Curvature Group Summary

Monday - Friday, July 15 - 19, 2013

This week, the class delved into the application for which the first two weeks have been preparation: the four-dimensional manifold "SpaceTime." Monday's topics included light cones, Galileo dropping things off the Leaning Tower of Pisa, special and general relativity, dust clouds, and the Einstein vacuum equation. Tuesday we derived Kepler's laws of planetary motion and considered an 1859 study of the advance of Mercury's perihelion (its closest point to the sun) that had astronomers looking for a small planet "Vulcan" even closer to the sun, when in fact the discrepancy was explained by Einstein in 1915 using the new general relativity.

In our debrief time, we continued exploring the triangulated manifolds we created the Friday before. For triangles on the plane, we determined the area of the k-ball to be 6k^2, breaking the area into six sectors and using the identity 1 + 3 + ... + (2n-1) = n^2. For the ruffly hyperbolic plane with 7 triangles around each vertex, we broke the k-ball area into 7 tilting wedges and found some very interesting patterns involving Lucas and Fibonacci numbers. In particular, the Lucas number begin L(1) = 2, , L(2) = 1, 3, 4, 7, 11, 18, 29, following the same recursion as Fibonacci numbers; the number of new triangles in each wedge of the k-ball is L(2n). An exploration of the recursion led to Fibonacci numbers.

We also spent time reflecting on our experience and preparing a short presentation to share Friday which is almost all pictures.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.