# Symmetries

## Overview

This page describes the different symmetries that a polyhedron with fixed center may have and the symmetry conventions followed by Scipion. There are five fundamental symmetry classes: cyclical, dihedral, tetrahedral, octahedral and icosahedral. Auxiliary data used to create this page is available at https://github.com/I2PC/testDataSym .

For each type we show in this page:

Surface rendering of a volume that displays the symmetry.

Atomic structure file (pb format) containing an atomic structure that satisfies the symmetry. (The PDB files may be visualized with viewers such as Chimera. Link to chimera axis file).

Definition including the elements of symmetry.

*X*,*Y*and*Z*axes are colored in red, yellow and blue respectively.Nomal vectors to the planes that define the unit cell.

Note: Is really hard to see the difference symmetries just looking to the images, instead download the PDB files, the axis.bild file and visualize them in chimera. In chimera command line type “represent sphere” to increase the atoms size.

Table of Contents

### Cyclic of order N (Cn)

Definition: N rotations of magnitude 360/N degrees around an axis.

Scipion Definition (CN): rotation axis =Zaxis.

PDB: link to C7 model link to C7 model link to C7 model <../../images/Conventions/Symmetry/c7.pdb> `link to C7 model

Plane normal vectors:v1 = 43.388373911755814 90.096886790241911 -0.0 v2 = 43.388373911755814 -90.096886790241911 0.0

### Dihedral of order N (Dn)

Definition: N rotations of magnitude 360/N degrees around an axis followed by a 180 rotation around a second axis. Both axes are perpendicular.

Scipion Definition (DNx): first axis =Z, second axis =X.

PDB: link to D7x model

Plane normal vectors:v1 = -90.096886790241911 43.388373911755835 0.000000000000000 v2 = 90.096886790241911 43.388373911755806 -0.000000000000000 v3 = 0 0 1

Scipion Definition (DNy): first axis =Z, second axis =Y.

PDB: link to D7y model

Plane normal vectors:I guess the result for DNx is valid here but I do not have the software to test this claim.

### Tetrahedral (T)

Definition: There are three orthogonal 2-fold rotation axes with in addition four 3-fold axes, centered between the three orthogonal directions

Scipion Definition (T222): two-fold symmetry axes along theX,Y, andZaxes, a three-fold along axis (1,1,1)

PDB: link to T222 model

Scipion Definition (Tz3): a three-fold symmetry axis alongZ, another three-fold axis in theYZplane such that rotation about theXaxis by ~110° is a symmetry operation

PDB: link to Tz3

### Octahedral (O)

Definition: There are three orthogonal 4-fold rotation axes with additional four 3-fold axes, centered between the three orthogonal directions

Scipion Definition (0): 3-fold symmetry axis around (.5773502, .5773502, .5773502) 4-fold rotation axis around (0 0 1).

PDB: link to O model

Plane normal vectors:.arrow 0 0 0 -60 60 0 0.200000 0.400000 0.750000 .arrow 0 0 0 60 60 0 0.200000 0.400000 0.750000 .arrow 0 0 0 0 -100 100 0.200000 0.400000 0.750000

### Icosahedral (I)

Definition: 60 elements of symmetry. 12 5-fold axes, 20 3-fold axes and 30 2-fold axes.

Scipion Definition (I222): 2-fold axes onX,YandZaxes. With the positiveZ-axis pointing at the viewer, the front-most 5-fold vertices are inYZplane, and the front-most 3-fold axis is in theXZplane. As known as no Crowther 222, standard in Heymman et al 2005 article).

PDB: link to I222 modelv1 = -9.56540190374910 -25.04254730006809 15.47714539631899 v2 = -9.56540190374910 -25.04254730006809 -15.47714539631899 v3 = 0.0 45.094037546245751 0.0

Scipion Definition (I222r): 2-fold axes onX,YandZaxes. With the positiveZ-axis pointing at the viewer, the front-most 5-fold vertices are inXZplane, and the front-most 3-fold axis is in theYZplane. As known as no Crowther 222, standard in Heymman et al 2005 article).

PDB: link to I222r modelv1 = -15.47714539631899 -25.04254730006809 9.56540190374910 v2 = 40.51969269638708 -1.54232144954710 25.04254730006809 v3 = 0.00000000000000 45.094037546245751 0.00000000000000

Scipion Definition (In25): 5fold axis inZand 2-fold inY. With the positiveZ-axis pointing at the viewer and without taken into account the 5-fold vertex inZ, there is one of the front-most 5-fold vertices in -XZplane (note the minusX)

PDB: link to In25 model

Scipion Definition (In25r): 5fold axis inZand 2-fold inY. With the positiveZ-axis pointing at the viewer and without taken into account the 5-fold vertex inZ, there is one of the front-most 5-fold vertices in +*XZ* plane (note the plusX)

PDB: link to In25r model

Scipion Definition (I2n3): 3-fold axis inZand 2-fold inX. With the positiveZ-axis pointing at the viewer and without taken into account the 3-fold vertex inZ, there is one of the front-most 3-fold vertices in -YZplane (note the minusY)

PDB: link to I2n3 model

Scipion Definition (I2n3r): 3-fold axis inZand 2-fold inX. With the positiveZ-axis pointing at the viewer and without taken into account the 3-fold vertex inZ, there is one of the front-most 3-fold vertices in +*YZ* plane (note the pluY)

PDB: link to I2n3 model