Simple Exercises to Learn Escher Tiling

1. A paint by number method may be a good start for young kids. The students are given a printed line drawing using designs from M. C. Escher or any historic examples. The students add color symmetry such that all adjacent tiles have different colors. The color should have enough contrast for the eyes to focus on one or the other. You can also view the colored pattern with color acetates (or color lamps).

2. One of the easiest ways to make good tiles is to chain together a group of squares or equilateral triangles. These are called polyomino and polyiamond. Use a sheet of paper with light gray grids. Any clump of square that follow the grid will fit without gaps. It is also easy to draw the clump because the pattern is repeated by counting the number of squares or triangles.

3. A lot of elementary schools students make tessellation with straight edged polygons. Polygons with 3, 4 or 6 sides will nest easily. Next, you apply the rules of symmetry to changing the edges to curves. The design looks progressively more appealing as the curved boundary suggests a familiar image. M.C. Escher demonstrated this technique with many of his "parquet deformation" prints. When this is done poorly, this technique often looks crude because the silhouette contradicts the content of the tile. This is like a face painted on a square box.

4. When the shapes overlap badly, you feel something is missing on one side. There are often empty spaces between two shapes that are difficult to hide or disguise. To make a good interlocking shape, the features on the boundary should make sense to the intended content on both sides. The two adjacent tiles should fit together with no gaps.

5. You can take an existing Escher tile, erase the color and lines inside the tile, and add new contents. Like changing the Chinese man into a Mexican man. Or changing a bat into a gorilla. The success of this method depends on the interpretation of the silhouette. When I was at University of Illinois, we used computers to generate silhouettes that fit well. However, most of these tiles look silly. After looking at 50 or 100 interlocking tiles, sometimes I can visualize a familiar object fitting into that tile. This is similar to making sense out of a cloud formation in the sky.

6. Here is a comment from David Williams who has made many impressive interlocking tiling. "The 'King Kong' design was a very early one of mine. I think about the second one I did. It was years later that I found in a book on Escher silk-screen prints that was pretty much the same. The Escher print has a 'Bats' design. I suppose there is a limited number of easily producible tiles that will actually suggest a familiar image." You can independently invent a design that is the same as an existing Escher print. They are the same because the grid, the vertices and the method of symmetry are the same. The differences in line curvature change a bat into a gorilla.

7. After you are at this for a while, you can embellish an existing Escher tile. Use the same vertices so that all the lines start and end in the same place. Just modify the curvatures of these lines to make a different image. If this method is practiced enough, you can finally break free and make a tile that nobody will say it is a knock-off from Escher.

8. The boundary of a tile should suggest a familiar image and the lines should be crisp and powerful. Escher commented that after looking at his "crab" and "spider" drawings, he did not like the confusing mesh of legs.

9. Here is a list of parameters needed to understand Escher's interlocking tiling:(1) method of color symmetry, (2) number of tiles if rotation and mirror are consider still the same tile, (3) number of tiles if rotation and mirror are consider different tiles, (4) the grid (triangle, square, rhombus, regular hexagon), (5) all the pivot points of rotation, (6) all the vertices where 3 or more shapes come to a point and not a pivot of rotation, (7) if the pivots and vertices are kept in same place, the degree of curvatures on the boundary possible for the tile to remain successful.