Symmetry and Antisymmetry in Maori Rafter Designs

Abstract

The Maori settled in New Zealand in the eighth or ninth centuries A.D.; the Europeans, many centuries later. Maori meeting houses are very ornate: besides carved wood figures, painted black-red-white designs embellish rafters and structural posts. These designs stand out as highly sophisticated examples of the twenty-four known band groups of 2-dimensional 1-translational symmetry G₂¹ and antisymmetry (G₂¹)′ The two independent symmetry elements, periodically repeated by lattice p or antilattice p′, are any two of the following: mirror m and antimirror m′, both either transverse or longitudinal; glide plane a or antiglide plane a′, longitudinal only; and two-fold rotation axis 2 or anti-axis 2′. Pseudosymmetry is common and obviously intentional: desymmetrization is achieved by added motifs, inserted along the edges of the band, at cell boundaries or inside the main design, which possess their own symmetries, always lower than that of the original design. The resulting symmetry of the whole is the symmetry common to the superposed patterns. Desymmetrization is also obtained by topological deformations of the design or deviations from the color scheme imposed by antisymmetry. Such irregularities led to Herbert Williams' hypothesis that the artist worked in two stages: tracing the outline and working in the color [1]. This conjecture finds support in group theory: the geometrical symmetry group involving no change of color (stage 1) is turned into an antisymmetry group (stage 2), as red goes to black and black goes to red under antisymmetry operations. The international symmetry notation of crystallographers is shown to be well suited to describe the Maori designs.