This chapter contains two investigations which use graphics: A, on spirographs and zigzags, and B, on the problem of determining the shape of a wire which gives the fastest time of descent for a bead sliding down the wire.

A Spirographs and zigzags

Aims of the project

The idea here is to draw a zigzag line determined by a simple rule. The interest lies in determining how many times the line must zig and zag before it closes up, and in working out how large the resulting picture is, so that it can be drawn fitting neatly on the screen. We shall also investigate the connection between the zigzags and certain spirograph curves (epicycloids). The basic idea for this project comes from the book [1].

Mathematical ideas used

Vectors in the plane, linear equations and 2 × 2 matrices, regular polygons, gcds, trigonometric formulae and ellipses all come into this project.

MATLAB techniques used

This is a project about drawing sequences of lines. You need to amend a given program to make it do successively more tasks automatically. One of these involves calculating gcds, and another involves setting the screen size to fit the zigzag.

Construction of the zigzag

This project involves drawing zigzag lines according to a simple rule, and examining the mathematics behind these constructions. There is an intimate connection between the zigzags and certain epicycloids, which are obtained by rolling one circle on another and drawing the path traced out by a point rigidly attached to the rolling circle.