Visual Perception of Symmetry - IV

Background:

Visual Perception of Symmetry III
Visual Perception of Symmetry II
Visual Perception of Symmetry I

                                         
In this run of the experiment, the colored flags were replaced by patterned flags, shown below. Everything else the same.


                                            A                   B                    C                   D  

A total of 59 undergraduate students ranked the flags in each orbit. The flags were printed in black and white and individually presented to the students.

The following table describes the frequency distribution for all 24 permutations embedded in the full set of rankings, thus allowing for a set of data indexed by the full symmetric group S_4. For example, the ranking ABCD = A first choice, ..., D last choice, appeared in 27 of the rankings.

The following tables show the transition frequency counts for first-to-second choices. The orbits are numbered sequentially top to bottom: Orbit 1 is the top row in the original set, and Orbit 6 is the bottom row in the original set above. Here, as in the previous experiments, each orbit is a symmetry orbit of K_4.

Here is the combined (across orbits) summary of first-to-second choices:

Does the patterning of the flags affect the rankings when compared with the original set of colored flags? It surely did. This is evident from the posterior densities, shown in Figure 1, where now the preference for vertically related first-to-second choices is less evident, relative to the colored-flags protocol referenced here.

Figure 1.

Posted: 01/11/2012

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)

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)