -m OC-6
if we want to compute the octahedral shape measure. For a file containing a perfect octahedron of carbon atoms:
6
C6_octa
C -1.0 0.0 0.0
C 1.0 0.0 0.0
C 0.0 1.0 0.0
C 0.0 -1.0 0.0
C 0.0 0.0 1.0
C 0.0 0.0 -1.0
the program will return:
Starting...
----------------------------------------------------------------------
COSYM v0.7.4
Electronic Structure Group, Universitat de Barcelona
----------------------------------------------------------------------
Structure OC-6
C6_octa, 0.000
End of cosym calculation
Indicating that it is indeed a perfect octahedron, S(OC-6) = 0.000. If we want to know how far this octahedron is from the reference triangular prism we may use:
$ shape input_file -m TPR-6
which returns a value of S(TPR-6) = 16.737. Note that since shape measures are
independent from size, position, or orientation of the problem structure, we would obtain
exactly the same values for any perfect octahedron in input_file
.
When studing the shape of the coordination sphere around a given atom, let us say a transition
metal atom M surrounded by n atoms L coming from the surrounding ligands, it is possible to consider
just the Ln polyhedron or a centered MLn “polyhedron”. We will obtain different
information from each calculation. While considering the Ln polyhedron, we will know
how different it is from the ideal references, but if we are interested in distortions due to displacements
of the central atom from the geometric center we will need to compare the centered MLn “polyhedron”
with the ideal references where the central atom is located at the geometric center of the object.
Since the central M atom and the n surrounding ligands are not equivalent (no M <-> L permutations are
allowed when computing the shape measure) it is necessary to indicate that the structure
in input_file
corresponds to a centered MLn polyhedron and not to a simple
Ln+1 polyhedron. This is achieved by including the -c N
flag in the shape command, where
N is an integer number indicating the position of the central atom in input_file
(for a file
with multiple structures the central atom should be in the same position for all of them). If one
uses the cocl6.xyz
file above as input_file
indicating that the first atom in the
structure (the Co atom) is in the center (-c 1
)
$ shape cocl6.xyz -l -c 1
we get the following valid labels:
Available reference structures with 6 Vertices:
Label Sym Info
HP-6 D6h Hexagon
PPY-6 C5v Pentagonal pyramid
OC-6 Oh Octahedron
TPR-6 D3h Trigonal prism
JPPY-6 C5v Johnson pentagonal pyramid J2
note that, although these labels the same as those for a structure with 6 atoms where we do
not include a central atom, a calculation including the -c N
flag is not equivalent
to a calculation where the central atom is ignored, that is just for the Ln polyhedron.
If one wants to calculate the shape measure just for the “empty” Ln shell one
needs to prepare a different input file deleting the line corresponding to the central atom
and reducing the number of atoms by 1.
If we try omitting the -c N
flag for the cocl6.xyz
file we get a different result.
Using
$ shape cocl6.xyz -l
we find:
Available reference structures with 7 Vertices:
Label Sym Info
HP-7 D7h Heptagon
HPY-7 C6v Hexagonal pyramid
PBPY-7 D5h Pentagonal bipyramid
COC-7 C3v Capped octahedron
CTPR-7 C2v Capped trigonal prism
JPBPY-7 D5h Johnson pentagonal bipyramid J13
JETPY-7 C3v Johnson elongated triangular pyramid J7
which are the possible reference structures for empty L7 polyhedra, since now the Co atom is being considered on equal foot to all other six Cl atoms, even if this might make no sense from a chemical point of view. The list of currently available reference structures in the cosymlib program is at the page end.
To calculate the octahedral shape measure for the CoCl6 structure contained in the
cocl6.xyz
file we will use:
$ shape cocl6.xyz -c 1 -m OC-6
which will return a S(OC-6)= 0.000 value since the six Cl atoms in the structure form
a perfect octahedron with the Co atom sitting exactly in its geometric center.
Note also that, as shown in this example, the position of the -c N
and
-m OC-6
flags, or the input_file
in the call to the shape script is totally irrelevant
and any combination such as:
$ shape cocl6.xyz -c 1 -m OC-6
$ shape -c 1 -m OC-6 cocl6.xyz
$ shape -m OC-6 cocl6.xyz -c 1
will result in exactly the same CShM calculation.
Somtimes we are not just interested in the shape measure, that is, how far our problem
shape is from the ideal reference, but also we would like to have the coordinates of
the ideal reference shape with the size, position, and orientation that is closest to
our problem shape. To achieve this we just need to include the -s
flag in our
call.
Let us consider a struct.xyz
file containing the geometry for an approximately square
H4 molecule.
4
H4 Quadrangle
H 1.1 0.9 0.0
H -1.0 1.1 0.0
H -0.9 -1.2 0.0
H 1.1 -1.0 0.0
If we want to know how far it is from having a perfectly square geometry and which is the closest square to its actual distorted structure we may use:
$ shape struct.xyz -m SP-4 -s
which will yield:
Starting...
----------------------------------------------------------------------
COSYM v0.7.4
Electronic Structure Group, Universitat de Barcelona
----------------------------------------------------------------------
Structure SP-4
H4, 0.520
4
H4
H 1.100000 0.900000 0.000000
H -1.000000 1.100000 0.000000
H -0.900000 -1.200000 0.000000
H 1.100000 -1.000000 0.000000
4
H4_SP-4
H 1.100000 1.000000 0.000000
H -0.975000 0.975000 0.000000
H -0.950000 -1.100000 0.000000
H 1.125000 -1.075000 0.000000
from which we find that the problem structure has an approximate square planar geometry with a small departure from the ideal shape, S(SP-4) = 0.520, together with the coordinates of the problem structure and its closest ideal (square) structure, which we can use to plot the superposition of problem structure (in red) and the ideal reference (in blue):
Other optional flags to control the execution of shape are:
shape -h
(no input file needed) returns a list of all available flags and their
use
Running shape
with the -o file_name
flag prints all output into the file_name
file
Running shape
with the -r
flag prints the coordinates of the reference shape in
a file named Ln.xyz
or MLn.xyz
where n is the number of vertices of the polyhedron.
The - info
flag may be used to print the coordinates of the input structure
You may use -fixp
to disable the minimization over the permutation of vertices while
searching for the shape measure. If you include the -fixp
in your call, the minimization
will be carried out considering only the distance between the i-th vertex in the problem
structure with the i-th vertex in the reference shape. Although this option allows a
drastic reduction of the computational cost, it should be used with care since the actual
shape measure is defined for the permutation thats gives the lowest value of S. For
large structures the -fixp
option will probably be the only way of obtaining a shape
measure, but this procedure is only justified for structures with small distortions from
the reference structure. Before doing the actual calculation it will be necessary to
run shape with the -r
flag to print the coordinates of the reference shape and order
the vertices in the problem structure accordingly.
A quite useful flag is -cref filename
that allows the user to specify a custom reference
structure in the filename file. Use this option if you want to use a reference structure
different from any of those provided by shape. To use this feature you will need to include
the -m custom
flag in your call:
$ shape input_file -m custom -cref filename
Besides the shorthand version of the flags described above, it is also possible to use
an explicit version by writing them preceded by a double --
sign. The explicit versions
of the flags are:
Short Flag | Explicit flag |
-h |
--help |
-m |
--measure |
-l |
--labels |
-s |
--structure |
-o |
--output_name |
-c |
--central_atom |
-r |
--references |
-cref |
--custom_ref |
-fixp |
--fix_permutation |
Sometimes, to avoid a cumbersome repetition of several flags in the call of the shape module we may write all flags in an input file and just call shape indicating the file with the structural input and the file with the options of the calculation. For example, if the original call is:
$ shape struct.xyz -c 1 -m OC-6 -s -o struct.out
You can create a new file called struct.yml
(the name for the file
can be freely chosen and does not need to be the same as for the structure)
containing the options in YAML format (http://en.wikipedia.org/wiki/YAML):
central_atom : 1
measure : OC-6
structure : True
output : struct.out
and then call shape just using:
$ shape struct.xyz struct.yml
Note that you must use the explicit version of the flags in the .yml
file. If a
flag such as -s
does not need any additional argument, you must include True
in the .yml
file.
shape_classic¶
To run shape_classic
you only need an old_shape.dat
input file containing both
the structural information and the necessary keywords to run a full CShM calculation as in
the old SHAPE
program:
$ shape_classic old_shape.dat
The script will perform all tasks indicated in the input file, creating the necessary output
files, normally old_shape.out
and old_shape.tab
with the same information as when using
our previous SHAPE
program. Follow the link below for a pdf version of the user guide for
SHAPE ver. 2.1 where you will find all information to perform a continuous shape analysis using
this option.
gsym¶
In the case of running a continuous symmetry measure (CSM), the gsym
script is required plus an input file containing
a geometric structure as the one used in the shape script. Since the main difference with the continuous shape measures
is that the reference structure now must contain one or more symmetry elements, the user will need to specify which
symmetry operation wants to analyse for the input geometry. The Th8molecule can be a good example to show the
S4symmetry that the molecule contains. The Th8.xyz file is shown below:
8
Th8
Th -16.80062 -0.55052 -13.74098
Th -12.80008 -0.09601 -14.54017
Th -15.57778 1.38797 -17.17300
Th -20.18823 -1.83274 -15.67222
Th -16.79762 -4.14442 -15.76983
Th -12.79709 -3.68990 -16.56902
Th -18.96539 0.10576 -19.10423
Th -15.57478 -2.20592 -19.20184
The simplest way to compute the S4CSM measure for Th8is to run the following command:
$ gsym Th8.xyz -m S4
which is equivalent to:
$ gsym -m S4 Th8.xyz
and will return the CSM result in the cosymlib format:
----------------------------------------------------------------------
COSYMLIB v0.9.5
Electronic Structure & Symmetry Group
Institut de Quimica Teorica i Computacional (IQTC)
Universitat de Barcelona
----------------------------------------------------------------------
Evaluating symmetry operation : S4
th8 0.000
----------------------------------------------------------------------
End of calculation
----------------------------------------------------------------------
If the user wants to save this information in a file, the -o (or --output)
flag should be added as well as the
output file name. For example, the following call will execute the previous CSM calculation and will store the
information in the Th8.txt file:
$ gsym Th8.xyz -m S4 -o Th8.txt
Additional commands (or flags) can be added to the command line to control the type of calculation that will be run.
For example, the -c N
command, where N is the position of an atom of the input file, explicitly tells the program
which atom acts as a central atom of the molecule and that only could permut by himself. Alternatively, the
--center x y z
, where x, y and z are the 3D coordinates of a point in space, will set the origin of a structure.
Similarly, the -s
and -l
commands work as in the shape case. The first returns the coordinates of the ideal
reference geometry with the size, position, and orientation that is closest to our problem shape and belongs to a
symmetry group. The second gives information of the available symmetry groups in cosymlib
Available symmetry groups
E Identity Symmetry
Ci Inversion Symmetry Group
Cs Reflection Symmetry Group
Cn Rotational Symmetry Group (n: rotation order)
Sn Rotation-Reflection Symmetry Group (n: rotation-reflection order)
The --info
flag may be used to print the coordinates of the input structure.
Supplementary, other flags are available to the user to control if the calculation should take into account the
connectivity --ignore_connectivity
, the atom nature --ignore_atoms_labels
, the connectivity threshold that
controls if two atoms are connected --connectivity_thresh
or the file that contains a custom connectivity
--connectivity_file
. In the last case, the format of the connectivity file should be as follow,
1 2 3 4 5
2 1
3 1
4 1
5 1
where each number is related to the position of the each atom on the input file. For example, in the first line, the connectivity file tells the program that the atom in position one of the the input file is connected to the second, third, fourth and fifth atoms, while the second line tells that the second atom is only connected to the first atom. For a methane molecule where the input file is written as follow,
5
Methane
C 0.0000 0.0000 0.0000
H 0.5288 0.1610 0.9359
H 0.2051 0.8240 -0.6786
H 0.3345 -0.9314 -0.4496
H -1.0685 -0.0537 0.1921
the connectivity file force the carbon atom to be connected to all hydrogen atoms and viceversa. Finally, a list of available flags and their uncontracted form is listed below.
Short Flag | Explicit flag |
-h |
--help |
-m |
--measure |
-s |
--structure |
-c |
--central_atom |
-o |
--output |
-l |
--labels |
-v |
--version |
Note: the actual program only runs for Cn, Cs, Ci and Sn symmetry groups as well as their symmetry operations.
cchir¶
The cchir
script allows the user to calculate the chirality of a structure by calculating the continuous symmetry
measure of the Sn improper rotation group. By default the -m
flag will measure the S1 symmetry
which is equivalent ot the Cs symmetry group. However, the user can control the order of the impropert
rotation by the -order n
flag where n=1,2,4,6,…
Other additional flags derived from the gsym script that have the same interaction with the chirality measure are
available and list below. For more information of these commands go to gsym subsection of this page.
Common cchir and gsym commands |
-o or --ouput |
-c or --central_atom |
-v or --version |
--info |
--center |
esym¶
We are currenly working on this feature of the program regarding the electronic structure symmetry of molecules, therefore the actual script is under construction.
cosym¶
This script is a general script that cover all the previous scripts.
Specific task scripts¶
In this section the specific task scripts are described.
shape_map¶
The shape_map
script calculate the continuous shape measures of a single or multiple structures with two reference
structures in the same way the shape script does. However, it computes additional information like the minimum
distortion pathway between the two reference structures, plus the deviation and the generalized coordinate of each
user’s structure.
The most common commands available in the script are similar to the commands found in the shape script. The required
commands are the -m_1 SH1` (or ``--measure_1
) and the -m_2 SH2
flags, where SH1 and SH2 are the reference
structure labels available in the program. Additionally, these flags can be substituted by the -m_custom_1 SH1
or
the -m_custom_2 SH2
to indicate the program that SH1 and/or SH2 are the files containing a custom reference
structure.
Moreover, a set of flags are available to control the different plot options on the shape_map. The
--min_dev MIN_DEV
and --max_dev MAX_DEV
will only show the structures that are between the minimum and maximum
deviation values (MIN_DEV and MAX_DEV), while the --min_gco MIN_GCO
and --max_gco MAX_GCO
show the structure
that are at the MIN_GCO to MAX-GCO range of the generalized coordinate. In addition, the user can plot more resolution
minimal distortion pathways by setting the number of structures needed to compute the curve with the
--n_points N_POINTS
flag.
Finally, a set of mutual flags found in all scripts is available and listed below:
Short Flag | Explicit flag |
-h |
--help |
-l |
--labels |
-o |
--output_name |
-c |
--central_atom |
-v |
--version |
Using cosymlib’s APIs¶
The current API’s are under construction and a set of tutorials will be provide in a near future.
Shape references¶
Here are the available shape reference’s labels and their symmetry that can be used by the shape program.
Vertices | Label | Shape | Symmetry |
---|---|---|---|
2 | L-2 | Linear | D∞h |
vT-2 | Divacant tetrahedron (V-shape, 109.47°) | C2v | |
vOC-2 | Tetravacant octahedron (L-shape, 90.00°) | C2v | |
3 | TP-3 | Trigonal planar | D3h |
vT-3 | Pyramidb (vacant tetrahedron) | C3v | |
fac-vOC-3 | fac-Trivacant octahedron | C3v | |
mer-vOC-3 | mer-Trivacant octahedron (T-shape) | C2v | |
4 | SP-4 | Square | D4h |
T-4 | Tetrahedron | Td | |
SS-4 | Seesaw or sawhorseb (cis-divacant octahedron) | C2v | |
vTBPY-4 | Axially vacant trigonal bipyramid | C3v | |
5 | PP-5 | Pentagon | D5h |
vOC-5 | Vacant octahedronb (Johnson square pyramid, J1) | C4v | |
TBPY-5 | Trigonal bipyramid | D3h | |
SPY-5 | Square pyramidc | C4v | |
JTBPY-5 | Johnson trigonal bipyramid (J12) | D3h | |
6 | HP-6 | Hexagon | D6h |
PPY-6 | Pentagonal pyramid | C5v | |
OC-6 | Octahedron | Oh | |
TPR-6 | Trigonal prism | D3h | |
JPPY-6 | Johnson pentagonal pyramid (J2) | C5v | |
7 | HP-7 | Heptagon | D7h |
HPY-7 | Hexagonal pyramid | C6v | |
PBPY-7 | Pentagonal bipyramid | D5h | |
COC-7 | Capped octahedrona | C3v | |
CTPR-7 | Capped trigonal prisma | C2v | |
JPBPY-7 | Johnson pentagonal bipyramid (J13) | D5h | |
JETPY-7 | Elongated triangular pyramid (J7) | C3v | |
8 | OP-8 | Octagon | D8h |
HPY-8 | Heptagonal pyramid | C7v | |
HBPY-8 | Hexagonal bipyramid | D6h | |
CU-8 | Cube | Oh | |
SAPR-8 | Square antiprism | D4d | |
TDD-8 | Triangular dodecahedron | D2d | |
JGBF-8 | Johnson - Gyrobifastigium (J26) | D2d | |
JETBPY-8 | Johnson - Elongated triangular bipyramid (J14) | D3h | |
JBTP-8 | Johnson - Biaugmented trigonal prism (J50) | C2v | |
BTPR-8 | Biaugmented trigonal prism | C2v | |
JSD-8 | Snub disphenoid (J84) | D2d | |
TT-8 | Triakis tetrahedron | Td | |
ETBPY-8 | Elongated trigonal bipyramid (see 8) | D3h | |
9 | EP-9 | Enneagon | D9h |
OPY-9 | Octagonal pyramid | C8v | |
HBPY-9 | Heptagonal bipyramid | D7h | |
JTC-9 | Triangular cupola (J3) = trivacant cuboctahedron | C3v | |
JCCU-9 | Capped cube (Elongated square pyramid, J8) | C4v | |
CCU-9 | Capped cube | C4v | |
JCSAPR-9 | Capped sq. antiprism (Gyroelongated square pyramid J10) | C4v | |
CSAPR-9 | Capped square antiprism | C4v | |
JTCTPR-9 | Tricapped trigonal prism (J51) | D3h | |
TCTPR-9 | Tricapped trigonal prism | D3h | |
JTDIC-9 | Tridiminished icosahedron (J63) | C3v | |
HH-9 | Hula-hoop | C2v | |
MFF-9 | Muffin | Cs | |
10 | DP-10 | Decagon | D10h |
EPY-10 | Enneagonal pyramid | C9v | |
OBPY-10 | Octagonal bipyramid | D8h | |
PPR-10 | Pentagonal prism | D5h | |
PAPR-10 | Pentagonal antiprism | D5d | |
JBCCU-10 | Bicapped cube (Elongated square bipyramid J15) | D4h | |
JBCSAPR-10 | Bicapped square antiprism (Gyroelongated square bipyramid J17) | D4d | |
JMBIC-10 | Metabidiminished icosahedron (J62) | C2v | |
JATDI-10 | Augmented tridiminished icosahedron (J64) | C3v | |
JSPC-10 | Sphenocorona (J87) | C2v | |
SDD-10 | Staggered dodecahedron (2:6:2)e | D2 | |
TD-10 | Tetradecahedron (2:6:2) | C2v | |
HD-10 | Hexadecahedron (2:6:2, or 1:4:4:1) | D4h | |
11 | HP-11 | Hendecagon | D11h |
DPY-11 | Decagonal pyramid | C10v | |
EBPY-11 | Enneagonal bipyramid | D9h | |
JCPPR-11 | Capped pent. Prism (Elongated pentagonal pyramid J9) | C5v | |
JCPAPR-11 | Capped pent. antiprism (Gyroelongated pentagonal pyramid J11) | C5v | |
JaPPR-11 | Augmented pentagonal prism (J52) | C2v | |
JASPC-11 | Augmented sphenocorona (J87) | Cs | |
12 | DP-12 | Dodecagon | D12h |
HPY-12 | Hendecagonal pyramid | C11v | |
DBPY-12 | Decagonal bipyramid | D10h | |
HPR-12 | Hexagonal prism | D6h | |
HAPR-12 | Hexagonal antiprism | D6d | |
TT-12 | Truncated tetrahedron | Td | |
COC-12 | Cuboctahedron | Oh | |
ACOC-12 | Anticuboctahedron (Triangular orthobicupola J27) | D3h | |
IC-12 | Icosahedron | Ih | |
JSC-12 | Square cupola (J4) | C4v | |
JEPBPY-12 | Elongated pentagonal bipyramid (J16) | D6h | |
JBAPPR-12 | Biaugmented pentagonal prism (J53) | C2v | |
JSPMC-12 | Sphenomegacorona (J88) | Cs | |
20 | DD-20 | Dodecahedrond | Ih |
24 | TCU-24 | Truncated cube | Oh |
TOC-24 | Truncated octahedron | Oh | |
48 | TCOC-48 | Truncated cuboctahedron | Oh |
60 | TRIC-60 | Truncated icosahedron (fullerene) | Ih |