Symmetry of "Improper Rotations"

On Jul 11, 4:51 pm, Seanpit <seanpitnos...@naturalselection.
0catch.com> wrote:

Again, the type of octahedron you described can be produced by simply
taking the reflected image of one of the pyramids, using that
reflected pyramid as the second pyramid, and rotating it 180 deg into
place so that the bases of each pyramid match each other. And,
presto, you have your octahedron via reflection and rotation. That is
a form of combined reflection and rotational symmetry. You can't do
this with objects that do not have exactly the same measurements on
both sides of a plane (i.e., both of the pyramids have exactly the
same measurements). The pyramids are in fact reflective images of
each other that have simply been stuck end-to-end via rotation of one
of the pyramids.

Just to explain a bit further, have you ever heard of "improper
rotations"?

"Just as two rotational symmetry elements can be combined to
generate a new symmetry element, rotational and mirror symmetry
elements can be combined to obtain new symmetry elements. The
symmetry element that combines a rotation with a reflection
perpendicular to the rotational axis is called an improper rotation.
The transformation matrix for an improper rotation can be calculated
by multiplying the matrices of the rotation and the mirror symmetry
elements . . .
Inversion and improper rotations are important symmetry elements
found in many molecules [like crystals] . . . Most molecules have
several symmetry elements. The combination of these symmetry elements
(their number and arrangement) allows them to be classified into
specific groups called point groups. Hence, point groups describe the
symmetry around a point [i.e., point symmetry - of which there are 32
classes or types]. Importantly, there are only 32 classes of point
groups, each of which has a specific notation. Classifying molecules
into their point groups is useful since it allows one to quickly
describe the symmetry of molecules . . .
Molecules which only have an improper axis are labeled Sn.
Interestingly, only those with n=even are found (S4, S6, and S8...) "

http://wang.biosci.ohio-state.edu/Classes/Chan's.slides/Lecture%2019.ppt

"In 3D geometry, an improper rotation, also called rotoreflection
or rotary reflection is, depending on context, a linear transformation
or affine transformation which is the combination of a rotation about
an axis and an inversion about the origin. Equivalently it is the
combination of a rotation and an inversion in a point on the axis.
Therefore it is also called a rotoinversion or rotary inversion. In
both cases the operations commute. Rotoreflection and rotoinversion
are the same if they differ in angle of rotation by 180°, and the
point of inversion is in the plane of reflection. An improper rotation
of an object thus produces a rotation of its mirror image. The axis is
called the rotation-reflection axis. This is called an n-fold improper
rotation if the angle of rotation is 360°/n. The notation Sn (S for
Spiegel, German for mirror) denotes the symmetry group generated by an
n-fold improper rotation (not to be confused with the same notation
for symmetric groups). The notation *n* is used for n-fold
rotoinversion, i.e. rotation by an angle of rotation of 360°/n with
inversion. In the wider sense, an improper rotation is an indirect
isometry."

http://en.wikipedia.org/wiki/Improper_rotation

Here are a few specific examples of what I'm talking about:

"The methane molecule has an reflection-rotation symmetry, S4,
which consists of a combination of first, a 90 degree of rotation
followed by a reflection which is perpendicular to this axis of
rotation."

http://jchemed.chem.wisc.edu/JCEDLib/symmath/collection/040/HOLSymmetry.pdf

Another example is ethane:

http://symmetry.otterbein.edu/jmol/improper.html

Sean Pitman
www.DetectingDesign.com



.

Relevant Pages

  • Re: Symmetry of "Improper Rotations"
    ... Of course it's a mirror image of what you started with. ... Rotation of a plane ... symmetry - i.e., reflective rotational symmetry. ... symmetry, and _no_ reflection symmetry. ...
    (talk.origins)
  • Re: Granite, Symmetry, and ID - Summary
    ... Of course it's a mirror image of what you started with. ... Rotation of a plane ... symmetry - i.e., reflective rotational symmetry. ... symmetry, and _no_ reflection symmetry. ...
    (talk.origins)
  • Re: isometries in E^3-
    ... Orientation preserving isometry in E^3 with unique fixed ... >>identity, reflection, rotation, translation, glide reflection (i.e. ... >>identity, plane reflection, rotation ...
    (sci.math)
  • Re: homomorphism from D6 to S5 or A5
    ... Dn is the group of all rigid symmetries of this n-gon. ... The element r corresponds to a rotation in the positive direction by ... The element s corresponds to the reflection about the ...
    (sci.math)
  • Re: Reflections on BitBoards
    ... one is the rotation or reflection of the other. ... Symmetry positions are, in effect, ... to a bit-permutation. ...
    (comp.programming)