Advances in Visual Psychophysics: Symmetry and Symmetry Groups

Dear Colleagues,

The aim of the present Special Issue is to compile a comprehensive psychophysics of the visual perception of 2D and 3D symmetries and symmetry groups. Stimulus features to be considered include the underlying mathematical parameters (e.g., unit cells, generating regions) as well as surface characteristics (e.g., shapes of tiles, pattern motifs, and space fillers). Of special interest are multiple symmetries as they are seen in tilings and patterns, and also in colored designs, in which the symmetry groups of elements and wholes, or shapes and colors, do not coincide. With regard to responses, the focus is on detection, discrimination, and identification, which can be measured using various psychophysical procedures, but which can also be described in terms of signal detection theory.

(1b) A more detailed exposition:

In mathematics, symmetry (isometry) is well defined for the Euclidean plane and Cartesian space. The concept can be extended to other surfaces (e.g., the surface of a sphere) and to higher-dimensional spaces. In any case, there are a limited number of operations that can algebraically be combined to yield symmetry groups. On planar surfaces, there are translations, rotations, reflections, and glide reflections, and in 3D space, in addition, there are screws (translational rotations) and rotoinversions (rotational reflections). All operations can be substituted by three or four consecutive reflections, respectively. For the Euclidean plane, there are 17 symmetry groups, and for Cartesian space, the number is 230 (11 of which are enantiomorphs, in which the mirror of one arrangement is congruent with an alternative, chiral arrangement). The plane and the space groups all exhibit the property of translational periodicity, that is, a unit cell of constant (but, across groups, variable) shape and size is repeated in two or three different directions. Unit cells in turn can most often be decomposed into fundamental domains (or generating regions), which, together with the symmetry operations, generate the groups.

In order to be visible, symmetries have to be materialized optically. For the Euclidean plane, Grünbaum and Shephard (1987) conceived two prototypical measures to create symmetric designs, in which measures appear relevant for both mathematical and perceptual purposes: tilings versus patterns. Tilings are gapless and nonoverlapping coverings of the Euclidean plane, whereas patterns are comprised of discrete, nonabutting motifs (topological disks), distributed over a homogeneous background. With regard to symmetry, patterns can be shown to be parasitic upon tilings. A third case are tiling patterns in which the tiles of a tiling are marked with a motif. The analogue of tiles in 3D space are space-fillers (typically, polyedra), and the analogue of patterns are arrangements of free-floating objects. In any case, as the symmetry groups of tilings and their tiles, or patterns and their motifs, need not coincide, there often will be layers of symmetry, and the empirical question is whether they can be discriminated, and which one dominates responses. Another feature is color (Coxeter, 1986). As exemplified in the graphical artwork of M.C. Escher (1898-1972), symmetry groups as defined by the shapes of the tiles of a plane tiling need not coincide with the groups defined by coloration. Again, the empirical question is whether observers are able to tell the different symmetry groups apart and which symmetries dominate responses.

Traditionally, psychophysics aims at the determination of absolute and difference thresholds, that is, the questions of whether a stimulus is perceptible at all, and whether two stimuli can reliably be discriminated. Concerning symmetry and symmetry groups, matters become a little more complicated, because observers may or may not be informed about the decisional criteria: Observers may be asked to detect the presence versus absence of unspecified symmetry, or they may be asked to report the presence of specific symmetries. Likewise, for discrimination, observers may be asked to report unspecified differences, or they may be asked to report on the kind or number of symmetries with regard to which one stimulus differs from another one (which presupposes implicit or explicit identification). Going beyond threshold measurements, signal detection theory has introduced the idea of observer operating characteristics; and with regard to methods, adaptive procedures have gained in popularity. Symmetry is an excellent test case for a comparative evaluation of these different approaches and methods.

Here, I invite research that looks at how the underlying mathematical parameters of symmetric designs, as well as their surface properties, as defined in the preceding paragraphs, influence symmetry perception. For example, do different sizes and shapes of unit cells and generating regions, or different shapes of tiles or pattern motifs, support or impede discrimination and identification of symmetries? In the case of multiple symmetries and symmetry groups present in one design, which symmetries or groups dominate perception? Although the focus of the present Special Issue is on 2D and 3D symmetry groups, work on delimited figures and shapes and on individual symmetry operations is also welcome. Submissions may contain ideas about neural mechanisms, but this is not mandatory.

Suggested readings:

Coxeter, H.S.M. Coloured symmetry. In M.C. Escher. Art and Science; Coxeter, H.S.M, Emmer, M, Penrose, R., Teuber, M.L., Eds.; North-Holland (Elsevier): Amsterdam, The Netherlands, 1986; pp. 15-33.
Grünbaum, B.; Shephard, G.C. Tilings and Patterns; Freeman: London, UK, 1987.
Hautus, M.J.; Macmillan, N.A.; Creelman, C.D. Detection Theory: A User’s Guide, 3rd ed.; Routledge: London, UK, 2022.
Schattschneider, D. In Black and White: How to Create Perfectly Colored Symmetric Patterns. Comp. & Maths. with Appls. 1986, 128(3/4), 673-695.

Dr. Klaus Landwehr
Guest Editor

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• symmetry
• symmetry groups
• vision
• psychophysics
• color