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Topics: Symmetry 18 images
tree :: Antiprism Examples and Visual Reference > Topics and Overviews > Topics: Symmetry
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    Several programs have features involving symmetry:
  • antiview, off2vrml, off2pov - display symmetry elements
  • poly_kscope - repeats a model symmetrically, used to make compounds
  • off_align - repeats a model symmetrically, used to augment polyhedra
  • off_trans - aligns a model according to symmetry
  • off_report - prints symmetry information for a model
  • off_color - colours by symmetry orbit
The program option parameters are organised around the following ideas
Symmetry group
A set of (Euclidean) transformations that carry a polyhedron onto itself, described in general form using Schoenflies notation (see below) e.g. Oh, D3v.

Full symmetry group of a polyhedron
The set of all (Euclidean) transformations that carry a polyhedron onto itself.

Symmetry subgroup, or subsymmetry
A set of transformations from a symmetry group which, considered alone, also form a symmetry group, e.g. a cube has Oh symmetry and has a 3-fold axis corresponding to a C3v subgroup.

Symmetry orbit of an element
A set of equivalent elements, those elements that this element is carried on to by a symmetry or subsymmetry of the model.

Standard alignment of a symmetry
A symmetry group could be aligned anywhere in the coordinate system, but there are particular alignments that fit nicely with the coordinate axes, and these are used as the 'standard' alignments in Antiprism. They are a way of associating a symbol like D3v with a fixed set of transformations.

Conjugation subtype of a subsymmetry
This is an integer used to distinguish subgroups which are not carried onto each other (by conjugation) by the transformations of the parent symmetry group. For example a cube has a 2 fold axis through mid-edge and a 2-fold axis through a face centre. There is no symmetry of the cube that carries one onto the other and so they will have different subtype numbers. Geometrically, they look different in the cube.

Symmetry realignment
If you align a polyhedron with the standard set of symmetries for its full symmetry group there is often more than one distinct way to achieve this (a transformation not in the symmetry group that transforms the symmetry group onto itself). For example, if you align a cereal box-like cuboid naturally with the coordinate axes there are 6 possibilities i.e. the centres of the three rectangle types can lie on any of the axes, with 3x2x1 = 6. Possibilities for some polyhedra are infinite e.g. the symmetry group of a pyramid does not change when it is translated along its principal axis. The realignment is given by a series of colon separated numbers, the first number selects from a finite set of realignments, and the following numbers are decimals to control rotations and translations as follows:
  • axial rotation: 1 number - degrees around principle axis
  • full rotation: 3 numbers - degrees around x, y and z axes
  • axial translation:i 1 number - distance to translate along principal axis
  • plane translation: 2 numbers - distance to translate along two orthogonal directions in (mirror) plane
  • full translation: 3 numbers - distance to translate along x, y and z axes

Schoenflies notation
Used to specify symmetry groups. The standard alignments have, preferentially, a centre (fixed point) on the origin a principal rotational axes on the z-axis, a dihedral axis on the x-axis, a mirror normal on the y-axis (except Cs has a mirror normal on the z-axis). The polyhedral symmetry types have a 3-fold axis on (1,1,1). In the following list of symbols, when a type contains 'n' this must be replaced by an integer (giving an n-fold axis), and for S this integer must be even.
  • C1 - identity
  • Cs - mirror
  • Ci - inversion
  • Cn - cyclic rotational
  • Cnv - cyclic rotational with vertical mirror
  • Cnh - cyclic rotational with horizontal mirror
  • Dn - dihedral rotational
  • Dnv - dihedral rotational with vertical mirror
  • Dnh - dihedral rotational with horizontal mirror
  • Sn - cyclic rotational (n/2-fold) with rotation-reflection
  • T - tetrahedral rotational
  • Td - full tetrahedral
  • Th - tetrahedral with inversion (pyritohedral)
  • O - octahedral rotational
  • Oh - full octahedral
  • I - icosahedral rotational
  • Ih - full icosahedral
There are two symmetry reports. off_report -S s gives a general listing of the full symmetry, subsymmetries with numer of types, realignment possibilities and the axes with their number
    off_report -S s rh_cubo
The subsymmetry and realignment possibilities indicate what will be valid in the symmetry options. There is also a list of orbits, with the total number of orbits for each element type, and the number of elements in each orbit.
    off_report -C O rh_cubo
A value of 1 for verts, edges, faces indicates the polyhedron is repsepctively isogonal, isotoxal, isohedral. The orbits can also be calculated for a subgroup, using the -y option
    off_report -y D2h -C O rh_cubo
    off_report -y D2h,1 -C O rh_cubo
These are the same orbits that are used for colouring. The first number indicates the number of colours that will be used, and the second numbers are the number of elements having each colour
    off_color -f S rh_cubo | antiview
    off_color -f S,D2h rh_cubo | antiview
    off_color -f S,D2h,1 rh_cubo | antiview
 
Image List

   
View symmetry  Med    Lrg  
Description : antiview, off2pov and off2vrml can display symmetry elements using the -s option. The axes (x), mirror planes (m) and rotation reflection planes (r) can be viewed independently and in combination. View them all with -s a, as in the example command. antiview can also cycle through the display of symmetry elements by pressing y while viewing a model.

View this model with the command

off_trans -R 0,20,0 -R 15,0,0 cube | antiview -s a
Click on one of the size names above to enlarge this image
 
Kaleidoscope  Med    Lrg  
Description : poly_kscope repeats a model symetrically, acting like a polyhedral kaleidoscope. Only one example is included here, there are more in the poly_kscope examples album.

When making a compound the component will generally have symmetries that are also symmetries of the compound. It could be difficult to align a component into a position where symmetries are shared using rotations and translations, as an alternative there is a facility to do this directly by symmetry. The option -y list will list all arrangements of the component and final symmetry according to how the component can be aligned to share a common symmetry group with the kaleidoscope. To make the model rerun the command but replace 'list' with the number and any optional realignment. Running this for the previous cube example, produces a list with 20 items

poly_kscope cube -s Ih -y list
The last number is 19 and corresponds to the compound of 5 cubes, as in the previous example
poly_kscope cube -s Ih -y 19 | antiview
In the following example the cube will be aligned on a C3v axis, and the optional realignment translates the cube 2 units along the icosahedral 3-fold axis.

View this model with the command

poly_kscope cube -m f,compound -s Ih -y 8,:2  | antiview -v 0.05
Click on one of the size names above to enlarge this image
 
Excavate Full  Med    Lrg  
Description : The -F option of off_align aligns a face on the brick with a face on the base and then uses it to augment or excavate the base model (the polyhedra are joined and the bonding faces removed). This operation can be repeated symmetrically with the -y option.

In this example the base polygon of a pentagonal pyramid is aligned with a face of a dodecahedron. An alignment number is chosen that causes the pyramid to lie inside the dodecahedron, producing an excavation. The bonding operation is repeated with the symmetry of the base model by specifying -y full.

View this model with the command

off_align -F pyr5,0,0,5 -y full dodecahedron  | antiview
Click on one of the size names above to enlarge this image
 
Augment Full  Med    Lrg  
Description : The -F option of off_align aligns a face on the brick with a face on the base and then uses it to augment or excavate the base model (the polyhedra are joined and the bonding faces removed). This operation can be repeated symmetrically with the -y option.

In this example a tetrahedron is aligned with a face of an icosahedron. The default alignment causes the tetrahedron to lie outside the icosahedron, producing an augmentation. The bonding operation is repeated with the symmetry of the base model by specifying -y full.

View this model with the command

off_align -F tet -y full icosahedron | antiview
Click on one of the size names above to enlarge this image
 
Augment Subsym  Med    Lrg  
Description : The -F option of off_align aligns a face on the brick with a face on the base and then uses it to augment or excavate the base model (the polyhedra are joined and the bonding faces removed). This operation can be repeated symmetrically with the -y option.

In this example a triangular prism is aligned with a face of an icosahedron. The bonding operation is repeated with tetrahedral symmetry, a subsymmetry of the base model, by specifying -y T.

View this model with the command

off_align -F pri3,5 -y T icosahedron | antiview
Click on one of the size names above to enlarge this image
 
Align full  Med    Lrg  
Description : Align a cube with the standard symmetry for its full symmetry. The square prism model, pri4, is a cube, but by construction is not aligned with the standard octahedral axes. This command shows how any model can be aligned with its standard symmetry group.

View this model with the command

off_trans -y full pri4 | antiview
Click on one of the size names above to enlarge this image
 
Align sub 0  Med    Lrg  
Description : Align the first C3v subgroup (subtype 0) of a hexagonal pyramid with the standard C3v symmetry

View this model with the command

polygon pyr 6 | off_trans -y C3v | antiview
Click on one of the size names above to enlarge this image
 
Align sub 1  Med    Lrg  
Description : The hexagonal pyramid has two C3v subgroups, align the second C3v subgroup (subtype 1) of a hexagonal pyramid with the standard C3v symmetry

View this model with the command

polygon pyr 6 | off_trans -y C3v,1 | antiview
Click on one of the size names above to enlarge this image
 
Align sub 1 2  Med    Lrg  
Description : Align the second C3v subgroup (subtype 1) of a hexagonal pyramid with the standard C3v symmetry. Then select an alternative alignment for this group using the third specifier, in this case realignment 2.

View this model with the command

polygon pyr 6 | off_trans -y C3v,1,2 | antiview
Click on one of the size names above to enlarge this image
 
Align cuboid  Med    Lrg  
Description : There are six ways to align a cuboid with its full symmetry group. Rerun the command with the last number running from 0 to 5 to see all the alternatives.

View this model with the command

off_trans cube -S 1,2,3 | off_trans -y full,0,0 | antiview
Click on one of the size names above to enlarge this image
 
Align with poly  Med    Lrg  
Description : Aligning one polyhedron with another is a two stage process. The polyhedron to be aligned is aligned with a standard symmetry using the -y option. The standard symmetry is then aligned with a subgroup of the same type in the fixed polyhedron using the -Y option.

In this case, a C3v subgroup of a hexagonal pyramid is aligned with a C3v subgroup of a cube.

View this model with the command

polygon pyr 6 | off_trans -y C3v -Y cube,C3v  | antiview - cube
Click on one of the size names above to enlarge this image
 
Align with poly  Med    Lrg  
Description : The cube only has one type of C3v subgroup but the hexagonal pyramid has two. In this example a C3v subgroup of a hexagonal pyramid is aligned with a C3v subgroup of a cube again, but a second alignment is specified.

View this model with the command

polygon pyr 6 | off_trans -y C3v,1 -Y cube,C3v  | antiview - cube
Click on one of the size names above to enlarge this image
 
Align with poly  Med    Lrg  
Description : Realignments also work when aligning one polyhedron with another, and may be applied using either the -y or -Y option.

In this case, a C3v subgroup of a hexagonal pyramid is aligned with a C3v subgroup of a cube, which is followed by a realignment by translating along the principal axis

View this model with the command

polygon pyr 6 | off_trans -y C3v -Y cube,C3v,0,0:-1.0 | antiview - cube
Click on one of the size names above to enlarge this image
 
Align with poly  Med    Lrg  
Description : When aligning one polyhedron with another, sometimes the fixed symmetry will have more than one type of the subsymmetry. Select between them with the third part of option -Y.

In this case, a cuboid is aligned on the second kind of D2h group of a cube (specified with the number 1).

View this model with the command

off_trans cube -S 0.5,1.5,3 | off_trans -y full -Y cube,D2h,1 | antiview - cube
Click on one of the size names above to enlarge this image
 
Align with sym  Med    Lrg  
Description : The -Y option will also work with a standard symmetry type, which will create an alignment suitable for use with poly_kscope. However, poly_kscope also includes support for this through its own -y option.

This example makes uniform compound UC40, consisting of 6 decagonal prisms

View this model with the command

off_trans -y D5v,1 -Y Ih,D5v pri10 | poly_kscope -s Ih -c f -m compound | antiview
Click on one of the size names above to enlarge this image
 
Colour Full  Med    Lrg  
Description : A symmetrical colouring can be made by colouring the elements according to symmetry orbits.

In this example, the square faces of the rhombicuboctahedron fall into two orbits under the full symmetry of the model (faces on 4-fold axes and faces on 2-fold axes). The following command assigns different colours to the different kinds of face.

View this model with the command

off_color -f S rh_cubo | antiview
Click on one of the size names above to enlarge this image
 
Colour Sub0  Med    Lrg  
Description : A symmetrical colouring can be made by colouring the elements according to symmetry orbits.

In this example, the square faces of the rhombicuboctahedron fall into six orbits under the default D2h sub-symmetry of the model. The following command assigns different colours to the different kinds of face.

View this model with the command

off_color -f S,D2h rh_cubo | antiview
Click on one of the size names above to enlarge this image
 
Colour Sub1  Med    Lrg  
Description : A symmetrical colouring can be made by colouring the elements according to symmetry orbits.

In this example, the square faces of the rhombicuboctahedron fall into five orbits under the second D2h sub-symmetry (specified as subsymmetry number 1) of the model. The following command assigns different colours to the different kinds of face.

View this model with the command

off_color -f S,D2h,1 rh_cubo | antiview
Click on one of the size names above to enlarge this image
 

 

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