Hyperbolic Footballs

We are all familiar with the basic Euclidean geometry of the plane, including the behavior of parallel lines and angles in triangles. This familiarity may lead us to think it is a law of nature that parallel lines are always the same distance apart and the sum of the angles of a triangle is 180°. The purpose of this activity is to disabuse us of this misconception. Included are materials and directions to build an attractive two-dimensional model of negatively curved (hyperbolic) space (called the hyperbolic football). We also describe an activity to use this model to explore the basic geometry of parallel lines, quadrilaterals, and triangles in the hyperbolic plane. This session is suitable for a math circle or classroom activity. This has been run at many levels from 5/6 grade on through University Lecturers.