To a math friend

I have been having the darnedest fun trying to understand geodesics on the surface z=x^2 + y^2. It turns out I got all the way through graduate school, including courses in differential geometry, without realizing that the book examples were cooked up to be do-able and that, for the most part, even simple expressions lead to unanswerable questions. For example you study curves using "arc-length" parametrization when, in simple cases like (t, t^2, t^3), the arc-length parametrization cannot be found easily.

Anyway, the geodesics on z=x^2 + y^2 that do not go through (0,0,0) can only spiral outward and will always have an infinite number of windings around the z axis. I don't know why it took a couple months to realize a geodesic cannot spiral inward without its curvature exceeding that of the surface and its velocity vector is always turning more in the horizontal direction than in the vertical direction - so it integrates faster in theta than in z - guaranteeing a winding. The lemma is that geodesic's velocity vector will always turn toward the direction of greatest bending of the surface.

Also, I find it interesting that winding number is a topological property of the geodesics but not of the surfaces. You can deform a surface just slightly and change the winding numbers dramatically. The lemma suggest other things to look at, like z=a*x^2+b*y^2 where the greatest amount of bending can be a horizontal direction in some places and vertical direction in others.

Anyway, lots of fun.

There was a hummingbird

I held up my hand and
it flew tight spirals around it
landed, and started pecking my finger
with its bill like a needle

Geodesics on z = F(x,y)

Here is a question about geodesics on the surface of the graph of

z = x^2 + y^2:

Excluding the case of geodesics that pass through x=y=0, do geodesics spiral towards or away from the point x=y=0 ? Or do they have a zero winding number about the z axis - remaining to one side?
Answer: Any geodesic on this surface spirals away from zero but not towards it.There is a lemma: For any unit tangent and normal at a point on a surface, the plane through the point containing the tangent and normal cuts the surface in an arc through the point with curvature that matches the curvature of any geodesic with that same tangent vector through the point. In other words the arc cut by this plane is representative of the geodesic.

On the surface z = x^2 + y^2 , as we get near 0, the radius of curvature of all such plane-cut-arcs approaches a constant. But a spiral getting closer to x=y=0 has curvature approaching infinity, so it could never be a geodesic without violating the lemma.
I still have no answer for the winding number question. Do all geodesics not through x=y=0 have a self crossing?

The latter question has me stumped but is lots of fun to think about. It is a growth rate question.

Update: Good news! The geodesics spiral outward on the parabaloid because of another simple lemma: On a geodesic, the tangent vector always turns more in the direction of greatest curvature. So for a horizontal tangent vector on z = x^2 + y^2, it will be turned in the horizontal more than in the vertical so the moving point moves horizontally faster than vertically. So you run out of angular dimension (<=2pi) long before you run out of vertical dimension (infinite).

Attacked by Adobe Flash Player Installer

This is my blog so I will just quietly mention that my computer had some serious problem booting just now, and when I had allowed Microsoft's utilities to complete the effort of recovery, the Adobe Flash Player - Installer dialog was sitting open, smack in the middle of the screen. I have no doubt at all that that is why my computer had trouble booting and, I am sorry, but this seems no better than a virus attack by Adobe. How did they get access to something that could cause me the boot up hassles? What if I was frightened by the experience of my computer not starting?

Some random carvings

A wood duck (acrylic on cedar)Truffle (oak)
A mother porpoise with baby (Catlinite pipe stone):

Too bad Moleman is un-readable

I am in the process of pasting typed words over the poor handwriting and cramped space that makes my Moleman comic book un-readable. I hope the whole thing gets revised.

The Mole Man Chronicles - Vol 1. The Origins of Mole Man

Chapter 1 here.
Chapter 2 here.
Chapter 3 here.
Chapter 4 here.
Chapter 5 here.