The frequency of my blog posts have been diminishing more and more, or should I say, the wave length has been getting longer. I attribute this mostly to my accession as math department chair. I just haven’t had the time and/or mental energy to do many math explorations even in the summer. I am also playing less golf. End of excuses.
Four years ago, I explored what happens at “infinity” for plane curves. This summer I was curious about what happens at infinity for three dimensional surfaces. The original idea was to project a curve onto a sphere by placing a plane containing the curve tangent to a sphere, projecting a line from a point on the curve through the center of the sphere and noting where the line intersected the surface of the sphere, effectively sketching the curve on the surface of the sphere like so,
The sketch on the sphere looks like this. Note the two symmetric curves. This was just to make the later 2D projection easier.
Now project the curve on the sphere onto a plane.
The disk is called the projective plane and opposite points on the edge are identified – are the same. To get a good look at what happens at infinity just rotate the sphere and project.
So why not try this one dimension up. Take a 3D figure, place it tangent to a 4D sphere, “sketch” the figure on the surface of a 4D sphere, and then project it in 3D. In theory this is not hard – just add a dimension. So I wrote a Mathematica script, built a crude 4D rotater and took a look. Thus a paraboloid in 3D.
Now projected onto the surface of a 4D sphere and projected in 3D.
This is what I expected. Since any cross-section of the paraboliod is parabola, all the cross-sections should “go” to the same point at infinity as shown. In this case, no rotation was needed. Note the distortion of the grid lines. They will help later to figure out what is going on.
A natural next shape to explore is the hyperbolic paraboloid or saddle.
Unrotated it looks like this.
Those two points at infinity should be able to be identified. Remember a line’s “ends” are the same point at infinity in the projective plane. The figure should be able to be rotated so they are (look like) the same point. This took some experimenting but here it is.
That is cool. A cone is next.
This looks a little odd. The important idea is that the surface extends to infinity along radial lines extending to different points of infinity.
The gird lines can be seen bunching up on the rim at all the different points of infinity. No matter how I try I shouldn’t be able to rotate the figure so that any of the points identify. I couldn’t. Here is an attempt.
I explored a few other surfaces to test my understanding. Here is a plane.
Unrotated, all the points at infinity can be seen on the rim.
Rotating just resulted in distortion.
A parabolic surface looks like this in 3D.
Projected in 4D like this.
The infinity point shown is where all the parabolas converge, but all the horizontal grid lines should also converge. I managed to rotate the figure to show this. The point indicated is the “horizontal” infinity.
Lastly, here is a cubic surface.
Unrotated in 4D it looks like this.
Those points should be able to be identified.
At the end of this exercise I was just beginning to get a feel for rotating objects in four dimensions. The shapes are aesthetically interesting and sometimes unexpected.