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Symmetry

Symmetry is a one of the most perceptually salient properties of visual images. Because of this, symmetry has been a recurring feature in art, architecture and other artifacts of human construction for centuries. The gestalt psychologists working in Germany in the early 20th century were among the first to recognize the importance of symmetry in visual perception, identifying symmetry as a fundamental principle of visual organization. More recent work in human psychophysics has reiterated this idea, and has lead to a surge in the interest in symmetry perception. Symmetry has been shown to be important for multiple aspects of visual perception, such as perceptual grouping and pattern recognition (Machilsen et al. 2009), face recognition and distinguishing living organisms from inanimate objects (Tyler 1995). Studies of the neural basis of symmetry perception have only recently begun; fMRI studies have found that symmetry appears to be represented late in the visual processing stream in cortical areas that are associated with object representation more generally. These areas include V4, an intermediate-level visual area and the lateral occipital complex (LOC), a high-level visual association area (LOC; Sasaki et al. 2005, Tyler et al. 2005). Another study found that trans-cranial magnetic stimulation (TMS) over LOC had an adverse effect on symmetry perception, suggestion a causal role for this area (Bona et al. 2014).

Almost all existing studies on symmetry, as well as the few attempts to build computation models of how the brain might encode symmetry (Cohen & Zaidi 2013, Dakin & Watt 1994), have focused on mirror symmetry, but there are several other types of symmetry that have yet to be studied. Crystallographers in the late 19th century discovered that any two-dimensional repetitive pattern can be classified as belonging to one of 17 “wallpaper” groups, based on unique combinations of the four fundamental types of symmetry: reflection, translation, rotation and glide reflection (Liu et al., 2010). Below, we show examples of all 17 groups where the repeating structure is a comma-like symbol (illustration by David Wade from Crystal & Dragon, Destiny Books, Vermont, 1993). The wallpaper groups are richly represented in architecture, and can be classified fairly well by naïve human observers (Clarke et al. 2011). Because all forms of symmetry can be exhaustively described in terms of the wallpaper groups, they provide a more complete and theoretically motivated framework for studying symmetry.

We have developed a technique that makes it possible to automatically generate exemplar patterns corresponding to each of the 17 wallpaper groups. The patterns are based on small “seed” regions comprised of 2-dimensional random noise patches. A new symmetric pattern emerges with every new seed region. These patterns are ideal for studying symmetry perception, because a near-infinite number of different images can be generated, each of which is tightly controlled in terms of low-level visual features, while still very being visually compelling and evoking a strong visual response.

 

Symmetry Wallpaper Group P6, illustration by David Wade from Crystal & Dragon (Destiny Books, Vermont, 1993)