A basic activity to help students understand finite geometry is to have them draw a shape. Give each student a sheet of graph paper and have them draw a shape. Don't be too specific about guidelines, but instead ask them to choose a starting point on the graph paper. Once they are done, ask them to answer a few questions about their shape. For example: How many points and lines does it have?
Have each student share their shape with the class, explaining how many points and lines each contains. Explain that each of these shapes can be considered finite geometry: each one contains a fixed number of points, lines and planes. Explain that by drawing the shapes as they did, the students defined the rules for their finite geometries. Have students exchange drawings with one another and try to create new shapes following the same rules.
Another important concept in finite geometry is axioms. Simply defined, these are the rules that govern finite geometry. For example, a set of axioms might say: "There are six points and two lines," "Each line contains at least two points" and "No line can contain more than four points." These axioms are what separate finite geometry from plane geometry, since the axioms of Euclidean geometry allow for an infinite number of points and lines.
Another finite geometry activity involves "Kirkman's Schoolgirl Problem," which states, "Fifteen young ladies in a school walk out three abreast for seven days in succession. It is required to arrange them daily so that no two shall walk twice abreast." This problem demonstrates finite geometry. To solve this problem, students must think of each girl as a point in a system. The axioms in this problem would be, "Each line must contain exactly three points" and "No line may contain the same three points."