# 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 .

• 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.

### Cyclic of order N (Cn)

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

Scipion Definition (CN): rotation axis = Z axis.

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.

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.

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 the X, Y, and Z axes, a three-fold along axis (1,1,1)

Scipion Definition (Tz3): a three-fold symmetry axis along Z, another three-fold axis in the YZ plane such that rotation about the X axis by ~110° is a symmetry operation

### 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).

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 on X, Y and Z axes. With the positive Z-axis pointing at the viewer, the front-most 5-fold vertices are in YZ plane, and the front-most 3-fold axis is in the XZ plane. As known as no Crowther 222, standard in Heymman et al 2005 article). Plane normal vectors:

```v1 = -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 on X, Y and Z axes. With the positive Z-axis pointing at the viewer, the front-most 5-fold vertices are in XZ plane, and the front-most 3-fold axis is in the YZ plane. As known as no Crowther 222, standard in Heymman et al 2005 article). Plane normal vectors:

```v1 = -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 in Z and 2-fold in Y. With the positive Z-axis pointing at the viewer and without taken into account the 5-fold vertex in Z, there is one of the front-most 5-fold vertices in -XZ plane (note the minus X)

Scipion Definition (In25r): 5fold axis in Z and 2-fold in Y. With the positive Z-axis pointing at the viewer and without taken into account the 5-fold vertex in Z, there is one of the front-most 5-fold vertices in +*XZ* plane (note the plus X)

Scipion Definition (I2n3): 3-fold axis in Z and 2-fold in X. With the positive Z-axis pointing at the viewer and without taken into account the 3-fold vertex in Z, there is one of the front-most 3-fold vertices in -YZ plane (note the minus Y)

Scipion Definition (I2n3r): 3-fold axis in Z and 2-fold in X. With the positive Z-axis pointing at the viewer and without taken into account the 3-fold vertex in Z, there is one of the front-most 3-fold vertices in +*YZ* plane (note the plu Y)