Challenges in Plane Symmetry: From Theory to Perception

TL;DR

The paper investigates the gap between mathematical plane symmetry and human perception by focusing on a single challenging Moroccan-like ornament. It combines a rigorous group-theoretic analysis (identifying the ornament's true symmetry as $cmm$ rather than the visually suggested $p6m$) with two perceptual experiments that collect similarity judgments using Kendall tau distances and visualize relationships via t-SNE. The main finding is that participants tend to favor visually salient higher-order symmetries and often misalign with the formal wallpaper-group classification, highlighting limits of purely group-theoretic approaches for perceptual symmetry. This work emphasizes the need for perceptually grounded models that account for prominence of certain motifs and the comparative evaluation people perform when assessing symmetry.

Abstract

The planar ornaments are created by repeating a base unit using a combination of four primitive geometric operations: translation, rotation, reflection, and glide reflection. According to group theory, different combinations of these four geometric operations lead to different symmetry groups. In this work, we select a single challenging ornament, and analyze it both from the theoretical point of view and perceptual point of view. We present the perceptual experiment results, where one can see that the symmetries that the participants perceived from the ornaments do not match to what the theory dictates.

Figures (17)

• Figure 1: Ornaments that belong to the same symmetry group in an uncolored case. When the color symmetry is allowed all four belong to different groups. The second row illustrates the generation rules for the ornaments given in the first row.
• Figure 2: Two quite similar ornaments produced using two different repetition rules. (a) An ornament of $p1$ group. Produced by translating two distinct figures, which are similar. The red box contains an image where the white and blue birds are superimposed. It shows the distinction between these two shapes. (b) An ornament of $pg$ group. Contains one bird figure of two different colors. A bird of one color is a glide reflection of a bird of another color.
• Figure 3: Moroccan Ornament
• Figure 4: Generation of $p6m$ ornament using two different fundamental domains (a) Obtained UC assuming that an ornament has 6-fold rotations and extracted two FDs. (b)-(c) Generated $p6m$ ornaments using two different FDs.
• Figure 5: Illustrating three-fold and four-fold rotations. (a) The center of three-fold rotation shown as red dot. (b) Rotation for $120 \degree$. Rotated image is laid on top of the original image. Observe the regions marked with black circles. These are the obvious mismatch places. (c) The center of four-fold rotation shown as green dot. (b) Rotation for $90 \degree$. Rotated image is laid on top of the original image. Obvious mismatch places marked as black circles.