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.
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.
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
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)