### Gudermannian Special Relativity, with a tip of the hat to Zigzag numbers

The day's running long, so I'm logging a few thoughts and notes on Gudermannian relativity here.  When I started trying to derive formulas in hyperbolic relativity, as I mentioned[1], I was inspired by Brian Greene's explanation that all objects move at the constant speed of light and can transfer their light speed from the time dimension, (where it points when an object is at rest), to the space dimension if they'd like, (leading to time dilation, slower time), in the rest frame.  I built diagrams like the following, (picture 1), which showed what I imagined the relationship to be in terms of circular trigonometry.

However, this never got me to the proper relationships which  I knew involved the hyperbolic tangent. What I was missing was the inverse Gudermannian which gives a relationship between circular and hyperbolic angles, (picture2).  The super-cool bit is that sigma shown in picture 2 corresponds to the actual speed that the travelling object would measure for itself, (referred to by Brehme as speedometer velocity and by Walter as proper speed [3]).

This brings us to a really nice mathematical time sink posted by +John Baez yesterday[2].  Check out the post that discusses the inverse Gudermannian and zigzag numbers, and don't forget the comments wherein you'll be treated to references to the Catalan numbers and the Borel transform!

References:
1.  Previous post on Brian Greene's constant speed interpretation of special relativity
http://copaseticflow.blogspot.com/2013/02/spheres-special-relativity-and-rotations.html

2.  John Baez's super-interesting post!

3.  Walter on proper speed[not open access]
http://www.sciencedirect.com/science/article/pii/S0094576506001433

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…