Skip to content

Symmetry

Jump to bottom
Adrian Quintana edited this page Dec 11, 2017 · 1 revision
<SCRIPT Language=Javascript SRC="http://conventions.cnb.uam.es/Submit/models.js"></SCRIPT><SCRIPT Language=Javascript SRC="http://conventions.cnb.uam.es/Submit/sym/model_viewer_xmipp.js"></SCRIPT>

Symmetry

This page describes the different symmetries that a polyhedron with fixed center may have. There are five fundamental symmetry classes: cyclical, dihedral, tetrahedral, octahedral, icosahedral. And 17 different symmetry groups. all of them are implemented in xmipp

For each type we show in this page:

  • surface rendering of a volume with such symmetry
  • java script model of a volume with such a symmetry. (The models can be viewed withJavaView which allows them to be rotated and seen from different angles.)
  • the sym file needed by the different xmipp programs to impose a given symmetry. IMPORTANT symmetry files must has a precision of 10-6. That is, in most cases 6 decimal digits are required (for more complex symmetries as icosahedral 7 are needed). Note that all the xmipp programs that require a symmetry file will also work supplying the symmetry acronym instead of the symmetry filename.

NOTE blue axis is z, yellow axis is y and red axis is x.

NOTE: should you want to repeat the test the volume use to generate the symmetries is available here /left_hand.spi

Symmetry file

Arbitrary symmetry files can be provided by describing the symmetry elements in the following syntax:


# The fold is the number of times that the volume can be rotated along
# the symmetry axis giving the same view from different view points.
# the structure for the rotational axis is
# rot_axis      <fold> <X0> <Y0> <Z0>
# mirror_plane         <X0> <Y0> <Z0>
# P4212
# inversion
# The following line is a 6-fold rotational symmetry axis along Z-axis.
rot_axis      6 0 0 1
mirror_plane    0 0 1

C1

No symmetry.

  • view model (close any open model before clicking here)
  • c1
    rot_axis 1 0 0 1 (rot=0, tilt=0, psi=0)
  • /c1.jpg

Ci

Symmetry#SymNote Inversion symmetry only.

  • view model (close any open model before clicking here)
  • ci
    inversion
  • /ci.jpg

Cs

Symmetry#SymNote Reflection symmetry only.

  • view model (close any open model before clicking here)
  • cs
    mirror_plane 0 0 1 (rot=0, tilt=0, psi=0)
  • /cs.jpg

Cn

Cyclic rotational symmetry only.

  • view model (close any open model before clicking here) The model shown has C5 symmetry with a vertical 5-fold axis of symmetry.
  • c5
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0)
  • /c5.jpg

Cnv

Symmetry#SymNote Cyclic rotational symmetry with reflection symmetry in a vertical mirror.

  • view model(close any open model before clicking here). The model shown has C5v symmetry with a vertical 5-fold axis of symmetry. * c5v
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0) mirror_plane 0 1 0 (rot=90, tilt=90, psi=0)
  • /c5v.jpg

Cnh

Symmetry#SymNote Cyclic rotational symmetry with reflection symmetry in a horizontal mirror.

  • view model (close any open model before clicking here). The model shown has C5h symmetry with a vertical 5-fold axis of symmetry.
  • c5h
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0) mirror_plane 0 0 1 (rot=0, tilt=0, psi=0)
    * /c5h.jpg

Sn

Symmetry#SymNote Cyclic rotational (n/2-fold) with reflection symmetry in a horizontal mirror.

  • view model (close any open model before clicking here). The model shown has S6 symmetry
  • s10
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0) inversion
  • /s10.jpg

Dn

Dihedral rotational symmetry only.

  • view model (close any open model before clicking here). The model shown has D5 symmetry with a vertical 5-fold axis of symmetry.
  • d5
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0) rot_axis 2 1 0 0 (rot=0, tilt=90, psi=0)
  • /d5.jpg

Dnv

Symmetry#SymNote Dihedral rotational symmetry with reflection symmetry in a vertical mirror.

  • view model (close any open model before clicking here). The model shown has D5v symmetry with a vertical 5-fold axis of symmetry.
  • d5v
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0) rot_axis 2 1 0 0 (rot=0, tilt=90, psi=0) mirror_plane 1 0 0 (rot=0, tilt=90, psi=0)
    * /d5v.jpg

Dnh

Symmetry#SymNote Dihedral rotational symmetry with reflection symmetry in a horizontal mirror.

  • view model (close any open model before clicking here). The model shown has D5h symmetry with a vertical 5-fold axis of symmetry.
  • d5h
    rot_axis 5 0 0 1 (rot=0, tilt=0, psi=0) rot_axis 2 1 0 0 (rot=0, tilt=90, psi=0) mirror_plane 0 0 1 (rot=0, tilt=0, psi=0)
  • /d5h.jpg

T

Tetrahedral rotational symmetry only

Td

Symmetry#SymNote Tetrahedral rotational symmetry with reflection symmetry.

  • view model (close any open model before clicking here).
  • td
    rot_axis 3 0 0 1 (rot=0, tilt=0, psi=0) rot_axis 2 0 0.816496 0.577350 (rot=90, tilt=54.7356, psi=0) mirror_plane 1.4142136 2.4494897 0 (rot=60, tilt=90, psi=0)
    * /td.jpg

Th

Symmetry#SymNote Tetrahedral rotational symmetry with inversion symmetry.

  • view model (close any open model before clicking here).
  • th
    rot_axis 3 0 0 1 (rot=0, tilt=0, psi=0) rot_axis 2 0. -0.816496 -0.577350 (rot=-90, tilt=125.264, psi=0) inversion
  • /th.jpg

O

Octahedral rotational symmetry only.

Oh

Symmetry#SymNote Octahedral rotational symmetry with reflection symmetry.

  • view model (close any open model before clicking here).
  • oh
    rot_axis 3 .5773502 .5773502 .5773502 (rot=45, tilt=54.7356, psi=0) rot_axis 4 0 0 1 (rot=0, tilt=0, psi=0) mirror_plane 0 1 1 (rot=90, tilt=45, psi=0)
  • /oh.jpg

I

Icosahedral rotational symmetry only. (NOTE blue axis is z, yellow axis is y and red axis is x.)

I1

  • view model (close any open model before clicking here).
  • i (no crowther 222 and standard in heyman et al 2005 article). 2-fold axes on x,y,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 axes are in the xz plane.
rot_axis 2 0 1 0 (rot=90, tilt=90, psi=0)
rot_axis 5 -0.85065080702670 0 0.5257311142635 (rot=0, tilt=-58.2825, psi=0)
rot_axis 3 -0.9341723640 0.3568220765 0 (rot=-20.9052, tilt=-90, psi=0)
  • /i.jpg

I2 (alternative notation I)

  • Alternative 222 (crowther) 2-fold axes on x,y,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 axes are in the yz plane.
rot_axis 2  0 0 1 (rot=0, tilt=0, psi=0)
rot_axis 5 -1.618033989  -1 0 (rot=-31.7175, tilt=-90, psi=0)
rot_axis 3 -0.53934467   -1.4120227   0 (rot=69.0948, tilt=-90, psi=0)

  • /i222crowther.jpg

  • To convert i2 to i1 apply a rotation with Euler angles 0 90 0

  • To convert i2 to i3 apply a rotation with Euler angles 0 31.7175 0

  • To convert i2 to i4 apply a rotation with Euler angles 0 -31.7175 0

  • To convert i2 to i5 apply a rotation with Euler angles 31.7175 90 0 (this is wrong)

I3

  • alternative 52 (as given by spider) 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)
   rot_axis 2  -0.5257311143 0 0.8506508070 (rot=0, tilt=-31.7175, psi=0)
   rot_axis 3  -0.9822469432 0 -0.1875924905 (rot=0, tilt=-100.812, psi=0)
   rot_axis 5  -0.7236067955 -0.5257311143 -0.4472135966 (rot=36, tilt=-116.565, psi=0)
  • /i532_b.jpg

I4

  • another 52 (as used by...) With the positive z-axis pointing at the viewer and without taken into account the 5-fold vertices in z, there is one of the front-most 5-fold vertices in +xz plane (note the plus x)
  rot_axis 2 0.5257311143 0 0.8506508070 (rot=0, tilt=3.7175, psi=0)
  rot_axis 3  0.9822469432 0 -0.1875924905 (rot=0, tilt=100.812, psi=0)
  rot_axis 5  0.7236067955 0.5257311143 -0.4472135966 (rot=36, tilt=116.565, psi=0)
  • /i532.jpg

I5

  • alternative 52 (used by EMBL-matfb). Two fold axis along Z (slow) Place 5 fold along Y (medium). incomplete description. The following two agree:

+x close to a 3 axis (placed in xy plane)

rot_axis 2 0.5257311143 0.8506508070 0 (rot=58.2825, tilt=90, psi=0)
rot_axis 3  0.9822469432 -0.1875924905 0 (rot=-10.8123, tilt=90, psi=0)
rot_axis 5  0.7236067955 -0.4472135966 0.5257311143 (rot=-31.7175, tilt=58.2825, psi=0)

+x close to a 5 axis (placed in xy plane)

rot_axis 2  -0.5257311143 0.8506508070 0 (rot=-58.2825, tilt=-90, psi=0)
rot_axis 3  -0.9822469432 -0.1875924905 0 (rot=10.8123, tilt=-90, psi=0)
rot_axis 5  -0.7236067955 -0.4472135966 -0.5257311143 (rot=31.7175, tilt=-121.717, psi=0)

Ih

Symmetry#SymNote Icosahedral rotational symmetry with reflection symmetry.

  • view model (close any open model before clicking here).
  • ih
    rot_axis 2 0 1 0 (rot=90, tilt=90, psi=0) rot_axis 5 -0.85065080702670 0 0.5257311142635 (rot=0, tilt=-58.2825, psi=0) rot_axis 3 -0.9341723640 0.3568220765 0 (rot=-20.9052, tilt=-90, psi=0) mirror_plane 1 0 0 (rot=0, tilt=90, psi=0)
  • /ih.jpg

(The java models contained in this page has been dowloaded from  http://www.freewebtown.com/adrian/geom/855_symmetry/imagelist with permission from the author)

USER's COMMENTS

In I4 tilt is 3.7175 or 31.7175 Main.RobertoMarabini 09 Nov 2007 - 10:06

-- Main.RobertoMarabini - 21 Apr 2007

Clone this wiki locally