Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and geometry of symmetry is controlled by numerical parameters, which act in analogy with a biological organism's DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional design parameters and shape symmetry for visual or tactile shape sensation, and for perception-based decision making in populations of experts and non-experts. Thereafter, results from a pilot study on the effects of fractal symmetry, referred to herein as the symmetry of things in a thing, on aesthetic judgments and visual preference are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (scale from 0 to 10) the perceived attractiveness of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from"symmetry of things in a thing"significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion.


27
Brain evolution has produced highly specialized processes which enable us to effectively 28 exploit the geometry of visual perceptual space. Some data suggest that the human brain is 29 equipped with an in-built sense of geometry (e.g. Amir et al., 2012;Amir et al., 2014), which is a key to 30 conscious knowledge about specific object properties and associations between two-dimensional 31 projections and their correlated three-dimensional structures in the real world (e.g. Biederman, 1987;

51 52
Gaudi's structures were largely inspired by nature, which abounds with curved shapes and 53 features (see also Ghyka, 1946), and our perception uses these features as cues to shape or object

63
curve symmetry can be directly linked to affine geometry (see also Gerbino and Zhang, 1991

106
Only once this symmetry is perceived by the expert or novice, will he/she be able to draw the 107 structure from memory into axonometric or topological reference frames provided to that effect 108 (Silvestri, Motro, Maurin and Dresp-Langley, 2010). Tensegrity structures have inspired current 109 biological models (e.g. Levin, 2002), from the level of single cells to that of the whole human body.

110
They posses what Mandelbrot (1982) called "fractal consistency across spatial scales", or "fractal 111 iterations", like those seen in large trees that appear composed of many smaller trees of the same 112 structure.

114
structures like those found in cells, trees, butterflies and flowers. A fractal may be defined as a 117 complex whole (object or pattern) that has the same structural characteristics as its constituent parts.

118
The structural symmetry that results from fractal iterations may be described as the symmetry of 119 things in a thing. The radial symmetry of a sunflower is a choice example of fractal symmetry as it 120 exists in nature. Behavioural studies have shown that various animal species are naturally attracted 121 to two-dimensional representations of objects exhibiting flower-like radial symmetry (Lehrer et al., 122 1995; Giurfa et al., 1996). In complex 3D fractal trees, single fractals ("things") have a symmetrical          In the aesthetic rating experiment, subjects were instructed to rate the beauty of each of the 182 fifteen individual images on a subjective psychophysical scale from 0 (zero) for "very ugly" to ten 183 (10) for "very beautiful". In the preference judgment experiment, subjects were instructed to    subjects. Individual responses were coded and written into text files, which were imported into the 203 data analysis software for further processing. The intervals between stimulus presentations were 204 observer controlled. They typically varied from one to three seconds, depending on the observer, 205 who initiated the next image presentation by striking a given response key ("1" for "left", "2" for 206 "right") on the computer keyboard.

209
The raw data from the two experiments were analyzed using Systat 11. Data plots showing 210 medians and variances of the rating distributions were generated. Means and their standard errors 211 of the subjective aesthetic ratings and the total number of "preferred" responses from the preference     (Figure 7), we see that none of the three figure types 235 produced an average score in the extremes ("very beautiful" or "very ugly"). The five symmetrical 236 ones (1 to 5 on the x-axis) produced average ratings between '5' and '8', the five with a small detail 237 missing on the right (6 to 10 on the x-axis) produced average scores between '4' and '6', and the five 238 asymmetrical figures (11 to 15 on the x-axis) scored between '3' and '4' on average.

254
As illustrated by examples from the introduction here above, shape sensation and perception 255 can be related to affine design geometry (e.g. Bahnsen, 1928;Koffka, 1935;Braitenberg, 1990; Gerbino 256 and Zhang, 1991; Dresp-Langley, 2015). Similarly, the topology and geometry of fractal objects may 257 be controlled by a few simple geometric parameters, as in the fractal mirror trees that were used as 258 stimuli here. The term "fractal" was first introduced by Mandelbrot (1982) based on the meaning 259 "broken" or "fractured" (fractus), with reference to geometric patterns existing in nature. The findings 260 from this study here show that the smallest "fractal" deviation from perfect symmetry of things in a

294
The visual attractiveness of 2D fractal design shapes closely depends on the symmetry of things in 295 a thing in configurations with simple geometry, as shown in this pilot study here on the example of a 296 few very basic fractal mirror-trees. In these simple displays, the smallest "fractal" deviation from a 297 perfect symmetry of things in a thing is shown to have significantly negative effects on subjectively 298 perceived beauty and preference judgments. These findings are to encourage further studies, using 299 more sophisticated fractal design objects with increasingly large number of fractal iterations, rotational symmetry in 3D. Such design objects are ideally suited for a numerically controlled 302 manipulation of the smallest of details in the symmetry of things in a thing, perfectly tailored for 303 investigating complex interactions between symmetry and complexity in their effects on visual 304 sensation and aesthetic perception.