I'm the student Principle Investigater for the Alice Group. We are currently working to visualize the nil geometry. One of the goals of the group is to write a program that navigates the sub nil geometry that runs in the cave.
Geoff
Sam
Katie
Here are the leture notes on the leture that Professor Tyson gave us about the Nil geometry. Leture Notes
So far I written several notebooks in Mathematica demonstrating properties of Nil as a sub-Riemannian Manifold. Note to open the notebooks, save as to your desktop and then open it with Mathematica.
A property of the sub-Nil geometry as a sub-Riemannian Manifold is that at any given point one can only travel along a tangent plane. Here is a notebook I wrote, showing what plane one can transverse for any given point in 3-space. subNilspace
This notebook shows how the tangent plane changes along the x, y and z axis subNil tangent planes
I also drew a "circle" in nil space. To approximate a circle in a plane, one can take a step of fixed length and turn a fixed degree, until one gets back to the starting point. The resulting shape will be a polygon with the number of sides equal to the number of steps taken. However, in sub-Nil space the plane you are allowed to travel on is consistly changing. There the after a step in taken that vector is projected on to the tangent plane in nil space and then rotated the fixed degrees. Nil Circle
Here are series of notebooks showing variation in the Nil circle
The shortest distance between the origin and a point on the z-axis projects a circle on the xy-plane. The height of a curve in sub-nil space is equal to the area of the closed curve that is projected onto the xy-plane. Therefore the projection that produces the minimum curve length for a given area is a circle. The notebooks gives evidence to suppport this conjecture and allows the user to experiment and try to disprove this conjecture. experiment
This note book demostates several properties of the sub-nil space including lifting a curve in Nil space and drawing a geodesic. Nil Space Properties
This is another notebook dealing with nil space. The first function in this notebook will draw a geodesic from the origin to any point. geodesic
This is a notebook dealing with the evolution of curves. The idea behind this notebook was to take any curve nil space and shorten it until it was the geodestic. The first part was to take a curve and sink in portion to the curavture and the second part was to resize so it would have the same area. Since the curve in nil space are uniquely determined by the projection on the xy plane, we just dealt with the projections. The first thing we tried was to get an ellipse to become circular over time, but we never got it to work. evolution
This notebook is part of the work we did with caveMathematica. This notebook shows how the tangent plane changes along the geodesic and runs in the cave. Cave Mathematica