By Adam Wood, La Trobe University
What is the shortest and straightest path between two points? The most common answer to this question is “Well, obviously it’s a straight line!”. This is not necessarily always the case; there are some factors that come into play that many people might not initially consider. One such factor that plays a key role is the space that you are in. What if it wasn’t flat? Then how could you move in a straight line? Well you couldn’t, but what you could do is move along the straightest possible path. These paths are called the geodesics. On a plane they are just the straight lines but if you were on the surface of a sphere (like we almost are on earth), then the geodesics are arcs of the great circles, which are the largest circles that can be drawn on the sphere.
My project focused on finding these curves in a group called the real three-dimensional Heisenberg group, named after the famous physicist and mathematician Werner Heisenberg. Heisenberg discovered that this group displayed the same structural properties as a collection of certain operators on a one dimensional quantum mechanical system. The Heisenberg group is actually a space of matrices, which can be related to three-dimensional space with the major difference being the way we move through the space. In standard Euclidean space we can move in each of the x, y, and z directions independently, however in the Heisenberg group how we move in the x and y directions directly influences how we move in the z direction.
Throughout the course of the project, I discovered that there were three types of geodesics in the Heisenberg group. The first of which was a parabolic curve, which if we took the union over all the possible constants in the equation would form a surface reminiscent of a saddle used for horse riding. The second type of geodesic I discovered was a helical type curve and the third type of geodesic is a special case of the second type; if some of the constants in the equation are taken to be 0 then we obtain a curve which is a straight line that runs along the z axis. It is really quite remarkable that a helical curve or a parabola could be the shortest distance between a pair of points because it is so counter intuitive. However, in the humble opinion of this author it is results like this that serve to show the beauty of non Euclidean geometry.
Adam Wood was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.