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eman2:symmetry [2025/07/04 17:47] – created steveludtkeeman2:symmetry [2025/07/04 17:58] (current) – steveludtke
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 C-symmetry occurs when an object is symmetric about a single axis. The degree of symmetry about this axis is referred to as //n//. A rotation of //2pi/n// about this axis will yield the object in an identical (but unique) conformation. Based on this fact it is possible to define the asymmetric unit for c-symmetries as shown in Figure 1 (left). However, as stated above, half of the asymmetric unit will contain the mirror projections of the other half, and the way in which the mirror projections are arranged respectfully in the asymmetric unit depends on whether //n// is even or odd. To illustrate this, the behavior of even and odd mirror projections in the asymmetric unit is shown in Figure 1 (middle and right, respectfully). C-symmetry occurs when an object is symmetric about a single axis. The degree of symmetry about this axis is referred to as //n//. A rotation of //2pi/n// about this axis will yield the object in an identical (but unique) conformation. Based on this fact it is possible to define the asymmetric unit for c-symmetries as shown in Figure 1 (left). However, as stated above, half of the asymmetric unit will contain the mirror projections of the other half, and the way in which the mirror projections are arranged respectfully in the asymmetric unit depends on whether //n// is even or odd. To illustrate this, the behavior of even and odd mirror projections in the asymmetric unit is shown in Figure 1 (middle and right, respectfully).
  
-{{attachment:csym_ai.png}}+{{http://blake.bcm.edu/dl/EMAN2/csym_ai.png}}
  
 //Figure 1. C-symmetry in EMAN: The asymmetric unit and location of mirror projections //for// even and odd **n**.// //Figure 1. C-symmetry in EMAN: The asymmetric unit and location of mirror projections //for// even and odd **n**.//
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 An object with d-symmetry is similar to an object c-symmetry in that a rotation around the //n//-fold axis of symmetry will yield the object in an identical conformation.  In addition to this the d-symmetric object exhibits (potentially numerous) 2-fold symmetries in the equatorial domain. These 2-fold axes of symmetry are mutually separated by an angle of //2pi/n //and the number of 2-fold symmetric axes a d-symmetric object exhibits is dependent on whether //n// is even or odd, and is //n/2// or //n//, respectfully. The asymmetric unit of d-symmetric objects is defined in terms of the //n//-fold and 2-fold axes of symmetry and is shown in Figure 2 (left). Similar to the behavior of the asymmetric unit for c-symmetries, in d-symmetries the behavior of the mirror projections is dependent on whether n is even or odd as shown in Figure 2 (middle and right, respectfully). An object with d-symmetry is similar to an object c-symmetry in that a rotation around the //n//-fold axis of symmetry will yield the object in an identical conformation.  In addition to this the d-symmetric object exhibits (potentially numerous) 2-fold symmetries in the equatorial domain. These 2-fold axes of symmetry are mutually separated by an angle of //2pi/n //and the number of 2-fold symmetric axes a d-symmetric object exhibits is dependent on whether //n// is even or odd, and is //n/2// or //n//, respectfully. The asymmetric unit of d-symmetric objects is defined in terms of the //n//-fold and 2-fold axes of symmetry and is shown in Figure 2 (left). Similar to the behavior of the asymmetric unit for c-symmetries, in d-symmetries the behavior of the mirror projections is dependent on whether n is even or odd as shown in Figure 2 (middle and right, respectfully).
  
-{{attachment:dsym_lowerres.png}}+{{http://blake.bcm.edu/dl/EMAN2/dsym_lowerres.png}}
  
 //Figure 2. D-symmetry in EMAN: The asymmetric unit and location of mirror projections for even and odd **n**.// //Figure 2. D-symmetry in EMAN: The asymmetric unit and location of mirror projections for even and odd **n**.//
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 A tetrahedron is an example of a platonic solid that has 4 3-fold axes of symmetry and 3 2-fold axes of symmetry. The asymmetric unit of the tetrahedron is defined in terms of neighboring 3- and 2-fold axis of symmetry and is depicted graphically in Figure 3 (left). The octahedron has 3 4-fold axes of symmetry, 4 3-fold axes of symmetry, and 6 2-fold axes of symmetry and its asymmetric unit (with respect to neighboring symmetric axes) is shown in Figure 3 (middle). Similarly, the icosahedron has 6 5-fold, 10 3-fold and  15 2-fold axes of symmetry and its asymmetric unit is shown (with respect to neighboring symmetric axes) in Figure 3 (right). A tetrahedron is an example of a platonic solid that has 4 3-fold axes of symmetry and 3 2-fold axes of symmetry. The asymmetric unit of the tetrahedron is defined in terms of neighboring 3- and 2-fold axis of symmetry and is depicted graphically in Figure 3 (left). The octahedron has 3 4-fold axes of symmetry, 4 3-fold axes of symmetry, and 6 2-fold axes of symmetry and its asymmetric unit (with respect to neighboring symmetric axes) is shown in Figure 3 (middle). Similarly, the icosahedron has 6 5-fold, 10 3-fold and  15 2-fold axes of symmetry and its asymmetric unit is shown (with respect to neighboring symmetric axes) in Figure 3 (right).
  
-{{attachment:platonic_asymm_unit2.png}}+{{http://blake.bcm.edu/dl/EMAN2/platonic_asymm_unit2.png}}
  
 //Figure 3. The asymmetric unit of the platonic solids (tetrahedron, octahedron, icosahedron).// //Figure 3. The asymmetric unit of the platonic solids (tetrahedron, octahedron, icosahedron).//
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 The mirror portion of the tetrahedral asymmetric is perhaps the most unusual and is shown graphically in terms of the neighboring symmetric axes in Figure 4 (left). This is in contrast to the mirror projection behavior of octahedral and icosahedral symmetries which is shown in Figure 4 (right). The mirror portion of the tetrahedral asymmetric is perhaps the most unusual and is shown graphically in terms of the neighboring symmetric axes in Figure 4 (left). This is in contrast to the mirror projection behavior of octahedral and icosahedral symmetries which is shown in Figure 4 (right).
  
-// **//' // +// {{http://blake.bcm.edu/dl/EMAN2/platonic_mirror3.png}} //
- +
-// {{attachment:platonic_mirror3.png}} //+
  
 Figure 4. //Location of mirror projections in the asymmetric unit of the tetrahedron, octahedron and icosahedron.// Figure 4. //Location of mirror projections in the asymmetric unit of the tetrahedron, octahedron and icosahedron.//
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 ===== Interactive asymmetric unit viewing using emimag3dsym.py in EMAN2 ===== ===== Interactive asymmetric unit viewing using emimag3dsym.py in EMAN2 =====
-<<Anchor(e2imagesym)>> 
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 EMAN2 includes a tool that will display the asymmetric unit of each of the symmetries in an interactive OpenGL interface. The distribution of projection orientations within the asymmetric unit is also shown. Pictures of this interface are shown below. EMAN2 includes a tool that will display the asymmetric unit of each of the symmetries in an interactive OpenGL interface. The distribution of projection orientations within the asymmetric unit is also shown. Pictures of this interface are shown below.
 || {{http://blake.bcm.edu/dl/EMAN2/groel_v1_d7_mirror_v3.png}} || {{http://blake.bcm.edu/dl/EMAN2/groel_v1_d7_nomirror_v3.png}}|| || {{http://blake.bcm.edu/dl/EMAN2/groel_v1_d7_mirror_v3.png}} || {{http://blake.bcm.edu/dl/EMAN2/groel_v1_d7_nomirror_v3.png}}||
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 ===== Python Programs Making use of Symmetry ===== ===== Python Programs Making use of Symmetry =====
  
-The //Symmetry3D// object in EMAN2's library represents a symmetry generator. The easiest way to construct one of these objects is through the **parsesym** function which is part of the core library you get when you //from EMAN2 import *". **sym=parsesym("c4")**, for example, will return a symmetry object representing the C4 symmetry group as described above. The symmetry object has a number of different useful methods:+The //Symmetry3D// object in EMAN2's library represents a symmetry generator. The easiest way to construct one of these objects is through the **parsesym** function which is part of the core library you get when you //from EMAN2 import *//. //sym=parsesym("c4")//, for example, will return a symmetry object representing the C4 symmetry group as described above. The symmetry object has a number of different useful methods:
   * **sym.gen_orientations(orientgen,parms)** - Will generate a list of Transform objects filling one asymmetric unit using the specified parameters. //e2help.py orientgen -v 2// for details   * **sym.gen_orientations(orientgen,parms)** - Will generate a list of Transform objects filling one asymmetric unit using the specified parameters. //e2help.py orientgen -v 2// for details
   * **sym.get_nsym()** - will return the symmetry number (ie- 60 for icos, 14 for D7, etc.)   * **sym.get_nsym()** - will return the symmetry number (ie- 60 for icos, 14 for D7, etc.)
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   * **reduce(xform,n)** - will return the the Transform //xform// mapped into the specified asymmetric unit, //n//   * **reduce(xform,n)** - will return the the Transform //xform// mapped into the specified asymmetric unit, //n//
 Use //help(Symmetry3D)// for full documentation. Use //help(Symmetry3D)// for full documentation.
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 **References** **References**
  
 //Baldwin, P.R. and Penczek, P.A. 2007. The Transform Class in SPARX and EMAN2. J. Struct. Biol. 157, 250-261. // //Baldwin, P.R. and Penczek, P.A. 2007. The Transform Class in SPARX and EMAN2. J. Struct. Biol. 157, 250-261. //
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eman2/symmetry.1751651271.txt.gz · Last modified: by steveludtke