A first look at handedness and symmetry

Handedness (chirality)  is a fundamental aspect in chemistry, physics and biology, and another beautiful application of symmetry.  In this elementary introduction we will  explore the handedness of simple planar images, making use of the reflection and double-reflection symmetries 

  D_2 = { 1, h, v, o } 

introduced earlier here.    

Chirality, or handedness, is a property of  pairs of entities related as single reflections of each other. To illustrate it we will look at the case of planar images. The figure below is a D_2 symmetry orbit of the solar-system symbol of  the Moon, shown in the top left corner, along with a vertical reflection on the top right corner. A horizontal  reflection and double-reflection are shown in the bottom row.  

 

We notice that its orbit is reduced to two distinct points, since the symbol has the symmetry of the subgroup { 1, h }. Both its horizontal and vertical reflections can be made to coincide with the original image after applying one of the elements in the subgroup {1, o}. That is, either the reflected image is already equal to the original image or it can be made to coincide by applying a double reflection o = hv.  If this is NOT the case we say that the planar image, object or entity is chiral. Otherwise we say that the entity is achiral. The moon symbol is therefore achiral.

Now consider the D_2 orbit of the symbol of Mercury, shown below:

The symbol has the symmetry of { 1 } alone, and  applying either of {1, o} will recover the initial image. Therefore, the symbol of Mercury is chiral.

The symbols of Pluto and Saturn,

are also chiral, whereas the symbols of the Sun,

the Earth,

Neptune,

 Mercury,

Venus,

and Uranus,

are also achiral. Note that the symbol of Mars,

is also achiral. However, here we need to adjust the reflection line  along a different direction, suggested by the image. Therefore, when inspecting for the handedness of an object we may have to check for the existence of reflection pairs that cannot be resolved by double reflections. In every case, indicating by X the image of interest, T a given reflection, and O a rotation (or double reflection) we want to compare the images

 O T X      ----------         X

or, iterating an arbitrary rotation on both components, we want to compare the images

O' T X      ------       O'' T X

where O' and O'' are arbitrary rotations.  It then says that we may inspect the handedness of a planar image by assessing its sensitivity to rotations and reversals, as introduced earlier on here. Chiral objects react to rotations reversals, whereas achiral objrcts are indifferent. Equivalenlty, a rotating chiral object gives off a different view when inspected from its two fronts, thus distinguishing them. Conversely, a standing chiral object would respond differently to different circular orientations. The key point here is the fact, introduced earlier on here, that reversals are rotations preceded or followed by a reflection.

 The animation shown here displays the rotations and reversals for the (chiral) symbol of Pluto, and the animation shown  here  displays the rotations and reversals for the (achiral) symbol of Mars.

The comparison of rotations and reversals is, ultimately, a particular type of diheral orbit invariant, in exact analogy with the D_2 orbit invariants described earlier on here. We will revisit those concepts in a future time. Sensitivity to rotations ans reversals is at the core of many molecular properties such as their optical activity.

Posted 11/09/2011

Last Revised 11/09/2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Symmetry orbits and space orientation

Here is a simple illustration connecting the notions of symmetry orbit  and  space orientation.  Imagine the following image pasted with its center at the crossing of the equator and the prime meridian of a transparent glass globe, matching its standard cardinal directions. The image is one-sided except in its central area, where it is two-sided.

An observer is allowed to survey the image by walking full-circle around the equator and full-circle around the prime meridian, so that what is seen on the left, right, up, down, front, and back sides of the image relative to the observer's position can be annotated. The globe is stationary relative to the surveyor. What is annotated is the number of disks and circles as the observer approaches the image from the cardinal positions. Here 

is what is the view from the other side of the globe, or the back side of the image, and here

is what the observer reported.

The arrows indicate the front / back side viewing of the image. For example, the observer approaching the image from the East would have seen 16 circles in the front central area, 16 disks up, 25 down, 9 on the left, and 4 on the right side of the image. When at the other side of the globe the observer would have seen 4 circles in back central area, 25 disks up, 16 down, 9 on the left, and 4 on the right side of the image, when again approaching the image from the East.

It turns out that the observer's  path has enough information to map it to a path along a symmetry orbit of the square. That is, a dihedral D_4 orbit. Each up/down, left/right, front/back observed data vector (U,D,L,R,F,B) with the reported number of disks and circles is then indexed by a point in that orbit. 

This will be (somewhat) more evident if we now hold the observer stationary and do a two-step mechanism (the dihedral trick):

• Rotate (say clockwise) the globe along the central axis through the equator-prime meridian intersection (call it the x-axis) by 90, 180, 270, 360 degrees, thus rotating the image around its center;
• Rotate the globe (either direction) 180 deg along the N-S (z) axis, then repeat the four-fold rotations described above. The projections of the 8 transformations onto the z-y plane reproduces the planar (D_4) rotations and reversals.  

The standard question, then, is: what are the summaries of the reported data that remain invariant under the D_4 relabeling? In the present context:  What are the summaries of the data that do not depend on the up/down, left/right, in front/back relabeling? Shortly: What are are the orbit invariants? Or yet: What are the invariants that resolve the arbitrariness in the labels? The corresponding questions formulated for D_2 were discussed in the context of  visual field data. 

We do not have all the tools to develop the complete set of invariants for D_4 yet. However, as we shall see coming along these postings,  the D_4 orbits have exactly 5 invariant summaries. In the present posting we will just enunciate 2 of them. Here they are:

The orbit invariant on the top combines within rotation variation and within-reversals variation, whereas the other one compares rotations with reversals, briefly stating it. Both invariants define one-dimensional subspaces for the data. The first invariant thus resolve the arbitrariness in the left-right, up-down orientation. The relabeling of the planar orientation has the effect of at most changing the sign of the summary (+/- 56). The second invariant resolves the arbitrariness in the front-back orientation. Again, the summary (+/- 48) stays in a one-dimensional subspace. 

These two invariant (subspaces) account for two of the eight dimensions afforded by D_4. As we move along the remaining invariants will be introduced. When all invariants are available, the inverse problem of recovering the original data along the orbit can be effected.  

First revised 06/08/2011
Text with this color was revised in  06/09/2011

These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)