Search This Blog

Wednesday, January 28, 2015

Visual Perception of Symmetry - V


Background:

Visual Perception of Symmetry IV.
Visual Perception of Symmetry III
Visual Perception of Symmetry II
Visual Perception of Symmetry I


A total of 24 undergraduate students participated in the survey, randomized to either Protocol 1 or Protocol 2 shown below.


Protocol 1:



Protocol 2:





In each protocol, students were presented with 6 color 4x4 prints individually enclosed in a sheet of paper. Each four-flag print is a dihedral D_2 orbit, as described in earlier blogs. Students were asked to simply rank the flags according to their attractiveness assigning the score 1 to the most attractive, 2 to the next to most attractive, 3 to the next, and 4 to the least attractive. The rankings were marked in the sheet of paper individually enclosing each 4x4 print, so that each students generated 6 rankings, accounting to the 24 distinct four-color flags.

In Protocol 1 rotations
and reversals are displayed side by side along the two rows, and the two rows are horizontal mirror image of each other. In Protocol 2 rotations and reversals are shown along the columns, with the columns vertical mirror image of each other. A total of 12 students were surveyed with Protocol 1 and the remaining 12 with Protocol 2.

Results: Figure 1 shows the posterior distribution for the proportion of transitions between first and second most attractive flags that are vertical mirror image of each other (red), horizontal mirror image of each other (green) and rotated image of each other (blue) for Protocol 1 (top) and Protocol 2 (bottom). As predicted from previous protocols, the preference to vertical mirror image was lessened with the transformed flags were not along the vertical mirror.


Figure 1



Raw Data:

Raw data for Protocol 1. Frequency counts for first (row) to second (column) choices. Vertical matches between first and second choices are accounted by (A,D)+(D,A)+(B,C)+(C,B) transitions, horizontal matches by (A,B)+(B,A)+(C,D)+(D,C) transitions, and rotational matches by the remaining transitions, as detailed earlier, for example, here and more generally here:


A
B
C
D
Total
A
0
8
5
5
18
B
4
0
4
5
13
C
2
15
0
10
27
D
6
3
5
0
14
Total
12
26
14
20
72




Recovered frequency counts for all S_4 rankings based on the six D_2 orbits:


SEQ
Frequency
Cumulative
Frequency
Percent
Cumulative
Percent
abcd
1
1
1.389
1.389
abdc
7
8
9.722
11.111
acbd
2
10
2.778
13.889
acdb
3
13
4.167
18.056
adbc
4
17
5.556
23.611
adcb
1
18
1.389
25.000
badc
4
22
5.556
30.556
bcad
3
25
4.167
34.722
bcda
1
26
1.389
36.111
bdac
1
27
1.389
37.500
bdca
4
31
5.556
43.056
cadb
2
33
2.778
45.833
cbad
4
37
5.556
51.389
cbda
11
48
15.278
66.667
cdab
4
52
5.556
72.222
cdba
6
58
8.333
80.556
dabc
1
59
1.389
81.944
dacb
5
64
6.944
88.889
dbac
1
65
1.389
90.278
dbca
2
67
2.778
93.056
dcab
3
70
4.167
97.222
dcba
2
72
2.778
100.000






Raw data for Protocol 2. Frequency counts for first (row) to second (column) choices. Vertical matches between first and second choices are accounted by (A,D)+(D,A)+(B,C)+(C,B) transitions, horizontal matches by (A,B)+(B,A)+(C,D)+(D,C) transitions, and rotational matches by the remaining transitions, as detailed earlier:




A
B
C
D
Total
A
0
4
1
19
24
B
2
0
7
3
12
C
2
5
0
4
11
D
13
4
8
0
25
Total
17
13
16
26
72


Recovered frequency counts for all S_4 rankings based on the six D_2 orbits:


SEQ
Frequency
Cumulative
Frequency
Percent
Cumulative
Percent
abdc
4
4
5.556
5.556
acbd
1
5
1.389
6.944
adbc
13
18
18.056
25.000
adcb
6
24
8.333
33.333
bacd
1
25
1.389
34.722
badc
1
26
1.389
36.111
bcad
4
30
5.556
41.667
bcda
3
33
4.167
45.833
bdac
3
36
4.167
50.000
cabd
1
37
1.389
51.389
cadb
1
38
1.389
52.778
cbad
4
42
5.556
58.333
cbda
1
43
1.389
59.722
cdab
3
46
4.167
63.889
cdba
1
47
1.389
65.278
dabc
3
50
4.167
69.444
dacb
10
60
13.889
83.333
dbac
2
62
2.778
86.111
dbca
2
64
2.778
88.889
dcab
2
66
2.778
91.667
dcba
6
72
8.333
100.000




First posted 01/28/2015
Most Recent revision: 01/29/2015 
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)
and on
 "Dihedral Fourier Analysis" Springer Lecture Notes in Statistics (2013)

Friday, February 24, 2012

Connecting Bridges





The following circuit is known as the Wheatstone Bridge:



The output voltage W in relation to the input voltage w is given by


$$ W=\frac{r_1r_4-r_2r_3}{(r_1+r_2)(r_3+r_4)} w. $$

Our simple project is the the classification of these circuits when we are allowed to shuffle the resistances around. We assume here that all four resistance values are distinct. We follow the notation and definitions  introduced in previous postings, such as in here.

We will shuffle the (indices in the) resistances in the circuit according to the symmetries of $$ K_4=\{1,(12)(34),(13)(24),(14)(23)\}\equiv \{1,h,v,o\}, $$ and then compare the resulting output voltages

$$W_h, W_v,W_o$$

with the original output voltage

$$W_1.$$

There are, however, several other circuits that can be used as a starting point of the K_4 orbits. So let's redraw the circuit in a simpler form and permute the resistances indexed by {1, 2, 3}  according to

$$
D_3=\{1,(123),(132),(12),(13),(23)\},
$$

thus obtaining the following six distinct circuits:


Indicate them, respectively, by


$$
\{\;W^1,\;\;W^{(123)},\;\;W^{(132)},\;\;W^{(12)},\;\;W^{(13)},\;\;W^{(23)}\;\}.
$$

As K_4 shuffle each one of these six generating circuits, we would then obtain the totality of the 24 distinct bridge circuits. This is in analogy to the sorting of the 24 flags used in the flag preference experiment.

Associated with K_4 we have the following table of characters  (to be made precise later):


$$
\boxed{ \begin {array}{c|rrrr}  &1&h&v&o\\\hline z_1&1&1&1&1
\\ z_2&1&1&-1&-1\\ z_3&1&-1&1&-1
\\ z_4&1&-1&-1&1\end {array} },
$$
Now a really important concept: We say that a circuit W reduces as z if

$$ W_\tau = z (\tau)\; W_1, $$

for all symmetries in K_4.

Direct calculation will show us that:

$$W^1\;\;\text{ and }\;\;W^{(23)} \text{ reduce as }\; z_4;$$

$$W^{(123)}\;\;\text{ and }\;\;W^{(12)} \text{ reduce as }\; z_3;$$
$$W^{(132)}\;\;\text{ and }\;\;W^{(13)} \text{ reduce as }\; z_2.$$


We have then obtained a classification of all (distinct- resistance) Wheatstone Bridge circuits according to the  character in $$\{z_1,z_2,z_2,z_4\}$$ to which its K_4 orbit belongs to.


Connected Bridges: If  $$W^{\sigma_j}$$ reduces as $$z_j$$ and $$W^{\sigma_i}$$ reduces as $$z_i$$ then the iterated bridge circuit, in which one circuit's input voltage is the other's output voltage, reduces as $$z_jz_i.$$

If we define the resonance of two iterated bridges by
$$<z_j,z_i>=\sum_\tau\; z_j(\tau)z\;_i(\tau),$$
then we will find out that two iterated bridges resonate, that is, $$<z_j,z_i>\neq 0,$$ if and only if they reduce according to the same character.

Only circuits that belong to the same character resonate. This is a simple sort of Selection Rule in that (current) transitions among circuits (states) can occur only when they share the same characters. This is a fundamental principle in Chemistry and Physics. 


First posted 02/24/2012
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Saturday, November 26, 2011

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)






Wednesday, November 9, 2011

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)




Sunday, October 30, 2011

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).

Tuesday, October 4, 2011

Visual Perception of Symmetry - III

Background:

Visual Perception of Symmetry II
Visual Perception of Symmetry I

The flag preference experiments described above were repeated in the Fall semester of 2011, with the same purpose of ranking the flags according to one's preference. The orbit structure of the flags was precisely the same. However, the flags were square in shape and the colors were the basic RGBY colors. Here are the flags presented for ranking:




The joint distribution of frequency counts for first-choice flag (rows) and second-choice flags (columns) is shown below. A total of 13 students ranked the flags in each of the 6 rows, thus together producing the 78 rankings for the joint distribution below.

In future postings we will be studying these data using the methods of symmetry orbits.

Posted: 10/04/2011
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)

Monday, June 20, 2011

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)