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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)