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)