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Hyperbolic Footballs

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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...