Lénart Spheres
What is the shortest distance between two points on a sphere?
Introduction
A Lenart sphere is a clear plastic ball used to draw and measure triangles directly on a curved surface, the way you would with a ruler and protractor on paper. On a sphere the 'straight lines' are great circles, and triangles drawn from three great-circle arcs behave differently from flat triangles: their angles add to more than 180 degrees, and larger triangles have a larger excess. This exploration asks you to draw spherical triangles, measure their angles, and find the relationship between the angle excess and the triangle's area. Non-Euclidean geometry itself is not part of the DP syllabus, so treat the sphere's geometry as new ground, using ordinary trigonometry - the sine rule and the triangle area formula - to take and check your measurements.
Guiding Questions
- On a sphere, what is the 'straightest' path between two points? Test your answer on a ball with string.
- Why do triangle angles add to more than 180 degrees on a sphere? Measure one and check.
- How does the angle excess relate to the triangle's area? Investigate with several triangles.
- Which everyday facts of flat geometry survive on the sphere, and which break?