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 molecular frameworks.

Here is a simple illustration of evaluating the dihedral orbits of D_2, introduced earlier on here, in analogy to the field orbits introduced here. We want to study the orbits generated by the symmetries of

$$

D_2 = {1,h,v,o}

$$

when applied to each one of the x- and y-  displacements shown in the 5-element framework below. We recall that o indicates the double reflection, that is,  o=hv=vh. The multiplication table of D_2 was introduced earlier here.

For example, applying the {1, h, v, o} symmetries, as planar transformations, to the displacement  x_1 gives the orbit

{ x_1, x_4, -x_2, -x_3 }.

There are two remaining orbits for x-displacements, specifically,

{ x_0, x_0, -x_0, -x_0 },   

{ x_2, x_3, -x_1, -x_4 };

and three orbits for y-displacements, specifically:

{ y_0, -y_0, y_0, -y_0 }, 

{ y_1, -y_4, y_2, -y_3 }, 

{ y_3, -y_2, y_4, -y_1 }.

These resulting displacements are then examples of data indexed by the symmetries of D_2, in analogy with the examples introduced here, or  here. The resulting orbit invariants, in each orbit, are the summaries of the data that were introduced earlier, in the same pages indexed above. In each orbit, therefore, we evaluate the invariants 

I_1 = d_1 + d_h + d_v + d_o,

I_h = d_1 + d_h - d_v - d_o,

I_v = d_1 - d_h + d_v - d_o,

I_o = d_1 - d_h - d_v + d_o,

where d_s indicated the result of applying the symmetry s to the displacement d.

For example, applying I_1 to the orbit

 { x_1, x_4, -x_2, -x_3 }

gives

 x_1 + x_4 - x_2 - x_3. 

When I_1 is applied to the remaining orbits we find that they become matched with (alternating or same-sign) companions of the same displacement type among the outer framework elements, and within the center-of-mass element. The same matching applies to the other invariants I_h, I_v, and I_o.

The resulting joint effect is called a normal mode of (potential) framework vibration, each mode being indexed by an orbit invariant (later on to be recognized as an irreducible representation) of D_2.

This simple construction retains, however, the main elements of several classically important areas in physics, chemistry and biology, and in particular in the foundations of vibrational spectroscopy.

Here is the superposition (d_x + d_y) of the modes associated with I_1:

Here is the mode associated with I_h:

Here is the mode associated with I_v:

And here is the mode associated with the I_o:

The animation of  1-mode is available here, the h-mode here, the v-mode here, and the o-mode  here.

In future postings we will revisit this mechanism to obtain the classification of normal modes of real molecules. The simple adaptation will require the introduction of 3D space symmetries operation on 3D frameworks, with x-, y-, and z-displacements. 

Posted: 10/30/2011

Last revised: 10/30/2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Note: The computer code to generate the graphical displacements and corresponding animations was developed with Mathematica (TM) and is available from the author (viana@uic.edu).

Symmetry orbits and curvature maps

The cornea is the main refracting surface of the human eye. Its front (anterior) surface is approximately 1.3 cm^2 with an average radius of curvature of about 7.8mm.  Typical computer algorithms for corneal curvature measurement (known as keratometry) are based on projecting a pattern of concentric rings of light onto the anterior surface of the cornea and numerically determining the relative separation between the images of these reflected rings of light. This models the  anterior surface of the cornea as a highly polished spherical mirror.  By sampling the curvature at specific circularly equidistant points, a numerical model for the surface curvature may be obtained.

The corneal curvature, (k), and its refractive index, (n), contribute to determining the surface's ptical refractive power k(n-n'), where n' is the reference refractive index of the air. Most of the light refraction takes place at the surface of the cornea, which has refractive index n=1.3376.

Light then passes through the aqueous humor (n=1.336, close to the refractive index of  water) to the lens (n=1.386 to 1.406, where it is refracted further) and through the vitreous humor to the retina at the back of the eye.

The standard unit of refractive optical power is the diopter (D) and is defined to the inverse of the radius of curvature, or approximately equal to the inverse of the focal length of the refracting element. One diopter equals one inverse meter (m^{-1}). For example, using the standard keratometric index n=1.3375, a cornea with a curvature of 7.50mm at 0 degrees has  power (1/0.0075)x0.3375=45D, whereas if the curvature at 90 degrees is 7.25mm, the power is 46.50D.

The difference between these steep (maximum) and the flat (minimum) curvatures, as illustrated in the diagram above (From W.F. Long), is the amount of regular astigmatism present in the optics of the eye, which interferes with a sharp formation of the image on the retina. In the present example, the regular astigmatism is 1.50D. 

The graph below shows a typical curvature contour near the apex of the cornea:

In analogy with the D_2 symmetry orbits derived  for the  visual field data we can evaluate the D_2 orbit

C, vC, hC, vhC

for the curvature contours by applying the corresponding planar reflections {v, h} and double reflection {vh} to the support of the contour. The following graph shows the contour for the difference

C-hC 

shifted by 7D (red contour) to allow its polar plotting relative to that reference value (black contour).

Recalling the multiplication table for D_2 we now know that

h(C-hC) = hC - h^2 C = hC - C = - (C - hC)

so that the contour is horizontally anti-symmetric. 

Here is the 7D-shifted contour for C - vC:

showing its vertical anti-symmetry, and here

is the contour for C - hvC, showing its central (double-reflection) anti-symmetry.

Again as introduced earlier in the study of the  visual field , we seek to describe the orbits invariants for D_2, namely:

I1 = C + vC + hC + hvC,

Iv = C + vC - hC - hvC, 

Ih = C - vC + hC - hvC,

Ihv = C - vC - hC + hvC. 

Here is the contour for the full symmetric invariant I1,

which is vertically, horizontally and centrally (double-reflection) symmetric. Here is the contour for the invariant Ih,

which is horizontally symmetric, vertically and centrally anti-symmetric.  Here

is the contour for Iv, which is vertically symmetric, horizontally and centrally anti-symmetric. And here

is the contour for Ihv, which is centrally symmetric, and vertically and horizontally anti-symmetric. These are then the invariant summaries of a single contour under the D_2 orbit applied to its support. Clearly, its support allows for many more refined groups of symmetries. We will, in the future, revisit these contours after learning the general method for determining the orbit invariants. 

First revised 06/20/2011
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)

Visual Perception of Symmetry - II

Let us now revisit the  flag experiment with the language we have introduced up to now. The experiment consisted of ranking the flags within each row in the figure below:

Each row, as we now recognize, is a symmetry orbit generated by

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

 The flags in columns B,C, and D are generated from the flag in column A by applying a horizontal reflection, a  double reflection, and a vertical reflection respectively.

The rankings were purely subjective, as there were no other directions given at the time of the   flag experiment.

The following table summarizes the frequency counts for the number of occasions in which the first choice is the flag with the row label and the second choice is the flag with the column label in that table. For example, in 14 occasions, flag B was the first choice and flag C was the second choice. I will refer to these frequency counts as transition counts, and write BC=14 to indicate the transition count just described.

With this is mind, a direct inspection of the flags in any one of the orbits shows that the transitions

• BC, CB, AD, DA are associated with vertical reflections,
• AB, BA, CD, DC are associated with  horizontal reflections,
• AC, CA, BD, DB are associated with  double reflections.

The posterior probability densities associated with these symmetry transitions of first-to-second preference, based on the joint frequency counts above, are shown in Figure 1. It is evident that the most likely pairing of first-to-second preferences is associated with those flags that are vertical reflection of each other.  

Figure 1.

Moreover, the vertical transitions form a symmetry orbit, as illustrated below:

For example, starting in (A,D) you move to (D,A) by a vertical reflection on each of the component flags; move to (B,C) by a horizontal reflection and to (C,B) by a double reflection.  The transitions were arranged in the diagram in a way that they are related by their relative positions in the square. The fellow diagram shows the corresponding transition counts, obtained from the summary data table at the top of the page.

These vertical transition data are, therefore, indexed by an orbit of  D_2. However, the assignment of data to the labels in D_2 is arbitrary, in the following way:

If we start the orbit in (A,D) then

• x_1 = 12;
• x_v = 16;
• x_o = 11;
• x_h = 14.

However, if we start the orbit in (B,C), then:

• x_1 = 14;
• x_v = 11;
• x_o = 16;
• x_h = 12.

Clearly, since each D_2 orbit has 4 points, there are 4 possible starting points and hence four relabelings of the data.

Similarly, the horizontal transition counts are labeled by a symmetry orbit:

And finally, the orbit for the the double reflection counts:

These transitions orbits, together, are illustrated here,

in relation to the joint frequency counts table shown at the top of this page.

The question to be investigated is this:

What are the summaries of the frequency counts that do NOT depend on a particular relabeling?

We have alluded to the answer, the orbit invariants for that specific orbit, here.  In all that is coming, the aim is exploring ways of systematically determining the invariants for most types of elementary orbits. Stay tuned.

Assignments:
[1] Following the discussion of the vertical transitions orbit, write the data assigned to the orbit with (A,D) as its starting point in the form of the symbolic sum

X=12*1 + 16*v + 11*o + 14*h. 

Then, referring to the multiplication table of D_2, evaluate the left multiplications

• h*X 
• v*X
• o*X

and verify that all regular relabellings of a D_2 orbit can be obtained that way.

Posted: 05/14/2011.
Last revised: 01/28/2015.
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Symmetries orbits and the visual field

The matrix

describes the output of an automated perimetry test used in the assessment of the visual field. The entries represent the sensitivity of the retina in detecting the light stimulus and is expressed in decibels units, with a maximal possible reading of 50db. A 50db target is the dimmest target the instrument can project. The smallest the reading the least the sensitivity at that retinal location given by reference to the rows and columns of the matrix. The matrix then describes a function

F( x, y )

defined over a symmetric array of 10x10 points. Here is a graphical display of F:

This says that we can explore the visual field data by letting certain symmetries act in the array where the field is defined. That is, we can study the transformed fields

F,  vF,  hF,  hvF 

forming the symmetry orbit of  F acted by the dihedral group

 D_2 = {1, v, h, o},

where the line reflections are with respect to rows and columns of F. It is opportune to remark that F  plays the role of coloring the underlying framework, in analogy with the colorings of flags and triangles introduced earlier. Here are the results:

  

• Horizontal reflection:

• Vertical reflection

• Double reflection:

The following fields, shown below, are obtained from the symmetry orbit as follows:

• I1 = F + vF + hF + hvF 
• Iv = F + vF - hF - hvF
• Ih = F - vF + hF - hvF
• Io = F - vF - hF + hvF 

They have some remarkable properties, and are referred to as orbit invariants: in the sense that each one of these matrices, indicated here as I,  is such that

wI = +/- I

for all w in  D_2 = { 1, v, h, o }.   Here they are, for your verification.

• The invariant field I1:

• The invariant field Iv:

• The invariant field Ih:

• The invariant field Io (double reflection):

Orbital invariants are important tools in the analysis of data associated with symmetry orbits, and we will often return to that notion in future postings.

Last revised  05/15/2011

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