Symmetry is less than meets the eye: Extra data and tables

Deborah Apthorp and Jason Bell

The material in this document represents further data and figures in relation to the paper published in Current Biology, “Symmetry is Less than Meets the Eye”. Due to limitations on the length of SI for the journal, we have posted the material here. Some explanatory paragraphs from the SI are also included to give context to the tables and Figure.

Experiment 4: Perceived structure comparisons.

To check that the difference in effects between the Glass patterns and the symmetrical patterns was not due to a difference in perceived structure between the two types of patterns (i.e., the effect might still have been due to perceived structure in the symmetrical patterns, but Glass patterns might not have been perceived as structured to the same extent), we tested eight participants in a perceived structure rating experiment. The stimuli were symmetrical, asymmetrical and obliquely symmetrical dot patterns containing 100 dots, and concentric and random Glass patterns containing 100 dots. Each stimulus type (randomly generated) appeared 20 times, in random order, for 300 ms. After each stimulus disappeared, participants were presented with a visual analogue scale on the computer screen, with a marker the participant could move using the left and right arrow keys on the keyboard. The scale read “How structured did the stimulus appear?” and the left and right ends of the scale read “Very unstructured” and “Very structured”. Custom MATLAB code converted the final marker position into a score ranging from 0 (extreme left end of the scale) to 10 (extreme right end of the scale), which was used as the perceived structure rating for that trial. Responses were untimed.

The overall repeated measures one-way ANOVA showed there were significant differences in perceived structure for the different patterns, F(4,28) = 35.375, p < .0001. Importantly, follow-up pairwise comparisons (Bonferroni corrected) showed there was no difference in perceived structure between symmetrical patterns and Glass patterns, with both patterns being rated as highly structured (see Table S1). In addition, symmetrical patterns were perceived as significantly more structured than asymmetrical patterns, and structured Glass patterns were also perceived as significantly more structured than random Glass patterns (see Table S1). Interestingly, here oblique symmetry patterns were perceived as significantly less structured than vertical symmetry patterns, and not different in perceived structure to the asymmetrical patterns; however, the decrease in perceived number for these patterns persisted (see Experiment 2), strongly suggesting that perceived structure was not the main driving force behind the effect. The full results are presented in Table S1.

Table S1: Pairwise comparisons between all pattern conditions, Bonferroni-corrected for multiple comparisons. The mean ratings being compared are shown in Figure S2.

Pattern comparison / t(7) / CI (diff) / P-value
Asym vs. Rand Glass / 1.524 / -1.779, .385 / >.9999
Asym vs. Struct Glass / 8.883 / -5.858, -3.395 / .0005***
Asym vs. Sym / 10.368 / 3.285, 5.226 / .0002***
Asym vs. Oblique Sym / 3.364 / -2.569, -.448 / 0.1202
Rand Glass vs. Struct Glass / 6.370 / -5.387, -2.471 / .0038**
Rand Glass vs. Sym / 5.740 / -5.024, -2.092 / .0071**
Rand Glass vs. Oblique Sym / 1.023 / -2.688, 1.065 / >.9999
Struct Glass vs. Sym / .568 / -1.173, 1.914 / >.9999
Struct Glass vs. Oblique Sym / 4.695 / 1.547, 4.688 / .0222*
Sym vs. Oblique Sym / 5.107 / 1.475, 4.019 / .0139*

P-values: * = <.05, ** = <.01, *** = <.001

Dot distance checks – histograms and statistical comparisons

For the symmetry experiments, the symmetrical and asymmetrical displays were generated slightly differently (i.e., symmetrical displays were mirrored, such that there were always an equal number of dots in each half of the display, whereas asymmetrical displays were generated randomly for the entire display). Thus it seemed possible that this might have affected the distances between the dots – for instance, asymmetrical displays, in the limit, may have had all the dots in one half of the display, resulting in a much more crowded image. Although we did not observe this during the experiment, it seemed sensible to check that there was no systematic difference in dot distances between the two types of display that might explain the effect.

Using custom code written in MATLAB Version R2013a, we generated 100 symmetrical and 100 asymmetrical dot patterns (these were generated with exactly the same code as was used in the experiments). We chose 100 as this was the number of trials participants completed in each condition. For each of these patterns, we calculated all the pairwise distances between the dots, using MATLAB’s function pdist. Example histograms are shown in Figure S3.

Since the distributions were slightly skewed, we checked the medians as well as the means and standard deviations of the dot distances for each type of pattern. In all cases (for sets of 50, 100 and 200 dots), there was no significant difference between any of these variables for symmetrical and asymmetrical patterns (see Table S2).

Figure S3. Example histograms of all pairwise dot distances for symmetrical and asymmetrical dot patterns containing 50 (a,b) and 100 (c,d) dots.

Table S2. Statistical results of comparisons between asymmetrical and symmetrical array dot distances for 50, 100 and 200 dot arrays across 100 iterations of the stimuli, showing comparisons of mean distance, median distance and variability (SD). All tests were independent 2-sample t-tests assuming equal variance; tests assuming unequal variance were also carried out, and produced very similar results.

Condition / t(198) / CI (diff) / p-value
50 dots mean / 1.250 / -2.242, 4.502 / .5095
100 dots mean / 0.9541 / -1.139. 3.274 / .3412
200 dots mean / 1.1003 / -0.643, 2.262 / .2727
50 dots median / 0.9290 / -1.932, 5.374 / .3540
100 dots median / 0.9541 / -0.913, 3.692 / .2355
200 dots median / 1.1866 / -0.609, 2.448 / .2369
50 dots SD / 0.1522 / -1.222, 1.427 / .8792
100 dots SD / 1.1124 / -0.379, 1.360 / .2763
200 dots SD / 0.8904 / -0.307, 0.810 / .3745

Supplemental Data: Individual data tables

Experiment 1 – Symmetry and perceived number

Number of Extra Symmetrical Dots
Observer / 50 / 100 / 200
am / 10.361 / 24.580
as / 3.604 / 3.694
da / 2.068 / 5.027 / 16.714
hb / 6.931 / 17.249 / 8.318
jb / 4.650 / 8.974
mc / 4.017 / 8.246 / 26.285
rj / 7.443 / 12.491 / 8.544
eb / 3.223 / 5.200 / 14.587
Percent Extra Symmetrical Dots
Observer / 50 / 100 / 200
am / 23.005 / 27.860
as / 7.257 / 3.797
da / 4.141 / 5.120 / 8.515
hb / 14.544 / 19.265
jc / 9.635 / 9.521 / 4.143
mc / 8.280 / 8.591 / 4.496
rj / 16.161 / 13.371 / 13.808
eb / 6.429 / 5.290 / 7.455

Experiment 2 – vertical vs. diagonal symmetry

Number of Extra Symmetrical Dots
Vertical / Diagonal
Observer / 50 / 100 / 50 / 100
da / 2.068 / 5.027 / 3.327 / 7.730
rj / 7.443 / 12.491 / 1.923 / 3.092
mc / 4.017 / 8.246 / 1.705 / 5.222
eb / 3.223 / 5.200 / 3.646 / 4.017
Percent Extra Symmetrical Dots
Vertical / Diagonal
Observer / 50 / 100 / 50 / 100
da / 4.141 / 5.340 / 6.720 / 4.775
rj / 16.161 / 11.442 / 3.868 / 7.357
mc / 8.280 / 6.158 / 3.427 / 5.168
eb / 6.429 / 4.233 / 7.636 / 7.448

Experiment 3 – Symmetry vs. Structure

Number of Extra Symmetrical Dots
Symmetry / Structure
Observer / 50 / 100 / 50 / 100
am / 10.361 / 24.580 / 0.825 / 0.587
as / 3.604 / 3.694 / 0.505 / 0.467
da / 2.068 / 5.027 / 1.819 / 2.490
hb / 6.931 / 17.249 / 3.337 / 10.579
jc / 4.650 / 8.974 / 2.113 / 1.045
rj / 7.443 / 12.491 / 0.714 / -4.947
Percent Extra Symmetrical Dots
Symmetry / Structure
Observer / 50 / 100 / 50 / 100
am / 23.005 / 27.860 / 1.653 / 0.587
as / 7.257 / 3.694 / 1.015 / 0.480
da / 4.141 / 5.120 / 3.756 / 2.539
hb / 14.544 / 19.265 / 6.849 / 10.987
jc / 9.635 / 9.521 / 4.277 / 1.050
rj / 16.161 / 13.371 / 1.442 / -4.804