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)

Field orbits

The following is a symmetry orbit in which the elements are planar vectors:

To generate the orbit, we may start with any one of its vectors in red, rotating it counterclockwise in steps of 90 degrees, producing the vectors in red; Then we may reflect the original starting vector with respect to the x-axis, and rotated it the same say, thus producing the remaining vectors (black color) in the orbit.  The resulting orbit thus has the symmetries of the square, introduced earlier on  here. Clearly, the orbit would have been the same regardless of its initial condition (generating vector) and direction of rotation.

The orbit above has the symmetry of the square. We will indicate those symmetries with the symbol D_4. We will refer to the symmetries of the regular polygons as dihedral symmetries,  We will write D_3 for the triangle, D_4 for the square and so on, for D_5, D_6, .... We extend the notation to D_2 as well, although in this case it  indicates the symmetries of the rectangle.

These symmetries are the planar line reflections and double reflections (or rotations), or equivalently, the n-fold rotations and rotations preceded or followed by a reflection. These always come in pairs, and referred to as rotations and reversals.

Rotations and reversals have, intrinsically, the same reflexivity found in the naming of left- and right-handed objects. It is perhaps better not to try to say it in words. It gets rather silly very fast.

Field orbits are simple designs to gather experimental results that are then indexed by the points in the orbit.

For example, dihedral symmetry aspects in a visual field can be studied by its response to a dihedral orbit of the square by embedding a set of rotations in the visual field, say:

0 0 0 26 27 24 21 0 0 0

0 0 23 23 26 28 25 22 0 0

0 27 29 27 27 28 29 29 28 0

28 29 29 29 31 20 20 28 28 29

26 26 29 30 33 34 28 27 28 27

28 30 30 31 32 33 33 4 27 28

27 29 31 31 32 32 30 29 28 27

0 30 31 31 28 29 32 30 30 0

0 0 28 32 29 28 29 28 0 0

0 0 0 26 27 28 29 0 0 0

and the companion reversals:

0 0 0 26 27 24 21 0 0 0

0 0 23 23 26 28 25 22 0 0

0 27 29 27 27 28 29 29 28 0

28 29 29 29 31 20 20 28 28 29

26 26 29 30 33 34 28 27 28 27

28 30 30 31 32 33 33 4 27 28

27 29 31 31 32 32 30 29 28 27

0 30 31 31 28 29 32 30 30 0

0 0 28 32 29 28 29 28 0 0

0 0 0 26 27 28 29 0 0 0

Any summary of the visual field over each dihedral field vector, such as its gradient, average, or extreme values, is then indexed by the dihedral symmetries. The summaries may, of course, be in more than one dimension. Visual fields are usually obtained from fellow eyes, so each orbit may have a fellow orbit obtained from the fellow eye, thus producing a pair of numbers in each point of the dihedral field. 

Similarly, below are the dihedral field orbits of D_5, D_6, and D_10.

The question to be addressed in later postings is that of determining the orbit invariants for those experimental data, their broad interpretations, and plausible methods of statistical inference.

Last revised: May 15th, 2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

A bit of (written) language for symmetries

Let's revisit the elements of the symmetry orbit of the triangle while replacing the three-coloring of their vertices with three-letter labeling using the letters  A, C, and T.  Let's first look at the single-reflections. 

We write (AC)  and say that

  label A goes to replace C and C goes to replace label A 

when referring to a transposition  of these two vertices, as another term indicating the reflection fixing the remaining vertex (T). Here is the diagram of the transposition (AC):

And here are the illustrations of the other two transpositions:

The double-reflections give rise to rotations:  Now we write (ACT) and say that

label A goes to replace label C, label C replaces T, and T replaces A.

Here is the illustration of the 2Pi/3 (clockwise) rotation (ACT):

The 4Pi/3 rotation in the same direction:

An additional rotation would bring us back to the original triangle.  This gives the identity transformation, and it is indicated by 1:

We now have language to write down the complete set of symmetries acting on the triangle:

{ 1, (ACT), (ATC), (AC), (AT), (TC) }. 

We also say that the symmetries were written using their cycles notation.

Last revised 02/23/11
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Symmetry orbits of the triangle

These six images

illustrate the symmetry orbit of the three-color triangle. Note that:

• Any two triangles in different rows are single-reflection images of each other;
• Any two triangles in the same row are double-reflection images of each other;
• Any two triangles in the same row are rotated images of each other.

As a consequence, the symmetry orbit of the triangle is generated by single and double reflections, or, equivalently, by reflections and rotations. 

The separation angle between adjacent reflection axes in the triangle is 2Pi/6 and is referred to as its dihedral angle. The resulting rotations are in angles that are multiple of twice the dihedral angle.

The rotations in one row are in the opposite direction relative to the rotations in the other row. This may hint us to connecting symmetry orbits with space orientation: For example, the arms of  a transparent clock hanging in a transparent wall rotate in opposite directions when viewed from opposing sides of its wall. Conversely, your position relative to the wall may be determined or label by the direction of rotating arms.
More to come on this.

Any one of the six triangles generates all the remaining ones after the reflections and iterated reflections are applied to that initial triangle. Therefore, the initial choice, or the generating element, is arbitrary. This property of arbitrariness is present in any symmetry orbit and will be explored later on in future postings.

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Reflections and reflection orbits

Reflections are very useful mechanisms to express symmetry relations among objects, images, ideas, narratives, poems, lyrics, melodies, movements, arguments, molecules, and many more.

The squares shown below illustrate the mechanism of line reflections.

Select any one of the squares. Can you then identify the square that is its image by a vertical line of reflection? Next, identify the other square that is its image by an horizontal line of reflection.  Now observe that the remaining square is the image of the originally selected one by:

• Iterating the two reflections (in any order) or;
• Directly rotating it (by 180 degrees).

Thus, it seems that here the iteration of two line reflections (with 90 degrees of separation) produces the same effect of a (planar 180 degree) rotation of the original square.

Let's indicate these symmetry operations with the letters:

v to indicate a reflection relative to a vertical line;

h to indicate a reflection relative to a horizontal  line;

o to indicate a double-reflection or the 180-degree central rotation.

You will observe that it really does not matter which square you select as your starting point: after applying  the reflections and double-reflections to that initial square you will always end up with the same set. This set is called an orbit of symmetry. We say that the orbit has the symmetry of those reflections and their iterations. To include the starting square in the orbit we add a transformation that just leaves any object as is. We call it the identity transformation, and write it as 1.

Together, then, the set of transformations, that we write as

D_2 = {1, v, h, o},

generates the reflection orbits. Shortly, we say that the resulting orbits have the symmetry of D_2.

In the Flag Experiment each row of flags is an orbit with the symmetry of D_2.

Last revised 02/02/2011
06/06/06
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)