> For the complete documentation index, see [llms.txt](https://guide.cryosparc.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://guide.cryosparc.com/symmetry-in-cryosparc.md). # Symmetry in CryoSPARC ## Symmetry Groups This page illustrates the various symmetry groups that are supported when enforcing symmetry during a 3D reconstruction in CryoSPARC. {% hint style="info" %} Note that simple 3D shapes like prisms are used in the examples below for illustration. These simple shapes also have mirror symmetry, but biological macromolecules are chiral and so cannot be mirrored; only rotations are allowed. {% endhint %} ### Cyclic (Cn) Symmetry

*This object is C5 symmetric. A single ASU is highlighted. It can be rotated by 1/5 turn about the Z axis and remain unchanged. It has no other symmetry operations.*

The simplest symmetry groups are the cyclic groups, designated Cn. They have the following properties: * N-fold symmetry about a single axis. By convention in CryoSPARC the symmetry axis is aligned to the Z-axis. * Symmetry order of N. {% hint style="info" %} Note that C1 symmetry (the only symmetry operation is a full 360° rotation) is equivalent to no symmetry. {% endhint %} ### Dihedral (Dn) Symmetry

*This object is D5 symmetric. A single ASU is highlighted. It can be rotated by 1/5 turn about the Z axis and/or a 1/2 turn about the Y axis and remain unchanged. It has no other symmetry operations.*

Dihedral symmetry groups have the following properties: * N-fold symmetry about one axis. By convention in CryoSPARC, this axis is aligned to the Z-axis. * 2-fold symmetry about a second, orthogonal axis. By convention in CryoSPARC, this axis is aligned to the Y-axis. * Symmetry order of 2N. ### Tetrahedral (T) Symmetry

This object is T symmetric. Note that the base is triangular. A single ASU is highlighted. The 3-fold axes pass through each vertex and the center of the opposite face, while the 2-fold axes pass through the centers of each opposing pair of edges.

Objects with tetrahedral symmetry have the following properties: * 3-fold symmetry around four axes. By convention in CryoSPARC, one of these axes is aligned to the Z-axis. * 2-fold symmetry around three axes. By convention in CryoSPARC, one of these axes lies in the YZ plane. * Symmetry order of 12. ### Octahedral (O) Symmetry

This object is O symmetric. A single ASU is highlighted. The 4-fold axes are aligned to X, Y, and Z. The 3-fold axes pass through the centers of each opposing pair of faces. The 2-fold axes pass through the centers of each opposing pair of edges.

Objects with octahedral symmetry have the following properties: * 4-fold symmetry along three orthogonal axes. By convention in CryoSPARC, the symmetry axes are aligned to the X, Y, and Z axes. * 3-fold symmetry along four axes. * 2-fold symmetry along six axes. * Symmetry order of 24. ### Icosahedral (I) Symmetry

These objects have icosahedral symmetry. A single ASU is highlighted. The left object is aligned according to the I or I1 conventions, while the object on the right is aligned to the I2 convention. The 5-fold axes pass through opposing pairs of vertices. The 3-fold axes pass through the centers of opposing pairs of faces. The 2-fold axes pass through the centers of opposing pairs of edges.

Objects with icosahedral symmetry have the following properties: * 5-fold symmetry along six axes. * 3-fold symmetry along ten axes. * 2-fold symmetry along 15 axes. * Symmetry order of 60. CryoSPARC defines two icosahedral conventions. Both of them align a 2-fold axis to each of the X, Y, and Z axes. * `I1` (or simply `I`) places the vertices with greatest Z value in the YZ plane. * `I2` places the vertices with greatest Z value in the XZ plane.