These worksheets contain a progressive sequence of exercises using the GeomLab programming environment. The first few worksheets introduce the idea of building up complex pictures by transforming and putting together simpler components. Then comes the idea of describing an increasingly complex sequence of pictures by a recurrence relation defining a recursive function. The final worksheet explores the structure of M. C. Escher's woodcut "Square Limit", which can also be described by a recurrence relation.
Each worksheet is provided as a web page that can be accessed by the links below. But if your screen is not huge, it's probably best to download and print out the entire collection of worksheets formatted as a PDF file. (Adobe PDF files require a reader program that is available free from the Adobe website.)
There's also a short PowerPoint presentation that I use to introduce the activity to small groups.
Worksheet 1: Above and Beside. Expressions can have pictures as their values, and there are operations for joining two pictures vertically or horizontally.
Worksheet 2: Rotations and Reflections. Pictures can be transformed by rotating them and flipping them over (or reflecting them). Complex pictures can be built up from simpler elements by using both these transformations and the combining operators introduced earlier.
Worksheet 3: Definitions and Functions. A specific operation on pictures can be captured by defining a function, which can then be applied to different arguments. More complex operations can be described by composing functions together.
Worksheet 4: Recurrences and Recursion. By analysing the structure of a series of pictures, we can capture the whole series in a recursive definition, so that a finite program can describe an infinite variety of behaviour. For example, we can write a GeomLab program that can draw a row of men of any length, or a crowd of men consisting of any number of rows.
Worksheet 5: Spirals and Zig-zags. By combining recursive definitions with rotations and reflections, it is possible to generate geometric figures like zig-zags and spirals. Again, a whole series of figures can be generated from one, finite definition.
Worksheet 6: Escher pictures. M. C. Escher's woodcut "Square Limit" shows a tessellation of fish-like figures that become smaller as they approach the edge of the picture, so that an infinite number of figures is (in principle) shown in a finite space. This picture can be made from a small number of fixed tiles that are assembled in a way that is described by a recurrence. As a final challenge, you are invited to find the recurrence that describes another picture, not drawn by Escher, in which the fish get smaller towards the centre.
Worksheet 7: Space-filling curves. These families of curves wiggle around so as to come close to every point in a square region. The curves on this sheet can be made with the same two tiles that were used in Worksheet 5, but the patterns of recursion are challengingly complex.
Worksheet 8: Turtle graphics. Other families of curves are difficult to make with tiles, and this worksheet introduces another way of describing them: as sequences of instructions for following the curve by turning to the left and to the right.