## Chasing Infinity – 3D Version

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,

Project a Parabola onto a 3D sphere

The sketch on the sphere looks like this.  Note the two symmetric curves.  This was just to make the later 2D projection easier.

Parabola Sketched on 3D Sphere

Now project the curve on the sphere onto a plane.

Parabola Sketched on 3D Sphere Projected onto the Projective 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.

Parabola on 3D Sphere Rotated and Projected on Projective Plane

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.

Paraboloid in 3D

Now projected onto the surface of a 4D sphere and projected in 3D.

Paraboloid Projected on 4D Sphere

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.

Hyperbolic Paraboloid in 3D

Unrotated it looks like this.

Hyperbolic Paraboloid Projected on 4D Sphere

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.

Hyperbolic Paraboloid Projected on 4D Sphere Rotated

That is cool.  A cone is next.

Cone in 3D

This looks a little odd.  The important idea is that the surface extends to infinity along radial lines extending to different points of infinity.

Cone Projected on 4D Sphere

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.

Cone Projected on 4D Sphere Rotated

I explored a few other surfaces to test my understanding.  Here is a plane.

Plane in 3D

Unrotated, all the points at infinity can be seen on the rim.

Rotating just resulted in distortion.

Plane Projected on 4D Sphere Rotated

A parabolic surface looks like this in 3D.

Parabolic Surface in 3D

Projected in 4D like this.

Parabolic Surface Projected on 4D Sphere

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.

Parabolic Surface Projected on 4D Sphere Rotated

Lastly, here is a cubic surface.

Cubic Surface in 3D

Unrotated in 4D it looks like this.

Cubic Surface projected on 4D sphere

Those points should be able to be identified.

Cubic Surface Projected on 4D Sphere Rotated

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.