Teaching Students to Illustrate Border and Wallpaper Patterns

Gwen L. Fisher

California Polytechnic State University, San Luis Obispo

In preparation for a new course in Mathematics and Visual Art, illustrations were designed to aid students in their abilities to produce their own examples of the 7 border symmetry groups and the 17 wallpaper symmetry groups. Certainly, with the aid of a computer and software such as KaleidoMania! by Key Curriculum Press or Tess by Pedagoguery Software, a student can easily generate numerous examples of any pre-programmed symmetry group. The software user can simply select a symmetry group, draw a scribble, and the computer will automatically perform the needed isometries to generate the selected pattern. However, students may develop a more robust understanding of the patterns and their generating isometries if they were to learn steps for drawing the designs by hand.

An interdisciplinary course on mathematics and visual art allows for the time to adequately teach students to illustrate the designs by hand. At the same time, the teacher must address a big obstacle: Many college students have difficulty organizing their hand drawings for even simple periodic patterns. Consequently, a complete set of illustrations were developed to guide students in drawing their own examples of the symmetry groups. All of the border and wallpaper groups are illustrated with the use of just two grids: standard square graph paper and isometric graph paper (i.e., the regular tiling by equilateral triangles).  Only two types of grids are needed because each of these symmetry groups is a subgroup of the symmetry group of one of these two grids.

The artist M. C. Escher writes of his work, "In Regular Division of the Plane 1, a process of development takes place. The viewer is invited to follow this by going through the phases, one after the other, of a band of images that… fills the image plane… through… stages of growth and metamorphosis…." Similarly, the images herein attempt to guide the viewer through their stages of development, from simple to complicated. The intention is to allow students to see, step by step, how each drawing was made.